Theorem lcdval2 42178

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 42177	. 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 7403	. 2 ⊢ (𝐷 ↾s 𝐵) = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
12	9, 11	eqtr4di 2814	1 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {crab 3413  ‘cfv 6517  (class class class)co 7392   ↾_s cress 17249  LFnlclfn 39645  LKerclk 39673  LDualcld 39711  LHypclh 40572  DVecHcdvh 41666  ocHcoch 41935  LCDualclcd 42174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-lcdual 42175

This theorem is referenced by:  lcdvbase  42181  lcdlss  42207