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Theorem lautlt 34854
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 473 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾𝐴)
2 simpr1 1065 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
3 simpr2 1066 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
4 simpr3 1067 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
5 lautlt.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2621 . . . . 5 (le‘𝐾) = (le‘𝐾)
7 lautlt.i . . . . 5 𝐼 = (LAut‘𝐾)
85, 6, 7lautle 34847 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
91, 2, 3, 4, 8syl22anc 1324 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
105, 7laut11 34849 . . . . . 6 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
111, 2, 3, 4, 10syl22anc 1324 . . . . 5 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
1211bicomd 213 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 = 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
1312necon3bid 2834 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝑌 ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
149, 13anbi12d 746 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋(le‘𝐾)𝑌𝑋𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
15 lautlt.s . . . 4 < = (lt‘𝐾)
166, 15pltval 16881 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
17163adant3r1 1271 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
185, 7lautcl 34850 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
191, 2, 3, 18syl21anc 1322 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
205, 7lautcl 34850 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
211, 2, 4, 20syl21anc 1322 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
226, 15pltval 16881 . . 3 ((𝐾𝐴 ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) < (𝐹𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
231, 19, 21, 22syl3anc 1323 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) < (𝐹𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
2414, 17, 233bitr4d 300 1 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝐹𝑋) < (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869  ltcplt 16862  LAutclaut 34748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-plt 16879  df-laut 34752
This theorem is referenced by:  lautcvr  34855
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