Theorem mapdval 41666

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)

Assertion

Ref	Expression
mapdval	⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)})

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

Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝑈(𝑓)   𝐻(𝑓)   𝐿(𝑓)   𝑀(𝑓)   𝑂(𝑓)   𝑋(𝑓)

Proof of Theorem mapdval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdval.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻))
2		mapdval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		mapdval.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		mapdval.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑈)
5		mapdval.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑈)
6		mapdval.l	. . . . 5 ⊢ 𝐿 = (LKer‘𝑈)
7		mapdval.o	. . . . 5 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
8		mapdval.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
9	2, 3, 4, 5, 6, 7, 8	mapdfval 41665	. . . 4 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}))
10	1, 9	syl 17	. . 3 ⊢ (𝜑 → 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}))
11	10	fveq1d 6824	. 2 ⊢ (𝜑 → (𝑀‘𝑇) = ((𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})‘𝑇))
12		mapdval.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
13	5	fvexi 6836	. . . 4 ⊢ 𝐹 ∈ V
14	13	rabex 5277	. . 3 ⊢ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} ∈ V
15		sseq2 3961	. . . . . 6 ⊢ (𝑠 = 𝑇 → ((𝑂‘(𝐿‘𝑓)) ⊆ 𝑠 ↔ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇))
16	15	anbi2d 630	. . . . 5 ⊢ (𝑠 = 𝑇 → (((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠) ↔ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)))
17	16	rabbidv 3402	. . . 4 ⊢ (𝑠 = 𝑇 → {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)} = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)})
18		eqid 2731	. . . 4 ⊢ (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}) = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})
19	17, 18	fvmptg 6927	. . 3 ⊢ ((𝑇 ∈ 𝑆 ∧ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)} ∈ V) → ((𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)})
20	12, 14, 19	sylancl 586	. 2 ⊢ (𝜑 → ((𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)})
21	11, 20	eqtrd 2766	1 ⊢ (𝜑 → (𝑀‘𝑇) = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑇)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ⊆ wss 3902   ↦ cmpt 5172  ‘cfv 6481  LSubSpclss 20862  LFnlclfn 39095  LKerclk 39123  LHypclh 40022  DVecHcdvh 41116  ocHcoch 41385  mapdcmpd 41662

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-mapd 41663

This theorem is referenced by:  mapdvalc  41667  mapddlssN  41678  mapdsn  41679  mapd1o  41686  mapd0  41703