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Theorem dicopelval2 36991
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 36987 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑆𝐸)))
11 dicval2.g . . . . 5 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
1211fveq2i 6335 . . . 4 (𝑆𝐺) = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄))
1312eqeq2i 2783 . . 3 (𝐹 = (𝑆𝐺) ↔ 𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1413anbi1i 610 . 2 ((𝐹 = (𝑆𝐺) ∧ 𝑆𝐸) ↔ (𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑆𝐸))
1510, 14syl6bbr 278 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆𝐺) ∧ 𝑆𝐸)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322   class class class wbr 4786  cfv 6031  crio 6753  lecple 16156  occoc 16157  Atomscatm 35072  LHypclh 35792  LTrncltrn 35909  TEndoctendo 36561  DIsoCcdic 36982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-dic 36983
This theorem is referenced by:  diclspsn  37004  cdlemn11a  37017  dihopelvalcqat  37056  dihopelvalcpre  37058  dihord6apre  37066
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