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Theorem dihopelvalbN 35545
 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 2610 . . . 4 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
61, 2, 3, 4, 5dihvalb 35544 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
76eleq2d 2673 . 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 35455 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
127, 11bitrd 267 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   ↾ cres 5040  ‘cfv 5804  Basecbs 15695  lecple 15775  LHypclh 34288  LTrncltrn 34405  trLctrl 34463  DIsoBcdib 35445  DIsoHcdih 35535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-disoa 35336  df-dib 35446  df-dih 35536 This theorem is referenced by:  dihmeetlem1N  35597  dihglblem5apreN  35598  dihmeetlem4preN  35613
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