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Theorem dicelval1sta 36302
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 2621 . . . . . 6 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
7 dicelval1sta.i . . . . . 6 𝐼 = ((DIsoC‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dicval 36291 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
98eleq2d 2686 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}))
109biimp3a 1431 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
11 eqeq1 2625 . . . . 5 (𝑓 = (1st𝑌) → (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ (1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
1211anbi1d 741 . . . 4 (𝑓 = (1st𝑌) → ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))))
13 fveq1 6188 . . . . . 6 (𝑠 = (2nd𝑌) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1413eqeq2d 2631 . . . . 5 (𝑠 = (2nd𝑌) → ((1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ (1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
15 eleq1 2688 . . . . 5 (𝑠 = (2nd𝑌) → (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ↔ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊)))
1614, 15anbi12d 747 . . . 4 (𝑠 = (2nd𝑌) → (((1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊))))
1712, 16elopabi 7228 . . 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 475 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 384  w3a 1037   = wceq 1482  wcel 1989   class class class wbr 4651  {copab 4710  cfv 5886  crio 6607  1st c1st 7163  2nd c2nd 7164  lecple 15942  occoc 15943  Atomscatm 34376  LHypclh 35096  LTrncltrn 35213  TEndoctendo 35866  DIsoCcdic 36287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-1st 7165  df-2nd 7166  df-dic 36288
This theorem is referenced by:  dicvaddcl  36305  dicvscacl  36306
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