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Theorem dihopelvalbN 38244
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 2826 . . . 4 ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
61, 2, 3, 4, 5dihvalb 38243 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
76eleq2d 2903 . 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 38154 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
127, 11bitrd 280 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  cop 4570   class class class wbr 5063  cmpt 5143   I cid 5458  cres 5556  cfv 6352  Basecbs 16473  lecple 16562  LHypclh 36990  LTrncltrn 37107  trLctrl 37164  DIsoBcdib 38144  DIsoHcdih 38234
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-disoa 38035  df-dib 38145  df-dih 38235
This theorem is referenced by:  dihmeetlem1N  38296  dihglblem5apreN  38297  dihmeetlem4preN  38312
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