Theorem lcdval 41884

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))

Assertion

Ref	Expression
lcdval	⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}))

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

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

Proof of Theorem lcdval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcdval.k	. 2 ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻))
2		lcdval.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		lcdval.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
4	3	lcdfval 41883	. . . . 5 ⊢ (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)})))
5	4	fveq1d 6835	. . . 4 ⊢ (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))
6	2, 5	eqtrid 2782	. . 3 ⊢ (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))
7		fveq2 6833	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8		lcdval.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9	7, 8	eqtr4di 2788	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10	9	fveq2d 6837	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = (LDual‘𝑈))
11		lcdval.d	. . . . . 6 ⊢ 𝐷 = (LDual‘𝑈)
12	10, 11	eqtr4di 2788	. . . . 5 ⊢ (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = 𝐷)
13	9	fveq2d 6837	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = (LFnl‘𝑈))
14		lcdval.f	. . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑈)
15	13, 14	eqtr4di 2788	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = 𝐹)
16		fveq2 6833	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
17		lcdval.o	. . . . . . . . 9 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2788	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ⊥ )
19	9	fveq2d 6837	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = (LKer‘𝑈))
20		lcdval.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LKer‘𝑈)
21	19, 20	eqtr4di 2788	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = 𝐿)
22	21	fveq1d 6835	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) = (𝐿‘𝑓))
23	18, 22	fveq12d 6840	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) = ( ⊥ ‘(𝐿‘𝑓)))
24	18, 23	fveq12d 6840	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))))
25	24, 22	eqeq12d 2751	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)))
26	15, 25	rabeqbidv 3416	. . . . 5 ⊢ (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	12, 26	oveq12d 7376	. . . 4 ⊢ (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 2735	. . . 4 ⊢ (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 7391	. . . 4 ⊢ (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) ∈ V
30	27, 28, 29	fvmpt 6940	. . 3 ⊢ (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	6, 30	sylan9eq 2790	. 2 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐶 = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}))
32	1, 31	syl 17	1 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3398   ↦ cmpt 5178  ‘cfv 6491  (class class class)co 7358   ↾_s cress 17159  LFnlclfn 39352  LKerclk 39380  LDualcld 39418  LHypclh 40279  DVecHcdvh 41373  ocHcoch 41642  LCDualclcd 41881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-lcdual 41882

This theorem is referenced by:  lcdval2  41885  lcdlvec  41886  lcdvadd  41892  lcdsca  41894  lcdvs  41898  lcd0v  41906  lcdlsp  41916