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Theorem mapdval 35735
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 → (𝐾𝑋𝑊𝐻))
mapdval.t (𝜑𝑇𝑆)
Assertion
Ref Expression
mapdval (𝜑 → (𝑀𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 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.
StepHypRef Expression
1 mapdval.k . . . 4 (𝜑 → (𝐾𝑋𝑊𝐻))
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)
92, 3, 4, 5, 6, 7, 8mapdfval 35734 . . . 4 ((𝐾𝑋𝑊𝐻) → 𝑀 = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))
101, 9syl 17 . . 3 (𝜑𝑀 = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))
1110fveq1d 6087 . 2 (𝜑 → (𝑀𝑇) = ((𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)})‘𝑇))
12 mapdval.t . . 3 (𝜑𝑇𝑆)
13 fvex 6095 . . . . 5 (LFnl‘𝑈) ∈ V
145, 13eqeltri 2680 . . . 4 𝐹 ∈ V
1514rabex 4732 . . 3 {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)} ∈ V
16 sseq2 3586 . . . . . 6 (𝑠 = 𝑇 → ((𝑂‘(𝐿𝑓)) ⊆ 𝑠 ↔ (𝑂‘(𝐿𝑓)) ⊆ 𝑇))
1716anbi2d 735 . . . . 5 (𝑠 = 𝑇 → (((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠) ↔ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)))
1817rabbidv 3160 . . . 4 (𝑠 = 𝑇 → {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)} = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
19 eqid 2606 . . . 4 (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}) = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)})
2018, 19fvmptg 6171 . . 3 ((𝑇𝑆 ∧ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)} ∈ V) → ((𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)})‘𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
2112, 15, 20sylancl 692 . 2 (𝜑 → ((𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)})‘𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
2211, 21eqtrd 2640 1 (𝜑 → (𝑀𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  {crab 2896  Vcvv 3169  wss 3536  cmpt 4634  cfv 5787  LSubSpclss 18696  LFnlclfn 33162  LKerclk 33190  LHypclh 34088  DVecHcdvh 35185  ocHcoch 35454  mapdcmpd 35731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-mapd 35732
This theorem is referenced by:  mapdvalc  35736  mapddlssN  35747  mapdsn  35748  mapd1o  35755  mapd0  35772
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