Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lauteq Structured version   Visualization version   GIF version

Theorem lauteq 34861
Description: A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)
Hypotheses
Ref Expression
lauteq.b 𝐵 = (Base‘𝐾)
lauteq.a 𝐴 = (Atoms‘𝐾)
lauteq.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lauteq (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (𝐹𝑋) = 𝑋)
Distinct variable groups:   𝐴,𝑝   𝐵,𝑝   𝐹,𝑝   𝐼,𝑝   𝐾,𝑝   𝑋,𝑝

Proof of Theorem lauteq
StepHypRef Expression
1 simpl1 1062 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ HL)
2 simpl2 1063 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝐹𝐼)
3 lauteq.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
4 lauteq.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
53, 4atbase 34056 . . . . . . . . 9 (𝑝𝐴𝑝𝐵)
65adantl 482 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
7 simpl3 1064 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
8 eqid 2621 . . . . . . . . 9 (le‘𝐾) = (le‘𝐾)
9 lauteq.i . . . . . . . . 9 𝐼 = (LAut‘𝐾)
103, 8, 9lautle 34850 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼) ∧ (𝑝𝐵𝑋𝐵)) → (𝑝(le‘𝐾)𝑋 ↔ (𝐹𝑝)(le‘𝐾)(𝐹𝑋)))
111, 2, 6, 7, 10syl22anc 1324 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → (𝑝(le‘𝐾)𝑋 ↔ (𝐹𝑝)(le‘𝐾)(𝐹𝑋)))
12 breq1 4616 . . . . . . 7 ((𝐹𝑝) = 𝑝 → ((𝐹𝑝)(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)(𝐹𝑋)))
1311, 12sylan9bb 735 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) ∧ (𝐹𝑝) = 𝑝) → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝐹𝑋)))
1413bicomd 213 . . . . 5 ((((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) ∧ (𝐹𝑝) = 𝑝) → (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋))
1514ex 450 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → ((𝐹𝑝) = 𝑝 → (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋)))
1615ralimdva 2956 . . 3 ((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) → (∀𝑝𝐴 (𝐹𝑝) = 𝑝 → ∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋)))
1716imp 445 . 2 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → ∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋))
18 simpl1 1062 . . 3 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → 𝐾 ∈ HL)
19 simpl2 1063 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → 𝐹𝐼)
20 simpl3 1064 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → 𝑋𝐵)
213, 9lautcl 34853 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
2218, 19, 20, 21syl21anc 1322 . . 3 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (𝐹𝑋) ∈ 𝐵)
233, 8, 4hlateq 34165 . . 3 ((𝐾 ∈ HL ∧ (𝐹𝑋) ∈ 𝐵𝑋𝐵) → (∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋) ↔ (𝐹𝑋) = 𝑋))
2418, 22, 20, 23syl3anc 1323 . 2 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋) ↔ (𝐹𝑋) = 𝑋))
2517, 24mpbid 222 1 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (𝐹𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869  Atomscatm 34030  HLchlt 34117  LAutclaut 34751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-lat 16967  df-clat 17029  df-oposet 33943  df-ol 33945  df-oml 33946  df-covers 34033  df-ats 34034  df-atl 34065  df-cvlat 34089  df-hlat 34118  df-laut 34755
This theorem is referenced by:  ltrnid  34901
  Copyright terms: Public domain W3C validator