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Theorem dicelval1sta 40584
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 2727 . . . . . 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 40573 . . . . 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 2814 . . . 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 1466 . . 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 2731 . . . . 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 6890 . . . . . 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 2738 . . . . 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 2816 . . . . 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 8058 . . 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 494 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5142  {copab 5204  β€˜cfv 6542  β„©crio 7369  1st c1st 7983  2nd c2nd 7984  lecple 17225  occoc 17226  Atomscatm 38659  LHypclh 39381  LTrncltrn 39498  TEndoctendo 40149  DIsoCcdic 40569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-1st 7985  df-2nd 7986  df-dic 40570
This theorem is referenced by:  dicvaddcl  40587  dicvscacl  40588
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