Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dicelval2N Structured version   Visualization version   GIF version
Theorem dicelval2N 40357
Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicval.l ≀ = (leβ€˜πΎ)
dicval.a 𝐴 = (Atomsβ€˜πΎ)
dicval.h 𝐻 = (LHypβ€˜πΎ)
dicval.p 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
dicval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dicval.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dicval.i 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
dicval2.g 𝐺 = (℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)
Assertion
Ref Expression
dicelval2N (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (π‘Œ ∈ (πΌβ€˜π‘„) ↔ (π‘Œ ∈ (V Γ— V) ∧ ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜πΊ) ∧ (2nd β€˜π‘Œ) ∈ 𝐸))))
Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,π‘Š   𝑄,𝑔
Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝐸(𝑔)   𝐺(𝑔)   𝐻(𝑔)   𝐼(𝑔)   ≀ (𝑔)   𝑉(𝑔)   π‘Œ(𝑔)
Proof of Theorem dicelval2N
StepHypRef Expression
1 dicval.l . . 3 ≀ = (leβ€˜πΎ)
2 dicval.a . . 3 𝐴 = (Atomsβ€˜πΎ)
3 dicval.h . . 3 𝐻 = (LHypβ€˜πΎ)
4 dicval.p . . 3 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
5 dicval.t . . 3 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
6 dicval.e . . 3 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
7 dicval.i . . 3 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
81, 2, 3, 4, 5, 6, 7dicelvalN 40353 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (π‘Œ ∈ (πΌβ€˜π‘„) ↔ (π‘Œ ∈ (V Γ— V) ∧ ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ (2nd β€˜π‘Œ) ∈ 𝐸))))
9 dicval2.g . . . . . 6 𝐺 = (℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)
109fveq2i 6894 . . . . 5 ((2nd β€˜π‘Œ)β€˜πΊ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄))
1110eqeq2i 2744 . . . 4 ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜πΊ) ↔ (1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)))
1211anbi1i 623 . . 3 (((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜πΊ) ∧ (2nd β€˜π‘Œ) ∈ 𝐸) ↔ ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ (2nd β€˜π‘Œ) ∈ 𝐸))
1312anbi2i 622 . 2 ((π‘Œ ∈ (V Γ— V) ∧ ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜πΊ) ∧ (2nd β€˜π‘Œ) ∈ 𝐸)) ↔ (π‘Œ ∈ (V Γ— V) ∧ ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ (2nd β€˜π‘Œ) ∈ 𝐸)))
148, 13bitr4di 289 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (π‘Œ ∈ (πΌβ€˜π‘„) ↔ (π‘Œ ∈ (V Γ— V) ∧ ((1st β€˜π‘Œ) = ((2nd β€˜π‘Œ)β€˜πΊ) ∧ (2nd β€˜π‘Œ) ∈ 𝐸))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473   class class class wbr 5148   Γ— cxp 5674  β€˜cfv 6543  β„©crio 7367  1st c1st 7977  2nd c2nd 7978  lecple 17209  occoc 17210  Atomscatm 38437  LHypclh 39159  LTrncltrn 39276  TEndoctendo 39927  DIsoCcdic 40347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7368  df-1st 7979  df-2nd 7980  df-dic 40348
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator