imasle

Description: The ordering of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015)

	Ref	Expression
Hypotheses	imasbas.u	⊢ ( 𝜑 → 𝑈 = ( 𝐹 “_s 𝑅 ) )
	imasbas.v	⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) )
	imasbas.f	⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 )
	imasbas.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑍 )
	imasle.n	⊢ 𝑁 = ( le ‘ 𝑅 )
	imasle.l	⊢ ≤ = ( le ‘ 𝑈 )
Assertion	imasle	⊢ ( 𝜑 → ≤ = ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		imasbas.u	⊢ ( 𝜑 → 𝑈 = ( 𝐹 “_s 𝑅 ) )
2		imasbas.v	⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) )
3		imasbas.f	⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 )
4		imasbas.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑍 )
5		imasle.n	⊢ 𝑁 = ( le ‘ 𝑅 )
6		imasle.l	⊢ ≤ = ( le ‘ 𝑈 )
7		eqid	⊢ ( +_g ‘ 𝑅 ) = ( +_g ‘ 𝑅 )
8		eqid	⊢ ( ._r ‘ 𝑅 ) = ( ._r ‘ 𝑅 )
9		eqid	⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ 𝑅 )
10		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑅 ) ) = ( Base ‘ ( Scalar ‘ 𝑅 ) )
11		eqid	⊢ ( ·_𝑠 ‘ 𝑅 ) = ( ·_𝑠 ‘ 𝑅 )
12		eqid	⊢ ( ·_𝑖 ‘ 𝑅 ) = ( ·_𝑖 ‘ 𝑅 )
13		eqid	⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 𝑅 )
14		eqid	⊢ ( dist ‘ 𝑅 ) = ( dist ‘ 𝑅 )
15		eqid	⊢ ( +_g ‘ 𝑈 ) = ( +_g ‘ 𝑈 )
16	1 2 3 4 7 15	imasplusg	⊢ ( 𝜑 → ( +_g ‘ 𝑈 ) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 ( +_g ‘ 𝑅 ) 𝑞 ) ) ⟩ } )
17		eqid	⊢ ( ._r ‘ 𝑈 ) = ( ._r ‘ 𝑈 )
18	1 2 3 4 8 17	imasmulr	⊢ ( 𝜑 → ( ._r ‘ 𝑈 ) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 ( ._r ‘ 𝑅 ) 𝑞 ) ) ⟩ } )
19		eqid	⊢ ( ·_𝑠 ‘ 𝑈 ) = ( ·_𝑠 ‘ 𝑈 )
20	1 2 3 4 9 10 11 19	imasvsca	⊢ ( 𝜑 → ( ·_𝑠 ‘ 𝑈 ) = ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) )
21		eqidd	⊢ ( 𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } )
22		eqid	⊢ ( TopSet ‘ 𝑈 ) = ( TopSet ‘ 𝑈 )
23	1 2 3 4 13 22	imastset	⊢ ( 𝜑 → ( TopSet ‘ 𝑈 ) = ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) )
24		eqid	⊢ ( dist ‘ 𝑈 ) = ( dist ‘ 𝑈 )
25	1 2 3 4 14 24	imasds	⊢ ( 𝜑 → ( dist ‘ 𝑈 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ inf ( ∪ 𝑢 ∈ ℕ ran ( 𝑧 ∈ { 𝑤 ∈ ( ( 𝑉 × 𝑉 ) ↑_m ( 1 ... 𝑢 ) ) ∣ ( ( 𝐹 ‘ ( 1^st ‘ ( 𝑤 ‘ 1 ) ) ) = 𝑥 ∧ ( 𝐹 ‘ ( 2^nd ‘ ( 𝑤 ‘ 𝑢 ) ) ) = 𝑦 ∧ ∀ 𝑣 ∈ ( 1 ... ( 𝑢 − 1 ) ) ( 𝐹 ‘ ( 2^nd ‘ ( 𝑤 ‘ 𝑣 ) ) ) = ( 𝐹 ‘ ( 1^st ‘ ( 𝑤 ‘ ( 𝑣 + 1 ) ) ) ) ) } ↦ ( ℝ^_𝑠 Σ_g ( ( dist ‘ 𝑅 ) ∘ 𝑧 ) ) ) , ℝ^ , < ) ) )
26		eqidd	⊢ ( 𝜑 → ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) = ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) )
27	1 2 7 8 9 10 11 12 13 14 5 16 18 20 21 23 25 26 3 4	imasval	⊢ ( 𝜑 → 𝑈 = ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ( ._r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ( ·_𝑠 ‘ 𝑈 ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑈 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } ) )
28		eqid	⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ( ._r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ( ·_𝑠 ‘ 𝑈 ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑈 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } ) = ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ( ._r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ( ·_𝑠 ‘ 𝑈 ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑈 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } )
29	28	imasvalstr	⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ( ._r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ( ·_𝑠 ‘ 𝑈 ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑈 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } ) Struct ⟨ 1 , ; 1 2 ⟩
30		pleid	⊢ le = Slot ( le ‘ ndx )
31		snsstp2	⊢ { ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ } ⊆ { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑈 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ }
32		ssun2	⊢ { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑈 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } ⊆ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ( ._r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ( ·_𝑠 ‘ 𝑈 ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑈 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } )
33	31 32	sstri	⊢ { ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ } ⊆ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ( ._r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ( ·_𝑠 ‘ 𝑈 ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( TopSet ‘ 𝑈 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } )
34		fof	⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → 𝐹 : 𝑉 ⟶ 𝐵 )
35	3 34	syl	⊢ ( 𝜑 → 𝐹 : 𝑉 ⟶ 𝐵 )
36		fvex	⊢ ( Base ‘ 𝑅 ) ∈ V
37	2 36	eqeltrdi	⊢ ( 𝜑 → 𝑉 ∈ V )
38	35 37	fexd	⊢ ( 𝜑 → 𝐹 ∈ V )
39	5	fvexi	⊢ 𝑁 ∈ V
40		coexg	⊢ ( ( 𝐹 ∈ V ∧ 𝑁 ∈ V ) → ( 𝐹 ∘ 𝑁 ) ∈ V )
41	38 39 40	sylancl	⊢ ( 𝜑 → ( 𝐹 ∘ 𝑁 ) ∈ V )
42		cnvexg	⊢ ( 𝐹 ∈ V → ^◡ 𝐹 ∈ V )
43	38 42	syl	⊢ ( 𝜑 → ^◡ 𝐹 ∈ V )
44		coexg	⊢ ( ( ( 𝐹 ∘ 𝑁 ) ∈ V ∧ ^◡ 𝐹 ∈ V ) → ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ∈ V )
45	41 43 44	syl2anc	⊢ ( 𝜑 → ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) ∈ V )
46	27 29 30 33 45 6	strfv3	⊢ ( 𝜑 → ≤ = ( ( 𝐹 ∘ 𝑁 ) ∘ ^◡ 𝐹 ) )

Metamath Proof Explorer

Theorem imasle

Proof