Metamath Proof Explorer

Theorem hdmap14lem13

Description: Lemma for proof of part 14 in Baer p. 50. (Contributed by NM, 6-Jun-2015)

Ref Expression
Hypotheses hdmap14lem12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hdmap14lem12.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmap14lem12.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
hdmap14lem12.t ⊢ · = ( ·𝑠 ‘ 𝑈 )
hdmap14lem12.r ⊢ 𝑅 = ( Scalar ‘ 𝑈 )
hdmap14lem12.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
hdmap14lem12.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmap14lem12.e ⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
hdmap14lem12.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
hdmap14lem12.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
hdmap14lem12.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
hdmap14lem12.p ⊢ 𝑃 = ( Scalar ‘ 𝐶 )
hdmap14lem12.a ⊢ 𝐴 = ( Base ‘ 𝑃 )
hdmap14lem12.o ⊢ 0 = ( 0g ‘ 𝑈 )
hdmap14lem12.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap14lem12.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐴 )
Assertion hdmap14lem13 ( 𝜑 → ( ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∀ 𝑦 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ) )

Proof

Step Hyp Ref Expression
1 hdmap14lem12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmap14lem12.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmap14lem12.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
4 hdmap14lem12.t ⊢ · = ( ·𝑠 ‘ 𝑈 )
5 hdmap14lem12.r ⊢ 𝑅 = ( Scalar ‘ 𝑈 )
6 hdmap14lem12.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
7 hdmap14lem12.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8 hdmap14lem12.e ⊢ ∙ = ( ·𝑠 ‘ 𝐶 )
9 hdmap14lem12.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
10 hdmap14lem12.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 hdmap14lem12.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
12 hdmap14lem12.p ⊢ 𝑃 = ( Scalar ‘ 𝐶 )
13 hdmap14lem12.a ⊢ 𝐴 = ( Base ‘ 𝑃 )
14 hdmap14lem12.o ⊢ 0 = ( 0g ‘ 𝑈 )
15 hdmap14lem12.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
16 hdmap14lem12.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐴 )
17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 hdmap14lem12 ⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∀ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ) )
18 velsn ⊢ ( 𝑦 ∈ { 0 } ↔ 𝑦 = 0 )
19 1 7 10 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
20 eqid ⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 )
21 12 8 13 20 lmodvs0 ⊢ ( ( 𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐴 ) → ( 𝐺 ∙ ( 0g ‘ 𝐶 ) ) = ( 0g ‘ 𝐶 ) )
22 19 16 21 syl2anc ⊢ ( 𝜑 → ( 𝐺 ∙ ( 0g ‘ 𝐶 ) ) = ( 0g ‘ 𝐶 ) )
23 1 2 14 7 20 9 10 hdmapval0 ⊢ ( 𝜑 → ( 𝑆 ‘ 0 ) = ( 0g ‘ 𝐶 ) )
24 23 oveq2d ⊢ ( 𝜑 → ( 𝐺 ∙ ( 𝑆 ‘ 0 ) ) = ( 𝐺 ∙ ( 0g ‘ 𝐶 ) ) )
25 1 2 10 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
26 5 4 6 14 lmodvs0 ⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ) → ( 𝐹 · 0 ) = 0 )
27 25 11 26 syl2anc ⊢ ( 𝜑 → ( 𝐹 · 0 ) = 0 )
28 27 fveq2d ⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 0 ) ) = ( 𝑆 ‘ 0 ) )
29 28 23 eqtrd ⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 0 ) ) = ( 0g ‘ 𝐶 ) )
30 22 24 29 3eqtr4rd ⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐹 · 0 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 0 ) ) )
31 oveq2 ⊢ ( 𝑦 = 0 → ( 𝐹 · 𝑦 ) = ( 𝐹 · 0 ) )
32 31 fveq2d ⊢ ( 𝑦 = 0 → ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝑆 ‘ ( 𝐹 · 0 ) ) )
33 fveq2 ⊢ ( 𝑦 = 0 → ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 0 ) )
34 33 oveq2d ⊢ ( 𝑦 = 0 → ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 0 ) ) )
35 32 34 eqeq12d ⊢ ( 𝑦 = 0 → ( ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ ( 𝐹 · 0 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 0 ) ) ) )
36 30 35 syl5ibrcom ⊢ ( 𝜑 → ( 𝑦 = 0 → ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ) )
37 18 36 biimtrid ⊢ ( 𝜑 → ( 𝑦 ∈ { 0 } → ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ) )
38 37 ralrimiv ⊢ ( 𝜑 → ∀ 𝑦 ∈ { 0 } ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) )
39 38 biantrud ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ { 0 } ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ) ) )
40 ralunb ⊢ ( ∀ 𝑦 ∈ ( ( 𝑉 ∖ { 0 } ) ∪ { 0 } ) ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ↔ ( ∀ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ∧ ∀ 𝑦 ∈ { 0 } ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ) )
41 39 40 bitr4di ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( 𝑉 ∖ { 0 } ) ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( ( 𝑉 ∖ { 0 } ) ∪ { 0 } ) ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ) )
42 3 14 lmod0vcl ⊢ ( 𝑈 ∈ LMod → 0 ∈ 𝑉 )
43 difsnid ⊢ ( 0 ∈ 𝑉 → ( ( 𝑉 ∖ { 0 } ) ∪ { 0 } ) = 𝑉 )
44 25 42 43 3syl ⊢ ( 𝜑 → ( ( 𝑉 ∖ { 0 } ) ∪ { 0 } ) = 𝑉 )
45 44 raleqdv ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( ( 𝑉 ∖ { 0 } ) ∪ { 0 } ) ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ) )
46 17 41 45 3bitrd ⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝐹 · 𝑋 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑋 ) ) ↔ ∀ 𝑦 ∈ 𝑉 ( 𝑆 ‘ ( 𝐹 · 𝑦 ) ) = ( 𝐺 ∙ ( 𝑆 ‘ 𝑦 ) ) ) )