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Theorem lautle 39551
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 39550 . . 3 (𝐾𝑉 → (𝐹𝐼 ↔ (𝐹:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))))
54simplbda 499 . 2 ((𝐾𝑉𝐹𝐼) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))
6 breq1 5145 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
7 fveq2 6891 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
87breq1d 5152 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑦)))
96, 8bibi12d 345 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)) ↔ (𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦))))
10 breq2 5146 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
11 fveq2 6891 . . . . 5 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1211breq2d 5154 . . . 4 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑌)))
1310, 12bibi12d 345 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦)) ↔ (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌))))
149, 13rspc2v 3619 . 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 205  wa 395   = wceq 1534  wcel 2099  wral 3057   class class class wbr 5142  1-1-ontowf1o 6541  cfv 6542  Basecbs 17173  lecple 17233  LAutclaut 39452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-laut 39456
This theorem is referenced by:  lautcnvle  39556  lautlt  39558  lautj  39560  lautm  39561  lauteq  39562  lautco  39564  ltrnle  39596
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