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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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