Theorem lcdval2 36879

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

Hypotheses

Ref	Expression
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))
lcdval2.b	⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}

Assertion

Ref	Expression
lcdval2	⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵))

Distinct variable groups:   𝑓,𝐾   𝑓,𝐹   𝑓,𝑊

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝑈(𝑓)   𝐻(𝑓)   𝐿(𝑓)   ⊥ (𝑓)   𝑋(𝑓)

Proof of Theorem lcdval2

Step	Hyp	Ref	Expression
1		lcdval.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcdval.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcdval.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
4		lcdval.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		lcdval.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
6		lcdval.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
7		lcdval.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
8		lcdval.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻))
9	1, 2, 3, 4, 5, 6, 7, 8	lcdval 36878	. 2 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}))
10		lcdval2.b	. . 3 ⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
11	10	oveq2i 6661	. 2 ⊢ (𝐷 ↾s 𝐵) = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
12	9, 11	syl6eqr 2674	1 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  ‘cfv 5888  (class class class)co 6650   ↾_s cress 15858  LFnlclfn 34344  LKerclk 34372  LDualcld 34410  LHypclh 35270  DVecHcdvh 36367  ocHcoch 36636  LCDualclcd 36875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-lcdual 36876

This theorem is referenced by:  lcdvbase  36882  lcdlss  36908