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Theorem lauteq 38135
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 1189 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝐾 ∈ HL)
2 simpl2 1190 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝐹𝐼)
3 lauteq.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
4 lauteq.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
53, 4atbase 37329 . . . . . . . . 9 (𝑝𝐴𝑝𝐵)
65adantl 481 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝑝𝐵)
7 simpl3 1191 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → 𝑋𝐵)
8 eqid 2733 . . . . . . . . 9 (le‘𝐾) = (le‘𝐾)
9 lauteq.i . . . . . . . . 9 𝐼 = (LAut‘𝐾)
103, 8, 9lautle 38124 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝐹𝐼) ∧ (𝑝𝐵𝑋𝐵)) → (𝑝(le‘𝐾)𝑋 ↔ (𝐹𝑝)(le‘𝐾)(𝐹𝑋)))
111, 2, 6, 7, 10syl22anc 835 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → (𝑝(le‘𝐾)𝑋 ↔ (𝐹𝑝)(le‘𝐾)(𝐹𝑋)))
12 breq1 5080 . . . . . . 7 ((𝐹𝑝) = 𝑝 → ((𝐹𝑝)(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)(𝐹𝑋)))
1311, 12sylan9bb 509 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) ∧ (𝐹𝑝) = 𝑝) → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝐹𝑋)))
1413bicomd 222 . . . . 5 ((((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) ∧ (𝐹𝑝) = 𝑝) → (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋))
1514ex 412 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ 𝑝𝐴) → ((𝐹𝑝) = 𝑝 → (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋)))
1615ralimdva 3158 . . 3 ((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) → (∀𝑝𝐴 (𝐹𝑝) = 𝑝 → ∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋)))
1716imp 406 . 2 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → ∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋))
18 simpl1 1189 . . 3 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → 𝐾 ∈ HL)
19 simpl2 1190 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → 𝐹𝐼)
20 simpl3 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → 𝑋𝐵)
213, 9lautcl 38127 . . . 4 (((𝐾 ∈ HL ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
2218, 19, 20, 21syl21anc 834 . . 3 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (𝐹𝑋) ∈ 𝐵)
233, 8, 4hlateq 37439 . . 3 ((𝐾 ∈ HL ∧ (𝐹𝑋) ∈ 𝐵𝑋𝐵) → (∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋) ↔ (𝐹𝑋) = 𝑋))
2418, 22, 20, 23syl3anc 1369 . 2 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (∀𝑝𝐴 (𝑝(le‘𝐾)(𝐹𝑋) ↔ 𝑝(le‘𝐾)𝑋) ↔ (𝐹𝑋) = 𝑋))
2517, 24mpbid 231 1 (((𝐾 ∈ HL ∧ 𝐹𝐼𝑋𝐵) ∧ ∀𝑝𝐴 (𝐹𝑝) = 𝑝) → (𝐹𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1537  wcel 2101  wral 3059   class class class wbr 5077  cfv 6447  Basecbs 16940  lecple 16997  Atomscatm 37303  HLchlt 37390  LAutclaut 38025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-riota 7252  df-ov 7298  df-oprab 7299  df-mpo 7300  df-map 8637  df-proset 18041  df-poset 18059  df-plt 18076  df-lub 18092  df-glb 18093  df-join 18094  df-meet 18095  df-p0 18171  df-lat 18178  df-clat 18245  df-oposet 37216  df-ol 37218  df-oml 37219  df-covers 37306  df-ats 37307  df-atl 37338  df-cvlat 37362  df-hlat 37391  df-laut 38029
This theorem is referenced by:  ltrnid  38175
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