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Theorem dicelval1sta 40046
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 2732 . . . . . 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 40035 . . . . 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 2819 . . . 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 1469 . . 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 2736 . . . . 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 630 . . . 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 6887 . . . . . 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 2743 . . . . 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 2821 . . . . 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 631 . . . 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 8044 . . 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 495 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5147  {copab 5209  β€˜cfv 6540  β„©crio 7360  1st c1st 7969  2nd c2nd 7970  lecple 17200  occoc 17201  Atomscatm 38121  LHypclh 38843  LTrncltrn 38960  TEndoctendo 39611  DIsoCcdic 40031
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-1st 7971  df-2nd 7972  df-dic 40032
This theorem is referenced by:  dicvaddcl  40049  dicvscacl  40050
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