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Theorem lautlt 35898
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 474 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾𝐴)
2 simpr1 1234 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
3 simpr2 1236 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
4 simpr3 1238 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
5 lautlt.b . . . . 5 𝐵 = (Base‘𝐾)
6 eqid 2760 . . . . 5 (le‘𝐾) = (le‘𝐾)
7 lautlt.i . . . . 5 𝐼 = (LAut‘𝐾)
85, 6, 7lautle 35891 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
91, 2, 3, 4, 8syl22anc 1478 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)𝑌 ↔ (𝐹𝑋)(le‘𝐾)(𝐹𝑌)))
105, 7laut11 35893 . . . . . 6 (((𝐾𝐴𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
111, 2, 3, 4, 10syl22anc 1478 . . . . 5 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) = (𝐹𝑌) ↔ 𝑋 = 𝑌))
1211bicomd 213 . . . 4 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 = 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
1312necon3bid 2976 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋𝑌 ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
149, 13anbi12d 749 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋(le‘𝐾)𝑌𝑋𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
15 lautlt.s . . . 4 < = (lt‘𝐾)
166, 15pltval 17181 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
17163adant3r1 1198 . 2 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
185, 7lautcl 35894 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
191, 2, 3, 18syl21anc 1476 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
205, 7lautcl 35894 . . . 4 (((𝐾𝐴𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
211, 2, 4, 20syl21anc 1476 . . 3 ((𝐾𝐴 ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
226, 15pltval 17181 . . 3 ((𝐾𝐴 ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) < (𝐹𝑌) ↔ ((𝐹𝑋)(le‘𝐾)(𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
231, 19, 21, 22syl3anc 1477 . 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 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  Basecbs 16079  lecple 16170  ltcplt 17162  LAutclaut 35792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-plt 17179  df-laut 35796
This theorem is referenced by:  lautcvr  35899
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