Theorem lcdval 42566

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 42565	. . . . 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 6875	. . . 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 2807	. . 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 6873	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8		lcdval.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9	7, 8	eqtr4di 2813	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
10	9	fveq2d 6877	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = (LDual‘𝑈))
11		lcdval.d	. . . . . 6 ⊢ 𝐷 = (LDual‘𝑈)
12	10, 11	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = 𝐷)
13	9	fveq2d 6877	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = (LFnl‘𝑈))
14		lcdval.f	. . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑈)
15	13, 14	eqtr4di 2813	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = 𝐹)
16		fveq2 6873	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
17		lcdval.o	. . . . . . . . 9 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
18	16, 17	eqtr4di 2813	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ⊥ )
19	9	fveq2d 6877	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = (LKer‘𝑈))
20		lcdval.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LKer‘𝑈)
21	19, 20	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = 𝐿)
22	21	fveq1d 6875	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) = (𝐿‘𝑓))
23	18, 22	fveq12d 6880	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) = ( ⊥ ‘(𝐿‘𝑓)))
24	18, 23	fveq12d 6880	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))))
25	24, 22	eqeq12d 2776	. . . . . 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 3429	. . . . 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 7426	. . . 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 2760	. . . 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 7441	. . . 4 ⊢ (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) ∈ V
30	27, 28, 29	fvmpt 6981	. . 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 2815	. 2 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐶 = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}))
32	1, 31	syl 18	1 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412   ↦ cmpt 5185  ‘cfv 6527  (class class class)co 7408   ↾_s cress 17370  LFnlclfn 40034  LKerclk 40062  LDualcld 40100  LHypclh 40961  DVecHcdvh 42055  ocHcoch 42324  LCDualclcd 42563

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-lcdual 42564

This theorem is used by:  lcdval2  42567  lcdlvec  42568  lcdvadd  42574  lcdsca  42576  lcdvs  42580  lcd0v  42588  lcdlsp  42598