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Theorem dibopelval3 37169
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 2799 . . 3 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7 dibval3.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibopelval2 37166 . 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 37054 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋)))
1110anbi1d 624 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → ((𝐹 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋) ∧ 𝑆 = 0 ) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 0 )))
128, 11bitrd 271 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cop 4374   class class class wbr 4843  cmpt 4922   I cid 5219  cres 5314  cfv 6101  Basecbs 16184  lecple 16274  LHypclh 36005  LTrncltrn 36122  trLctrl 36179  DIsoAcdia 37049  DIsoBcdib 37159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-disoa 37050  df-dib 37160
This theorem is referenced by:  dihord2cN  37242  dihord11b  37243  dihopelvalbN  37259  dihopelvalcpre  37269  dihjatcclem4  37442
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