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Theorem lautle 40066
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 40065 . . 3 (𝐾𝑉 → (𝐹𝐼 ↔ (𝐹:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))))
54simplbda 499 . 2 ((𝐾𝑉𝐹𝐼) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))
6 breq1 5150 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
7 fveq2 6906 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
87breq1d 5157 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑦)))
96, 8bibi12d 345 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)) ↔ (𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦))))
10 breq2 5151 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
11 fveq2 6906 . . . . 5 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1211breq2d 5159 . . . 4 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑌)))
1310, 12bibi12d 345 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦)) ↔ (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌))))
149, 13rspc2v 3632 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)) → (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌))))
155, 14mpan9 506 1 (((𝐾𝑉𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058   class class class wbr 5147  1-1-ontowf1o 6561  cfv 6562  Basecbs 17244  lecple 17304  LAutclaut 39967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-laut 39971
This theorem is referenced by:  lautcnvle  40071  lautlt  40073  lautj  40075  lautm  40076  lauteq  40077  lautco  40079  ltrnle  40111
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