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Theorem dicopelval2 41164
Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 20-Feb-2014.)
Hypotheses
Ref Expression
dicval.l = (le‘𝐾)
dicval.a 𝐴 = (Atoms‘𝐾)
dicval.h 𝐻 = (LHyp‘𝐾)
dicval.p 𝑃 = ((oc‘𝐾)‘𝑊)
dicval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicval.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dicval.i 𝐼 = ((DIsoC‘𝐾)‘𝑊)
dicval2.g 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
dicelval2.f 𝐹 ∈ V
dicelval2.s 𝑆 ∈ V
Assertion
Ref Expression
dicopelval2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆𝐺) ∧ 𝑆𝐸)))
Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,𝑊   𝑄,𝑔
Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝑆(𝑔)   𝐸(𝑔)   𝐹(𝑔)   𝐺(𝑔)   𝐻(𝑔)   𝐼(𝑔)   (𝑔)   𝑉(𝑔)

Proof of Theorem dicopelval2
StepHypRef Expression
1 dicval.l . . 3 = (le‘𝐾)
2 dicval.a . . 3 𝐴 = (Atoms‘𝐾)
3 dicval.h . . 3 𝐻 = (LHyp‘𝐾)
4 dicval.p . . 3 𝑃 = ((oc‘𝐾)‘𝑊)
5 dicval.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
6 dicval.e . . 3 𝐸 = ((TEndo‘𝐾)‘𝑊)
7 dicval.i . . 3 𝐼 = ((DIsoC‘𝐾)‘𝑊)
8 dicelval2.f . . 3 𝐹 ∈ V
9 dicelval2.s . . 3 𝑆 ∈ V
101, 2, 3, 4, 5, 6, 7, 8, 9dicopelval 41160 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑆𝐸)))
11 dicval2.g . . . . 5 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
1211fveq2i 6910 . . . 4 (𝑆𝐺) = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄))
1312eqeq2i 2748 . . 3 (𝐹 = (𝑆𝐺) ↔ 𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1413anbi1i 624 . 2 ((𝐹 = (𝑆𝐺) ∧ 𝑆𝐸) ↔ (𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑆𝐸))
1510, 14bitr4di 289 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆𝐺) ∧ 𝑆𝐸)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148  cfv 6563  crio 7387  lecple 17305  occoc 17306  Atomscatm 39245  LHypclh 39967  LTrncltrn 40084  TEndoctendo 40735  DIsoCcdic 41155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-dic 41156
This theorem is referenced by:  diclspsn  41177  cdlemn11a  41190  dihopelvalcqat  41229  dihopelvalcpre  41231  dihord6apre  41239
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