Theorem lcdfval 38739

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

Hypothesis

Ref	Expression
lcdval.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
lcdfval	⊢ (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)})))

Distinct variable groups:   𝑤,𝐻   𝑤,𝑓,𝐾

Allowed substitution hints:   𝐻(𝑓)   𝑋(𝑤,𝑓)

Proof of Theorem lcdfval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3512	. 2 ⊢ (𝐾 ∈ 𝑋 → 𝐾 ∈ V)
2		fveq2 6670	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		lcdval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	syl6eqr 2874	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6670	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
6	5	fveq1d 6672	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
7	6	fveq2d 6674	. . . . 5 ⊢ (𝑘 = 𝐾 → (LDual‘((DVecH‘𝑘)‘𝑤)) = (LDual‘((DVecH‘𝐾)‘𝑤)))
8	6	fveq2d 6674	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LFnl‘((DVecH‘𝑘)‘𝑤)) = (LFnl‘((DVecH‘𝐾)‘𝑤)))
9		fveq2 6670	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (ocH‘𝑘) = (ocH‘𝐾))
10	9	fveq1d 6672	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((ocH‘𝑘)‘𝑤) = ((ocH‘𝐾)‘𝑤))
11	6	fveq2d 6674	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (LKer‘((DVecH‘𝑘)‘𝑤)) = (LKer‘((DVecH‘𝐾)‘𝑤)))
12	11	fveq1d 6672	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))
13	10, 12	fveq12d 6677	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) = (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)))
14	10, 13	fveq12d 6677	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))))
15	14, 12	eqeq12d 2837	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ↔ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)))
16	8, 15	rabeqbidv 3485	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ (((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)} = {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)})
17	7, 16	oveq12d 7174	. . . 4 ⊢ (𝑘 = 𝐾 → ((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)}) = ((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)}))
18	4, 17	mpteq12dv 5151	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ ((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)})))
19		df-lcdual 38738	. . 3 ⊢ LCDual = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ((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)})))
20	18, 19, 3	mptfvmpt 6990	. 2 ⊢ (𝐾 ∈ V → (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)})))
21	1, 20	syl 17	1 ⊢ (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)})))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  {crab 3142  Vcvv 3494   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156   ↾_s cress 16484  LFnlclfn 36208  LKerclk 36236  LDualcld 36274  LHypclh 37135  DVecHcdvh 38229  ocHcoch 38498  LCDualclcd 38737

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 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pr 5330

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-lcdual 38738

This theorem is referenced by:  lcdval  38740