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Theorem lautlt 38105
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 1193 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
3 simpr2 1194 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
4 simpr3 1195 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
5 lautlt.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2738 . . . . 5 (le‘𝐾) = (le‘𝐾)
7 lautlt.i . . . . 5 𝐼 = (LAut‘𝐾)
85, 6, 7lautle 38098 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
91, 2, 3, 4, 8syl22anc 836 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
105, 7laut11 38100 . . . . . 6 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
111, 2, 3, 4, 10syl22anc 836 . . . . 5 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
1211bicomd 222 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 = 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
1312necon3bid 2988 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝑌 ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
149, 13anbi12d 631 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋(le‘𝐾)𝑌𝑋𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
15 lautlt.s . . . 4 < = (lt‘𝐾)
166, 15pltval 18050 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
17163adant3r1 1181 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
185, 7lautcl 38101 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
191, 2, 3, 18syl21anc 835 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
205, 7lautcl 38101 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
211, 2, 4, 20syl21anc 835 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
226, 15pltval 18050 . . 3 ((𝐾𝐴 ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) < (𝐹𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
231, 19, 21, 22syl3anc 1370 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) < (𝐹𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
2414, 17, 233bitr4d 311 1 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝐹𝑋) < (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943   class class class wbr 5074  cfv 6433  Basecbs 16912  lecple 16969  ltcplt 18026  LAutclaut 37999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-plt 18048  df-laut 38003
This theorem is referenced by:  lautcvr  38106
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