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Theorem dicelval1sta 40712
Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 16-Feb-2014.)
Hypotheses
Ref Expression
dicelval1sta.l ≀ = (leβ€˜πΎ)
dicelval1sta.a 𝐴 = (Atomsβ€˜πΎ)
dicelval1sta.h 𝐻 = (LHypβ€˜πΎ)
dicelval1sta.p 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
dicelval1sta.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dicelval1sta.i 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dicelval1sta (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘„)) β†’ (1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)))
Distinct variable groups:   𝑔,𝐾   𝑄,𝑔   𝑇,𝑔   𝑔,π‘Š
Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝐻(𝑔)   𝐼(𝑔)   ≀ (𝑔)   𝑉(𝑔)   π‘Œ(𝑔)
Proof of Theorem dicelval1sta
Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicelval1sta.l . . . . . 6 ≀ = (leβ€˜πΎ)
2 dicelval1sta.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
3 dicelval1sta.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
4 dicelval1sta.p . . . . . 6 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
5 dicelval1sta.t . . . . . 6 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
6 eqid 2725 . . . . . 6 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
7 dicelval1sta.i . . . . . 6 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
81, 2, 3, 4, 5, 6, 7dicval 40701 . . . . 5 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))})
98eleq2d 2811 . . . 4 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (π‘Œ ∈ (πΌβ€˜π‘„) ↔ π‘Œ ∈ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))}))
109biimp3a 1465 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘„)) β†’ π‘Œ ∈ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))})
11 eqeq1 2729 . . . . 5 (𝑓 = (1st β€˜π‘Œ) β†’ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ↔ (1st β€˜π‘Œ) = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄))))
1211anbi1d 629 . . . 4 (𝑓 = (1st β€˜π‘Œ) β†’ ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) ↔ ((1st β€˜π‘Œ) = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))))
13 fveq1 6889 . . . . . 6 (𝑠 = (2nd β€˜π‘Œ) β†’ (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)))
1413eqeq2d 2736 . . . . 5 (𝑠 = (2nd β€˜π‘Œ) β†’ ((1st β€˜π‘Œ) = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ↔ (1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄))))
15 eleq1 2813 . . . . 5 (𝑠 = (2nd β€˜π‘Œ) β†’ (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↔ (2nd β€˜π‘Œ) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)))
1614, 15anbi12d 630 . . . 4 (𝑠 = (2nd β€˜π‘Œ) β†’ (((1st β€˜π‘Œ) = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) ↔ ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ (2nd β€˜π‘Œ) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))))
1712, 16elopabi 8060 . . 3 (π‘Œ ∈ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))} β†’ ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ (2nd β€˜π‘Œ) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)))
1810, 17syl 17 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘„)) β†’ ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ (2nd β€˜π‘Œ) ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)))
1918simpld 493 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ π‘Œ ∈ (πΌβ€˜π‘„)) β†’ (1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5144  {copab 5206  β€˜cfv 6543  β„©crio 7368  1st c1st 7985  2nd c2nd 7986  lecple 17234  occoc 17235  Atomscatm 38787  LHypclh 39509  LTrncltrn 39626  TEndoctendo 40277  DIsoCcdic 40697
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 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-1st 7987  df-2nd 7988  df-dic 40698
This theorem is referenced by:  dicvaddcl  40715  dicvscacl  40716
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