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Theorem lautlt 37761
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 486 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾𝐴)
2 simpr1 1195 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
3 simpr2 1196 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
4 simpr3 1197 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
5 lautlt.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2739 . . . . 5 (le‘𝐾) = (le‘𝐾)
7 lautlt.i . . . . 5 𝐼 = (LAut‘𝐾)
85, 6, 7lautle 37754 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
91, 2, 3, 4, 8syl22anc 838 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
105, 7laut11 37756 . . . . . 6 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
111, 2, 3, 4, 10syl22anc 838 . . . . 5 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
1211bicomd 226 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 = 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
1312necon3bid 2979 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝑌 ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
149, 13anbi12d 634 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋(le‘𝐾)𝑌𝑋𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
15 lautlt.s . . . 4 < = (lt‘𝐾)
166, 15pltval 17699 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
17163adant3r1 1183 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
185, 7lautcl 37757 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
191, 2, 3, 18syl21anc 837 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
205, 7lautcl 37757 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
211, 2, 4, 20syl21anc 837 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
226, 15pltval 17699 . . 3 ((𝐾𝐴 ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) < (𝐹𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
231, 19, 21, 22syl3anc 1372 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) < (𝐹𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
2414, 17, 233bitr4d 314 1 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝐹𝑋) < (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2935   class class class wbr 5040  cfv 6350  Basecbs 16599  lecple 16688  ltcplt 17680  LAutclaut 37655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-map 8452  df-plt 17697  df-laut 37659
This theorem is referenced by:  lautcvr  37762
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