Theorem lcdval2 38728

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 38727	. 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 7169	. 2 ⊢ (𝐷 ↾s 𝐵) = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
12	9, 11	syl6eqr 2876	1 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3144  ‘cfv 6357  (class class class)co 7158   ↾_s cress 16486  LFnlclfn 36195  LKerclk 36223  LDualcld 36261  LHypclh 37122  DVecHcdvh 38216  ocHcoch 38485  LCDualclcd 38724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-lcdual 38725

This theorem is referenced by:  lcdvbase  38731  lcdlss  38757