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Metamath Proof Explorer

Theorem psrvscafval

Description: The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014) (Revised by Mario Carneiro, 2-Oct-2015) (Proof shortened by AV, 2-Nov-2024)

Ref Expression
Hypotheses psrvsca.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
psrvsca.n ⊢ ∙ = ( ·𝑠 ‘ 𝑆 )
psrvsca.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
psrvsca.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
psrvsca.m ⊢ · = ( .r ‘ 𝑅 )
psrvsca.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
Assertion psrvscafval ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) )

Proof

Step Hyp Ref Expression
1 psrvsca.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psrvsca.n ⊢ ∙ = ( ·𝑠 ‘ 𝑆 )
3 psrvsca.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
4 psrvsca.b ⊢ 𝐵 = ( Base ‘ 𝑆 )
5 psrvsca.m ⊢ · = ( .r ‘ 𝑅 )
6 psrvsca.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
7 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
8 eqid ⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 𝑅 )
9 simpl ⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐼 ∈ V )
10 1 3 6 4 9 psrbas ⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ( 𝐾 ↑m 𝐷 ) )
11 eqid ⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
12 1 4 7 11 psrplusg ⊢ ( +g ‘ 𝑆 ) = ( ∘f ( +g ‘ 𝑅 ) ↾ ( 𝐵 × 𝐵 ) )
13 eqid ⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
14 1 4 5 13 6 psrmulr ⊢ ( .r ‘ 𝑆 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )
15 eqid ⊢ ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) )
16 eqidd ⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) = ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) )
17 simpr ⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑅 ∈ V )
18 1 3 7 5 8 6 10 12 14 15 16 9 17 psrval ⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑆 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) )
19 18 fveq2d ⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) ) )
20 3 fvexi ⊢ 𝐾 ∈ V
21 4 fvexi ⊢ 𝐵 ∈ V
22 20 21 mpoex ⊢ ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ∈ V
23 psrvalstr ⊢ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) Struct ⟨ 1 , 9 ⟩
24 vscaid ⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx )
25 snsstp2 ⊢ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ } ⊆ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ }
26 ssun2 ⊢ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } )
27 25 26 sstri ⊢ { ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } )
28 23 24 27 strfv ⊢ ( ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ∈ V → ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) ) )
29 22 28 ax-mp ⊢ ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( ·𝑠 ‘ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ⟩ } ) )
30 19 2 29 3eqtr4g ⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) )
31 eqid ⊢ ∅ = ∅
32 fn0 ⊢ ( ∅ Fn ∅ ↔ ∅ = ∅ )
33 31 32 mpbir ⊢ ∅ Fn ∅
34 reldmpsr ⊢ Rel dom mPwSer
35 34 ovprc ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mPwSer 𝑅 ) = ∅ )
36 1 35 eqtrid ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑆 = ∅ )
37 36 fveq2d ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ ∅ ) )
38 24 str0 ⊢ ∅ = ( ·𝑠 ‘ ∅ )
39 37 2 38 3eqtr4g ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ∅ )
40 34 1 4 elbasov ⊢ ( 𝑓 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) )
41 40 con3i ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ¬ 𝑓 ∈ 𝐵 )
42 41 eq0rdv ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ )
43 42 xpeq2d ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐾 × 𝐵 ) = ( 𝐾 × ∅ ) )
44 xp0 ⊢ ( 𝐾 × ∅ ) = ∅
45 43 44 eqtrdi ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐾 × 𝐵 ) = ∅ )
46 39 45 fneq12d ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ∙ Fn ( 𝐾 × 𝐵 ) ↔ ∅ Fn ∅ ) )
47 33 46 mpbiri ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ Fn ( 𝐾 × 𝐵 ) )
48 fnov ⊢ ( ∙ Fn ( 𝐾 × 𝐵 ) ↔ ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( 𝑥 ∙ 𝑓 ) ) )
49 47 48 sylib ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( 𝑥 ∙ 𝑓 ) ) )
50 41 pm2.21d ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑓 ∈ 𝐵 → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( 𝑥 ∙ 𝑓 ) ) )
51 50 a1d ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑥 ∈ 𝐾 → ( 𝑓 ∈ 𝐵 → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( 𝑥 ∙ 𝑓 ) ) ) )
52 51 3imp ⊢ ( ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑥 ∈ 𝐾 ∧ 𝑓 ∈ 𝐵 ) → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( 𝑥 ∙ 𝑓 ) )
53 52 mpoeq3dva ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( 𝑥 ∙ 𝑓 ) ) )
54 49 53 eqtr4d ⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) )
55 30 54 pm2.61i ⊢ ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) )