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Theorem lautlt 37107
Description: Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
lautlt.b 𝐵 = (Base‘𝐾)
lautlt.s < = (lt‘𝐾)
lautlt.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautlt ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝐹𝑋) < (𝐹𝑌)))

Proof of Theorem lautlt
StepHypRef Expression
1 simpl 483 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾𝐴)
2 simpr1 1186 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
3 simpr2 1187 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
4 simpr3 1188 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
5 lautlt.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2818 . . . . 5 (le‘𝐾) = (le‘𝐾)
7 lautlt.i . . . . 5 𝐼 = (LAut‘𝐾)
85, 6, 7lautle 37100 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
91, 2, 3, 4, 8syl22anc 834 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
105, 7laut11 37102 . . . . . 6 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
111, 2, 3, 4, 10syl22anc 834 . . . . 5 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
1211bicomd 224 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 = 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
1312necon3bid 3057 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝑌 ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
149, 13anbi12d 630 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋(le‘𝐾)𝑌𝑋𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
15 lautlt.s . . . 4 < = (lt‘𝐾)
166, 15pltval 17558 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
17163adant3r1 1174 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
185, 7lautcl 37103 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
191, 2, 3, 18syl21anc 833 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
205, 7lautcl 37103 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
211, 2, 4, 20syl21anc 833 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
226, 15pltval 17558 . . 3 ((𝐾𝐴 ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) < (𝐹𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
231, 19, 21, 22syl3anc 1363 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) < (𝐹𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
2414, 17, 233bitr4d 312 1 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝐹𝑋) < (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  ltcplt 17539  LAutclaut 37001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-plt 17556  df-laut 37005
This theorem is referenced by:  lautcvr  37108
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