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Theorem dibopelval3 38356
Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)
Hypotheses
Ref Expression
dibval3.b 𝐵 = (Base‘𝐾)
dibval3.l = (le‘𝐾)
dibval3.h 𝐻 = (LHyp‘𝐾)
dibval3.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibval3.r 𝑅 = ((trL‘𝐾)‘𝑊)
dibval3.o 0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
dibval3.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibopelval3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 0 )))
Distinct variable groups:   𝑔,𝐾   𝑔,𝑊   𝑇,𝑔
Allowed substitution hints:   𝐵(𝑔)   𝑅(𝑔)   𝑆(𝑔)   𝐹(𝑔)   𝐻(𝑔)   𝐼(𝑔)   (𝑔)   𝑉(𝑔)   𝑋(𝑔)   0 (𝑔)

Proof of Theorem dibopelval3
StepHypRef Expression
1 dibval3.b . . 3 𝐵 = (Base‘𝐾)
2 dibval3.l . . 3 = (le‘𝐾)
3 dibval3.h . . 3 𝐻 = (LHyp‘𝐾)
4 dibval3.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
5 dibval3.o . . 3 0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
6 eqid 2824 . . 3 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7 dibval3.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibopelval2 38353 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ (𝐹 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑆 = 0 )))
9 dibval3.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
101, 2, 3, 4, 9, 6diaelval 38241 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋)))
1110anbi1d 632 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝐹 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑆 = 0 ) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 0 )))
128, 11bitrd 282 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cop 4556   class class class wbr 5053  cmpt 5133   I cid 5447  cres 5545  cfv 6344  Basecbs 16481  lecple 16570  LHypclh 37192  LTrncltrn 37309  trLctrl 37366  DIsoAcdia 38236  DIsoBcdib 38346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-disoa 38237  df-dib 38347
This theorem is referenced by:  dihord2cN  38429  dihord11b  38430  dihopelvalbN  38446  dihopelvalcpre  38456  dihjatcclem4  38629
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