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Theorem lautle 37100
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 37099 . . 3 (𝐾𝑉 → (𝐹𝐼 ↔ (𝐹:𝐵1-1-onto𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))))
54simplbda 500 . 2 ((𝐾𝑉𝐹𝐼) → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)))
6 breq1 5060 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
7 fveq2 6663 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
87breq1d 5067 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑦)))
96, 8bibi12d 347 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)) ↔ (𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦))))
10 breq2 5061 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
11 fveq2 6663 . . . . 5 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1211breq2d 5069 . . . 4 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑌)))
1310, 12bibi12d 347 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦 ↔ (𝐹𝑋) (𝐹𝑦)) ↔ (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌))))
149, 13rspc2v 3630 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝐹𝑥) (𝐹𝑦)) → (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌))))
155, 14mpan9 507 1 (((𝐾𝑉𝐹𝐼) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 𝑌 ↔ (𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135   class class class wbr 5057  1-1-ontowf1o 6347  cfv 6348  Basecbs 16471  lecple 16560  LAutclaut 37001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-laut 37005
This theorem is referenced by:  lautcnvle  37105  lautlt  37107  lautj  37109  lautm  37110  lauteq  37111  lautco  37113  ltrnle  37145
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