Metamath Proof Explorer

Theorem lcdval

Description: Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015)

Ref Expression
Hypotheses lcdval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
lcdval.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
lcdval.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
lcdval.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
lcdval.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
lcdval.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
lcdval.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
lcdval.k ⊢ ( 𝜑 → ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) )
Assertion lcdval ( 𝜑 → 𝐶 = ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) )

Proof

Step Hyp Ref Expression
1 lcdval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 lcdval.o ⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3 lcdval.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
4 lcdval.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 lcdval.f ⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6 lcdval.l ⊢ 𝐿 = ( LKer ‘ 𝑈 )
7 lcdval.d ⊢ 𝐷 = ( LDual ‘ 𝑈 )
8 lcdval.k ⊢ ( 𝜑 → ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) )
9 1 lcdfval ⊢ ( 𝐾 ∈ 𝑋 → ( LCDual ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) )
10 9 fveq1d ⊢ ( 𝐾 ∈ 𝑋 → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) ‘ 𝑊 ) )
11 3 10 syl5eq ⊢ ( 𝐾 ∈ 𝑋 → 𝐶 = ( ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) ‘ 𝑊 ) )
12 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
13 12 4 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) = 𝑈 )
14 13 fveq2d ⊢ ( 𝑤 = 𝑊 → ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ( LDual ‘ 𝑈 ) )
15 14 7 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝐷 )
16 13 fveq2d ⊢ ( 𝑤 = 𝑊 → ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ( LFnl ‘ 𝑈 ) )
17 16 5 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝐹 )
18 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) )
19 18 2 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) = ⊥ )
20 13 fveq2d ⊢ ( 𝑤 = 𝑊 → ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = ( LKer ‘ 𝑈 ) )
21 20 6 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝐿 )
22 21 fveq1d ⊢ ( 𝑤 = 𝑊 → ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) = ( 𝐿 ‘ 𝑓 ) )
23 19 22 fveq12d ⊢ ( 𝑤 = 𝑊 → ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) = ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) )
24 19 23 fveq12d ⊢ ( 𝑤 = 𝑊 → ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) )
25 24 22 eqeq12d ⊢ ( 𝑤 = 𝑊 → ( ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ↔ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ) )
26 17 25 rabeqbidv ⊢ ( 𝑤 = 𝑊 → { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } )
27 15 26 oveq12d ⊢ ( 𝑤 = 𝑊 → ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) = ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) )
28 eqid ⊢ ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) = ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) )
29 ovex ⊢ ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) ∈ V
30 27 28 29 fvmpt ⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ↾s { 𝑓 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑤 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) ) ‘ 𝑓 ) } ) ) ‘ 𝑊 ) = ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) )
31 11 30 sylan9eq ⊢ ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) → 𝐶 = ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) )
32 8 31 syl ⊢ ( 𝜑 → 𝐶 = ( 𝐷 ↾s { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ) )