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Theorem lautle 35842
Description: Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
Hypotheses
Ref Expression
lautset.b 𝐵 = (Base‘𝐾)
lautset.l = (le‘𝐾)
lautset.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautle (((𝐾𝑉𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌)))

Proof of Theorem lautle
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lautset.b . . . 4 𝐵 = (Base‘𝐾)
2 lautset.l . . . 4 = (le‘𝐾)
3 lautset.i . . . 4 𝐼 = (LAut‘𝐾)
41, 2, 3islaut 35841 . . 3 (𝐾𝑉 → (𝐹𝐼 ↔ (𝐹:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))))
54simplbda 655 . 2 ((𝐾𝑉𝐹𝐼) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))
6 breq1 4795 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
7 fveq2 6340 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
87breq1d 4802 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑦)))
96, 8bibi12d 334 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)) ↔ (𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦))))
10 breq2 4796 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
11 fveq2 6340 . . . . 5 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1211breq2d 4804 . . . 4 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑌)))
1310, 12bibi12d 334 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦)) ↔ (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌))))
149, 13rspc2v 3449 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)) → (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌))))
155, 14mpan9 487 1 (((𝐾𝑉𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  wral 3038   class class class wbr 4792  1-1-ontowf1o 6036  cfv 6037  Basecbs 16030  lecple 16121  LAutclaut 35743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-map 8013  df-laut 35747
This theorem is referenced by:  lautcnvle  35847  lautlt  35849  lautj  35851  lautm  35852  lauteq  35853  lautco  35855  ltrnle  35887
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