imasip

Description: The inner product of an image structure. (Contributed by Thierry Arnoux, 16-Jun-2019)

	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 )
	imasip.i	⊢ , = ( ·_𝑖 ‘ 𝑅 )
	imasip.w	⊢ 𝐼 = ( ·_𝑖 ‘ 𝑈 )
Assertion	imasip	⊢ ( 𝜑 → 𝐼 = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } )

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		imasip.i	⊢ , = ( ·_𝑖 ‘ 𝑅 )
6		imasip.w	⊢ 𝐼 = ( ·_𝑖 ‘ 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	⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 𝑅 )
13		eqid	⊢ ( dist ‘ 𝑅 ) = ( dist ‘ 𝑅 )
14		eqid	⊢ ( le ‘ 𝑅 ) = ( le ‘ 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 , 𝑞 ) ⟩ } = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } )
22		eqidd	⊢ ( 𝜑 → ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) = ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) )
23		eqid	⊢ ( dist ‘ 𝑈 ) = ( dist ‘ 𝑈 )
24	1 2 3 4 13 23	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^ , < ) ) )
25		eqidd	⊢ ( 𝜑 → ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) = ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) )
26	1 2 7 8 9 10 11 5 12 13 14 16 18 20 21 22 24 25 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 , 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } ) )
27		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 , 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 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 , 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } )
28	27	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 , 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } ) Struct ⟨ 1 , ; 1 2 ⟩
29		ipid	⊢ ·_𝑖 = Slot ( ·_𝑖 ‘ ndx )
30		snsstp3	⊢ { ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ⟩ } ⊆ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ( ·_𝑠 ‘ 𝑈 ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ⟩ }
31		ssun2	⊢ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ( ·_𝑠 ‘ 𝑈 ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ⟩ } ⊆ ( { ⟨ ( 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 , 𝑞 ) ⟩ } ⟩ } )
32	30 31	sstri	⊢ { ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ⟩ } ⊆ ( { ⟨ ( 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 , 𝑞 ) ⟩ } ⟩ } )
33		ssun1	⊢ ( { ⟨ ( 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 , 𝑞 ) ⟩ } ⟩ } ) ⊆ ( ( { ⟨ ( 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 , 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } )
34	32 33	sstri	⊢ { ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ⟩ } ⊆ ( ( { ⟨ ( 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 , 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } )
35		fvex	⊢ ( Base ‘ 𝑅 ) ∈ V
36	2 35	eqeltrdi	⊢ ( 𝜑 → 𝑉 ∈ V )
37		snex	⊢ { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ∈ V
38	37	rgenw	⊢ ∀ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ∈ V
39		iunexg	⊢ ( ( 𝑉 ∈ V ∧ ∀ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ∈ V ) → ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ∈ V )
40	36 38 39	sylancl	⊢ ( 𝜑 → ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ∈ V )
41	40	ralrimivw	⊢ ( 𝜑 → ∀ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ∈ V )
42		iunexg	⊢ ( ( 𝑉 ∈ V ∧ ∀ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ∈ V ) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ∈ V )
43	36 41 42	syl2anc	⊢ ( 𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } ∈ V )
44	26 28 29 34 43 6	strfv3	⊢ ( 𝜑 → 𝐼 = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 , 𝑞 ) ⟩ } )

Metamath Proof Explorer

Theorem imasip

Proof