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Metamath Proof Explorer

Theorem prdsmulr

Description: Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015) (Revised by Mario Carneiro, 15-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Revised by Zhi Wang, 18-Aug-2024)

Ref Expression
Hypotheses prdsbas.p ⊢ 𝑃 = ( 𝑆 Xs 𝑅 )
prdsbas.s ⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
prdsbas.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
prdsbas.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
prdsbas.i ⊢ ( 𝜑 → dom 𝑅 = 𝐼 )
prdsmulr.t ⊢ · = ( .r ‘ 𝑃 )
Assertion prdsmulr ( 𝜑 → · = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 prdsbas.p ⊢ 𝑃 = ( 𝑆 Xs 𝑅 )
2 prdsbas.s ⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
3 prdsbas.r ⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
4 prdsbas.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
5 prdsbas.i ⊢ ( 𝜑 → dom 𝑅 = 𝐼 )
6 prdsmulr.t ⊢ · = ( .r ‘ 𝑃 )
7 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
8 1 2 3 4 5 prdsbas ⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
9 eqid ⊢ ( +g ‘ 𝑃 ) = ( +g ‘ 𝑃 )
10 1 2 3 4 5 9 prdsplusg ⊢ ( 𝜑 → ( +g ‘ 𝑃 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) )
11 eqidd ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) )
12 eqidd ⊢ ( 𝜑 → ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) = ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) )
13 eqidd ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) )
14 eqidd ⊢ ( 𝜑 → ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) )
15 eqidd ⊢ ( 𝜑 → { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } )
16 eqidd ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) )
17 eqidd ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) )
18 eqidd ⊢ ( 𝜑 → ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) = ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) )
19 1 7 5 8 10 11 12 13 14 15 16 17 18 2 3 prdsval ⊢ ( 𝜑 → 𝑃 = ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ⟩ , ⟨ ( le ‘ ndx ) , { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ⟩ , ⟨ ( dist ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ) )
20 mulridx ⊢ .r = Slot ( .r ‘ ndx )
21 ovssunirn ⊢ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ ∪ ran ( .r ‘ ( 𝑅 ‘ 𝑥 ) )
22 20 strfvss ⊢ ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ( 𝑅 ‘ 𝑥 )
23 fvssunirn ⊢ ( 𝑅 ‘ 𝑥 ) ⊆ ∪ ran 𝑅
24 rnss ⊢ ( ( 𝑅 ‘ 𝑥 ) ⊆ ∪ ran 𝑅 → ran ( 𝑅 ‘ 𝑥 ) ⊆ ran ∪ ran 𝑅 )
25 uniss ⊢ ( ran ( 𝑅 ‘ 𝑥 ) ⊆ ran ∪ ran 𝑅 → ∪ ran ( 𝑅 ‘ 𝑥 ) ⊆ ∪ ran ∪ ran 𝑅 )
26 23 24 25 mp2b ⊢ ∪ ran ( 𝑅 ‘ 𝑥 ) ⊆ ∪ ran ∪ ran 𝑅
27 22 26 sstri ⊢ ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran 𝑅
28 rnss ⊢ ( ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran 𝑅 → ran ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ran ∪ ran ∪ ran 𝑅 )
29 uniss ⊢ ( ran ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ran ∪ ran ∪ ran 𝑅 → ∪ ran ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran ∪ ran 𝑅 )
30 27 28 29 mp2b ⊢ ∪ ran ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
31 21 30 sstri ⊢ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran ∪ ran 𝑅
32 ovex ⊢ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ∈ V
33 32 elpw ⊢ ( ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↔ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ⊆ ∪ ran ∪ ran ∪ ran 𝑅 )
34 31 33 mpbir ⊢ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅
35 34 a1i ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ∈ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 )
36 35 fmpttd ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) : 𝐼 ⟶ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 )
37 rnexg ⊢ ( 𝑅 ∈ 𝑊 → ran 𝑅 ∈ V )
38 uniexg ⊢ ( ran 𝑅 ∈ V → ∪ ran 𝑅 ∈ V )
39 3 37 38 3syl ⊢ ( 𝜑 → ∪ ran 𝑅 ∈ V )
40 rnexg ⊢ ( ∪ ran 𝑅 ∈ V → ran ∪ ran 𝑅 ∈ V )
41 uniexg ⊢ ( ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran 𝑅 ∈ V )
42 39 40 41 3syl ⊢ ( 𝜑 → ∪ ran ∪ ran 𝑅 ∈ V )
43 rnexg ⊢ ( ∪ ran ∪ ran 𝑅 ∈ V → ran ∪ ran ∪ ran 𝑅 ∈ V )
44 uniexg ⊢ ( ran ∪ ran ∪ ran 𝑅 ∈ V → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V )
45 42 43 44 3syl ⊢ ( 𝜑 → ∪ ran ∪ ran ∪ ran 𝑅 ∈ V )
46 45 pwexd ⊢ ( 𝜑 → 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ∈ V )
47 3 dmexd ⊢ ( 𝜑 → dom 𝑅 ∈ V )
48 5 47 eqeltrrd ⊢ ( 𝜑 → 𝐼 ∈ V )
49 46 48 elmapd ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∈ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ↔ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) : 𝐼 ⟶ 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ) )
50 36 49 mpbird ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∈ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
51 50 ralrimivw ⊢ ( 𝜑 → ∀ 𝑔 ∈ 𝐵 ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∈ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
52 51 ralrimivw ⊢ ( 𝜑 → ∀ 𝑓 ∈ 𝐵 ∀ 𝑔 ∈ 𝐵 ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∈ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
53 eqid ⊢ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) )
54 53 fmpo ⊢ ( ∀ 𝑓 ∈ 𝐵 ∀ 𝑔 ∈ 𝐵 ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∈ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ↔ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) : ( 𝐵 × 𝐵 ) ⟶ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
55 52 54 sylib ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) : ( 𝐵 × 𝐵 ) ⟶ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) )
56 4 fvexi ⊢ 𝐵 ∈ V
57 56 56 xpex ⊢ ( 𝐵 × 𝐵 ) ∈ V
58 ovex ⊢ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ∈ V
59 fex2 ⊢ ( ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) : ( 𝐵 × 𝐵 ) ⟶ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ∧ ( 𝐵 × 𝐵 ) ∈ V ∧ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) ∈ V ) → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ∈ V )
60 57 58 59 mp3an23 ⊢ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) : ( 𝐵 × 𝐵 ) ⟶ ( 𝒫 ∪ ran ∪ ran ∪ ran 𝑅 ↑m 𝐼 ) → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ∈ V )
61 55 60 syl ⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ∈ V )
62 snsstp3 ⊢ { ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ }
63 ssun1 ⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ⟩ } )
64 62 63 sstri ⊢ { ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ⟩ } )
65 ssun1 ⊢ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ⊆ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ⟩ , ⟨ ( le ‘ ndx ) , { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ⟩ , ⟨ ( dist ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } ) )
66 64 65 sstri ⊢ { ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ⊆ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ⟩ } ) ∪ ( { ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ⟩ , ⟨ ( le ‘ ndx ) , { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ⟩ , ⟨ ( dist ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) ⟩ } ∪ { ⟨ ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( ⟨ ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) ⟩ ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ⟩ } ) )
67 19 6 20 61 66 prdsbaslem ⊢ ( 𝜑 → · = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) )