Theorem mapdvalc 42217

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 42216	. 2 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)})
11		anass 472	. . . 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 42079	. . . . . . 7 ⊢ (𝑓 ∈ 𝐶 ↔ (𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)))
14	13	anbi1i 633	. . . . . 6 ⊢ ((𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇) ↔ ((𝑓 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))
15	14	bicomi 226	. . . . 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 287	. . 3 ⊢ (𝜑 → ((𝑓 ∈ 𝐹 ∧ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)) ↔ (𝑓 ∈ 𝐶 ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)))
18	17	rabbidva2 3415	. 2 ⊢ (𝜑 → {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇})
19	10, 18	eqtrd 2796	1 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {crab 3413   ⊆ wss 3904  ‘cfv 6517  LSubSpclss 20978  LFnlclfn 39645  LKerclk 39673  LHypclh 40572  DVecHcdvh 41666  ocHcoch 41935  mapdcmpd 42212

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-pw 4556  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-mapd 42213

This theorem is referenced by:  mapdval2N  42218  mapdordlem2  42225  mapdrval  42235