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Theorem dicelval1sta 38476
 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 2801 . . . . . 6 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
7 dicelval1sta.i . . . . . 6 𝐼 = ((DIsoC‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dicval 38465 . . . . 5 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
98eleq2d 2878 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}))
109biimp3a 1466 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
11 eqeq1 2805 . . . . 5 (𝑓 = (1st𝑌) → (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ (1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
1211anbi1d 632 . . . 4 (𝑓 = (1st𝑌) → ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))))
13 fveq1 6648 . . . . . 6 (𝑠 = (2nd𝑌) → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1413eqeq2d 2812 . . . . 5 (𝑠 = (2nd𝑌) → ((1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ (1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
15 eleq1 2880 . . . . 5 (𝑠 = (2nd𝑌) → (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ↔ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊)))
1614, 15anbi12d 633 . . . 4 (𝑠 = (2nd𝑌) → (((1st𝑌) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ ((TEndo‘𝐾)‘𝑊))))
1712, 16elopabi 7746 . . 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 498 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   class class class wbr 5033  {copab 5095  ‘cfv 6328  ℩crio 7096  1st c1st 7673  2nd c2nd 7674  lecple 16567  occoc 16568  Atomscatm 36552  LHypclh 37273  LTrncltrn 37390  TEndoctendo 38041  DIsoCcdic 38461 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-1st 7675  df-2nd 7676  df-dic 38462 This theorem is referenced by:  dicvaddcl  38479  dicvscacl  38480
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