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Theorem dihopelvalbN 39252
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 2738 . . . 4 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
61, 2, 3, 4, 5dihvalb 39251 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
76eleq2d 2824 . 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 39162 . 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 396   = wceq 1539  wcel 2106  cop 4567   class class class wbr 5074  cmpt 5157   I cid 5488  cres 5591  cfv 6433  Basecbs 16912  lecple 16969  LHypclh 37998  LTrncltrn 38115  trLctrl 38172  DIsoBcdib 39152  DIsoHcdih 39242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-disoa 39043  df-dib 39153  df-dih 39243
This theorem is referenced by:  dihmeetlem1N  39304  dihglblem5apreN  39305  dihmeetlem4preN  39320
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