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Theorem dihopelvalbN 40715
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 2727 . . . 4 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
61, 2, 3, 4, 5dihvalb 40714 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
76eleq2d 2814 . 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 40625 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
127, 11bitrd 278 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  cop 4636   class class class wbr 5150  cmpt 5233   I cid 5577  cres 5682  cfv 6551  Basecbs 17185  lecple 17245  LHypclh 39461  LTrncltrn 39578  trLctrl 39635  DIsoBcdib 40615  DIsoHcdih 40705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-disoa 40506  df-dib 40616  df-dih 40706
This theorem is referenced by:  dihmeetlem1N  40767  dihglblem5apreN  40768  dihmeetlem4preN  40783
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