imasmulr

Description: The ring multiplication in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015) (Revised by Mario Carneiro, 11-Jul-2015) (Revised 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 )
	imasmulr.p	⊢ · = ( ._r ‘ 𝑅 )
	imasmulr.t	⊢ ∙ = ( ._r ‘ 𝑈 )
Assertion	imasmulr	⊢ ( 𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 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		imasmulr.p	⊢ · = ( ._r ‘ 𝑅 )
6		imasmulr.t	⊢ ∙ = ( ._r ‘ 𝑈 )
7		eqid	⊢ ( +_g ‘ 𝑅 ) = ( +_g ‘ 𝑅 )
8		eqid	⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ 𝑅 )
9		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑅 ) ) = ( Base ‘ ( Scalar ‘ 𝑅 ) )
10		eqid	⊢ ( ·_𝑠 ‘ 𝑅 ) = ( ·_𝑠 ‘ 𝑅 )
11		eqid	⊢ ( ·_𝑖 ‘ 𝑅 ) = ( ·_𝑖 ‘ 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		eqidd	⊢ ( 𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } )
18		eqidd	⊢ ( 𝜑 → ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) = ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) )
19		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 ) ⟩ } )
20		eqidd	⊢ ( 𝜑 → ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) = ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) )
21		eqid	⊢ ( dist ‘ 𝑈 ) = ( dist ‘ 𝑈 )
22	1 2 3 4 13 21	imasds	⊢ ( 𝜑 → ( dist ‘ 𝑈 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ { ℎ ∈ ( ( 𝑉 × 𝑉 ) ↑_m ( 1 ... 𝑛 ) ) ∣ ( ( 𝐹 ‘ ( 1^st ‘ ( ℎ ‘ 1 ) ) ) = 𝑥 ∧ ( 𝐹 ‘ ( 2^nd ‘ ( ℎ ‘ 𝑛 ) ) ) = 𝑦 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑛 − 1 ) ) ( 𝐹 ‘ ( 2^nd ‘ ( ℎ ‘ 𝑖 ) ) ) = ( 𝐹 ‘ ( 1^st ‘ ( ℎ ‘ ( 𝑖 + 1 ) ) ) ) ) } ↦ ( ℝ^_𝑠 Σ_g ( ( dist ‘ 𝑅 ) ∘ 𝑔 ) ) ) , ℝ^ , < ) ) )
23		eqidd	⊢ ( 𝜑 → ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) = ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) )
24	1 2 7 5 8 9 10 11 12 13 14 16 17 18 19 20 22 23 3 4	imasval	⊢ ( 𝜑 → 𝑈 = ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } ) )
25		eqid	⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 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 ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } )
26	25	imasvalstr	⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } ) Struct ⟨ 1 , ; 1 2 ⟩
27		mulrid	⊢ ._r = Slot ( ._r ‘ ndx )
28		snsstp3	⊢ { ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ }
29		ssun1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } )
30	28 29	sstri	⊢ { ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } )
31		ssun1	⊢ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ⊆ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } )
32	30 31	sstri	⊢ { ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ⊆ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +_g ‘ ndx ) , ( +_g ‘ 𝑈 ) ⟩ , ⟨ ( ._r ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , ( Scalar ‘ 𝑅 ) ⟩ , ⟨ ( ·_𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ ( Scalar ‘ 𝑅 ) ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·_𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·_𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·_𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } ) ∪ { ⟨ ( TopSet ‘ ndx ) , ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) ⟩ , ⟨ ( le ‘ ndx ) , ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ^◡ 𝐹 ) ⟩ , ⟨ ( dist ‘ ndx ) , ( dist ‘ 𝑈 ) ⟩ } )
33		fvex	⊢ ( Base ‘ 𝑅 ) ∈ V
34	2 33	eqeltrdi	⊢ ( 𝜑 → 𝑉 ∈ V )
35		snex	⊢ { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ∈ V
36	35	rgenw	⊢ ∀ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ∈ V
37		iunexg	⊢ ( ( 𝑉 ∈ V ∧ ∀ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ∈ V ) → ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ∈ V )
38	34 36 37	sylancl	⊢ ( 𝜑 → ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ∈ V )
39	38	ralrimivw	⊢ ( 𝜑 → ∀ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ∈ V )
40		iunexg	⊢ ( ( 𝑉 ∈ V ∧ ∀ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ∈ V ) → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ∈ V )
41	34 39 40	syl2anc	⊢ ( 𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } ∈ V )
42	24 26 27 32 41 6	strfv3	⊢ ( 𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ⟩ } )

Metamath Proof Explorer

Theorem imasmulr

Proof