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Theorem dicelval1sta 40692
Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 16-Feb-2014.)
Hypotheses
Ref Expression
dicelval1sta.l = (le‘𝐾)
dicelval1sta.a 𝐴 = (Atoms‘𝐾)
dicelval1sta.h 𝐻 = (LHyp‘𝐾)
dicelval1sta.p 𝑃 = ((oc‘𝐾)‘𝑊)
dicelval1sta.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicelval1sta.i 𝐼 = ((DIsoC‘𝐾)‘𝑊)
Assertion
Ref Expression
dicelval1sta (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
Distinct variable groups:   𝑔,𝐾   𝑄,𝑔   𝑇,𝑔   𝑔,𝑊
Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝐻(𝑔)   𝐼(𝑔)   (𝑔)   𝑉(𝑔)   𝑌(𝑔)

Proof of Theorem dicelval1sta
Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicelval1sta.l . . . . . 6 = (le‘𝐾)
2 dicelval1sta.a . . . . . 6 𝐴 = (Atoms‘𝐾)
3 dicelval1sta.h . . . . . 6 𝐻 = (LHyp‘𝐾)
4 dicelval1sta.p . . . . . 6 𝑃 = ((oc‘𝐾)‘𝑊)
5 dicelval1sta.t . . . . . 6 𝑇 = ((LTrn‘𝐾)‘𝑊)
6 eqid 2728 . . . . . 6 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
7 dicelval1sta.i . . . . . 6 𝐼 = ((DIsoC‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dicval 40681 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
98eleq2d 2815 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}))
109biimp3a 1465 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
11 eqeq1 2732 . . . . 5 (𝑓 = (1st𝑌) → (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ (1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
1211anbi1d 629 . . . 4 (𝑓 = (1st𝑌) → ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))))
13 fveq1 6901 . . . . . 6 (𝑠 = (2nd𝑌) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1413eqeq2d 2739 . . . . 5 (𝑠 = (2nd𝑌) → ((1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ (1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
15 eleq1 2817 . . . . 5 (𝑠 = (2nd𝑌) → (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ↔ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊)))
1614, 15anbi12d 630 . . . 4 (𝑠 = (2nd𝑌) → (((1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊))))
1712, 16elopabi 8072 . . 3 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} → ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊)))
1810, 17syl 17 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊)))
1918simpld 493 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5152  {copab 5214  cfv 6553  crio 7381  1st c1st 7997  2nd c2nd 7998  lecple 17247  occoc 17248  Atomscatm 38767  LHypclh 39489  LTrncltrn 39606  TEndoctendo 40257  DIsoCcdic 40677
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-1st 7999  df-2nd 8000  df-dic 40678
This theorem is referenced by:  dicvaddcl  40695  dicvscacl  40696
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