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Theorem lautle 40086
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 40085 . . 3 (𝐾𝑉 → (𝐹𝐼 ↔ (𝐹:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))))
54simplbda 499 . 2 ((𝐾𝑉𝐹𝐼) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))
6 breq1 5146 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
7 fveq2 6906 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
87breq1d 5153 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑦)))
96, 8bibi12d 345 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)) ↔ (𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦))))
10 breq2 5147 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
11 fveq2 6906 . . . . 5 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1211breq2d 5155 . . . 4 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑌)))
1310, 12bibi12d 345 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦)) ↔ (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌))))
149, 13rspc2v 3633 . 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 1540  wcel 2108  wral 3061   class class class wbr 5143  1-1-ontowf1o 6560  cfv 6561  Basecbs 17247  lecple 17304  LAutclaut 39987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-laut 39991
This theorem is referenced by:  lautcnvle  40091  lautlt  40093  lautj  40095  lautm  40096  lauteq  40097  lautco  40099  ltrnle  40131
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