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Theorem lauteq 34399
 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 1057 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ HL)
2 simpl2 1058 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝐹𝐼)
3 lauteq.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
4 lauteq.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
53, 4atbase 33594 . . . . . . . . 9 (𝑝𝐴𝑝𝐵)
65adantl 481 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
7 simpl3 1059 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
8 eqid 2610 . . . . . . . . 9 (le‘𝐾) = (le‘𝐾)
9 lauteq.i . . . . . . . . 9 𝐼 = (LAut‘𝐾)
103, 8, 9lautle 34388 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼) ∧ (𝑝𝐵𝑋𝐵)) → (𝑝(le‘𝐾)𝑋 ↔ (𝐹𝑝)(le‘𝐾)(𝐹𝑋)))
111, 2, 6, 7, 10syl22anc 1319 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → (𝑝(le‘𝐾)𝑋 ↔ (𝐹𝑝)(le‘𝐾)(𝐹𝑋)))
12 breq1 4586 . . . . . . 7 ((𝐹𝑝) = 𝑝 → ((𝐹𝑝)(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)(𝐹𝑋)))
1311, 12sylan9bb 732 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) ∧ (𝐹𝑝) = 𝑝) → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝐹𝑋)))
1413bicomd 212 . . . . 5 ((((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) ∧ (𝐹𝑝) = 𝑝) → (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋))
1514ex 449 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → ((𝐹𝑝) = 𝑝 → (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋)))
1615ralimdva 2945 . . 3 ((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) → (∀𝑝𝐴 (𝐹𝑝) = 𝑝 → ∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋)))
1716imp 444 . 2 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → ∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋))
18 simpl1 1057 . . 3 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → 𝐾 ∈ HL)
19 simpl2 1058 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → 𝐹𝐼)
20 simpl3 1059 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → 𝑋𝐵)
213, 9lautcl 34391 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
2218, 19, 20, 21syl21anc 1317 . . 3 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (𝐹𝑋) ∈ 𝐵)
233, 8, 4hlateq 33703 . . 3 ((𝐾 ∈ HL ∧ (𝐹𝑋) ∈ 𝐵𝑋𝐵) → (∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋) ↔ (𝐹𝑋) = 𝑋))
2418, 22, 20, 23syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋) ↔ (𝐹𝑋) = 𝑋))
2517, 24mpbid 221 1 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (𝐹𝑋) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  Atomscatm 33568  HLchlt 33655  LAutclaut 34289 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-laut 34293 This theorem is referenced by:  ltrnid  34439
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