This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem imassca

Description: The scalar field of an image structure. (Contributed by Mario Carneiro, 23-Feb-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 )
imassca.g ⊢ 𝐺 = ( Scalar ‘ 𝑅 )
Assertion imassca ( 𝜑 → 𝐺 = ( Scalar ‘ 𝑈 ) )

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 imassca.g ⊢ 𝐺 = ( Scalar ‘ 𝑅 )
6 5 fvexi ⊢ 𝐺 ∈ V
7 eqid ⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑈 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) ⟩ } )
8 7 imasvalstr ⊢ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑈 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ⟩
9 scaid ⊢ Scalar = Slot ( Scalar ‘ ndx )
10 snsstp1 ⊢ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ } ⊆ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ }
11 ssun2 ⊢ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } )
12 10 11 sstri ⊢ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑈 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·𝑠 ‘ 𝑅 ) 𝑞 ) ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝑝 ( ·𝑖 ‘ 𝑅 ) 𝑞 ) ⟩ } ⟩ } )
13 ssun1 ⊢ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑈 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) ⟩ } )
14 12 13 sstri ⊢ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ } ⊆ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑈 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) ⟩ } )
15 8 9 14 strfv ⊢ ( 𝐺 ∈ V → 𝐺 = ( Scalar ‘ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑈 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) ⟩ } ) ) )
16 6 15 ax-mp ⊢ 𝐺 = ( Scalar ‘ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑈 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) ⟩ } ) )
17 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
18 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
19 eqid ⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
20 eqid ⊢ ( ·𝑠 ‘ 𝑅 ) = ( ·𝑠 ‘ 𝑅 )
21 eqid ⊢ ( ·𝑖 ‘ 𝑅 ) = ( ·𝑖 ‘ 𝑅 )
22 eqid ⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 𝑅 )
23 eqid ⊢ ( dist ‘ 𝑅 ) = ( dist ‘ 𝑅 )
24 eqid ⊢ ( le ‘ 𝑅 ) = ( le ‘ 𝑅 )
25 eqid ⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 )
26 1 2 3 4 17 25 imasplusg ⊢ ( 𝜑 → ( +g ‘ 𝑈 ) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 ( +g ‘ 𝑅 ) 𝑞 ) ) ⟩ } )
27 eqid ⊢ ( .r ‘ 𝑈 ) = ( .r ‘ 𝑈 )
28 1 2 3 4 18 27 imasmulr ⊢ ( 𝜑 → ( .r ‘ 𝑈 ) = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 { ⟨ ⟨ ( 𝐹 ‘ 𝑝 ) , ( 𝐹 ‘ 𝑞 ) ⟩ , ( 𝐹 ‘ ( 𝑝 ( .r ‘ 𝑅 ) 𝑞 ) ) ⟩ } )
29 eqidd ⊢ ( 𝜑 → ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·𝑠 ‘ 𝑅 ) 𝑞 ) ) ) = ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 ( ·𝑠 ‘ 𝑅 ) 𝑞 ) ) ) )
30 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 ) ⟩ } )
31 eqidd ⊢ ( 𝜑 → ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) = ( ( TopOpen ‘ 𝑅 ) qTop 𝐹 ) )
32 eqid ⊢ ( dist ‘ 𝑈 ) = ( dist ‘ 𝑈 )
33 1 2 3 4 23 32 imasds ⊢ ( 𝜑 → ( dist ‘ 𝑈 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ inf ( ∪ 𝑛 ∈ ℕ ran ( 𝑔 ∈ { ℎ ∈ ( ( 𝑉 × 𝑉 ) ↑m ( 1 ... 𝑛 ) ) ∣ ( ( 𝐹 ‘ ( 1st ‘ ( ℎ ‘ 1 ) ) ) = 𝑥 ∧ ( 𝐹 ‘ ( 2nd ‘ ( ℎ ‘ 𝑛 ) ) ) = 𝑦 ∧ ∀ 𝑖 ∈ ( 1 ... ( 𝑛 − 1 ) ) ( 𝐹 ‘ ( 2nd ‘ ( ℎ ‘ 𝑖 ) ) ) = ( 𝐹 ‘ ( 1st ‘ ( ℎ ‘ ( 𝑖 + 1 ) ) ) ) ) } ↦ ( ℝ*𝑠 Σg ( ( dist ‘ 𝑅 ) ∘ 𝑔 ) ) ) , ℝ* , < ) ) )
34 eqidd ⊢ ( 𝜑 → ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ◡ 𝐹 ) = ( ( 𝐹 ∘ ( le ‘ 𝑅 ) ) ∘ ◡ 𝐹 ) )
35 1 2 17 18 5 19 20 21 22 23 24 26 28 29 30 31 33 34 3 4 imasval ⊢ ( 𝜑 → 𝑈 = ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑈 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) ⟩ } ) )
36 35 fveq2d ⊢ ( 𝜑 → ( Scalar ‘ 𝑈 ) = ( Scalar ‘ ( ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ 𝑈 ) ⟩ , ⟨ ( .r ‘ ndx ) , ( .r ‘ 𝑈 ) ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝐺 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ ( Base ‘ 𝐺 ) , 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 ) ⟩ } ) ) )
37 16 36 eqtr4id ⊢ ( 𝜑 → 𝐺 = ( Scalar ‘ 𝑈 ) )