Theorem mapdvalc 41134

Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015.)

Hypotheses

Ref	Expression
mapdval.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdval.s	⊢ 𝑆 = (LSubSp‘𝑈)
mapdval.f	⊢ 𝐹 = (LFnl‘𝑈)
mapdval.l	⊢ 𝐿 = (LKer‘𝑈)
mapdval.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
mapdval.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdval.k	⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻))
mapdval.t	⊢ (𝜑 → 𝑇 ∈ 𝑆)
mapdvalc.c	⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}

Assertion

Ref	Expression
mapdvalc	⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇})

Distinct variable groups:   𝑓,𝐾   𝑓,𝐹   𝑓,𝑊   𝑓,𝑔,𝐹   𝑔,𝐿   𝑔,𝑂   𝑇,𝑓   𝜑,𝑓

Allowed substitution hints:   𝜑(𝑔)   𝐶(𝑓,𝑔)   𝑆(𝑓,𝑔)   𝑇(𝑔)   𝑈(𝑓,𝑔)   𝐻(𝑓,𝑔)   𝐾(𝑔)   𝐿(𝑓)   𝑀(𝑓,𝑔)   𝑂(𝑓)   𝑊(𝑔)   𝑋(𝑓,𝑔)

Proof of Theorem mapdvalc

Step	Hyp	Ref	Expression
1		mapdval.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdval.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdval.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑈)
4		mapdval.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
5		mapdval.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
6		mapdval.o	. . 3 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
7		mapdval.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
8		mapdval.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻))
9		mapdval.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mapdval 41133	. 2 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)})
11		anass 467	. . . 4 ⊢ (((𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ↔ (𝑓 ∈ 𝐹 ∧ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)))
12		mapdvalc.c	. . . . . . . 8 ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
13	12	lcfl1lem 40996	. . . . . . 7 ⊢ (𝑓 ∈ 𝐶 ↔ (𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)))
14	13	anbi1i 622	. . . . . 6 ⊢ ((𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ↔ ((𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))
15	14	bicomi 223	. . . . 5 ⊢ (((𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ↔ (𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (((𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ↔ (𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)))
17	11, 16	bitr3id 284	. . 3 ⊢ (𝜑 → ((𝑓 ∈ 𝐹 ∧ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)) ↔ (𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)))
18	17	rabbidva2 3432	. 2 ⊢ (𝜑 → {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇})
19	10, 18	eqtrd 2768	1 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3430   ⊆ wss 3949  ‘cfv 6553  LSubSpclss 20822  LFnlclfn 38561  LKerclk 38589  LHypclh 39489  DVecHcdvh 40583  ocHcoch 40852  mapdcmpd 41129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-mapd 41130

This theorem is referenced by:  mapdval2N  41135  mapdordlem2  41142  mapdrval  41152