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Theorem dihopelvalbN 37308
Description: Ordered pair member of the partial isomorphism H for argument under 𝑊. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihval3.b 𝐵 = (Base‘𝐾)
dihval3.l = (le‘𝐾)
dihval3.h 𝐻 = (LHyp‘𝐾)
dihval3.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihval3.r 𝑅 = ((trL‘𝐾)‘𝑊)
dihval3.o 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
dihval3.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
Assertion
Ref Expression
dihopelvalbN (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,𝑊
Allowed substitution hints:   𝐵(𝑔)   𝑅(𝑔)   𝑆(𝑔)   𝐹(𝑔)   𝐻(𝑔)   𝐼(𝑔)   (𝑔)   𝑂(𝑔)   𝑉(𝑔)   𝑋(𝑔)

Proof of Theorem dihopelvalbN
StepHypRef Expression
1 dihval3.b . . . 4 𝐵 = (Base‘𝐾)
2 dihval3.l . . . 4 = (le‘𝐾)
3 dihval3.h . . . 4 𝐻 = (LHyp‘𝐾)
4 dihval3.i . . . 4 𝐼 = ((DIsoH‘𝐾)‘𝑊)
5 eqid 2825 . . . 4 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
61, 2, 3, 4, 5dihvalb 37307 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
76eleq2d 2892 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋)))
8 dihval3.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
9 dihval3.r . . 3 𝑅 = ((trL‘𝐾)‘𝑊)
10 dihval3.o . . 3 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
111, 2, 3, 8, 9, 10, 5dibopelval3 37218 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
127, 11bitrd 271 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  cop 4405   class class class wbr 4875  cmpt 4954   I cid 5251  cres 5348  cfv 6127  Basecbs 16229  lecple 16319  LHypclh 36054  LTrncltrn 36171  trLctrl 36228  DIsoBcdib 37208  DIsoHcdih 37298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-disoa 37099  df-dib 37209  df-dih 37299
This theorem is referenced by:  dihmeetlem1N  37360  dihglblem5apreN  37361  dihmeetlem4preN  37376
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