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Theorem lautle 39557
Description: Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
Hypotheses
Ref Expression
lautset.b 𝐡 = (Baseβ€˜πΎ)
lautset.l ≀ = (leβ€˜πΎ)
lautset.i 𝐼 = (LAutβ€˜πΎ)
Assertion
Ref Expression
lautle (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋 ≀ π‘Œ ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘Œ)))

Proof of Theorem lautle
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lautset.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 lautset.l . . . 4 ≀ = (leβ€˜πΎ)
3 lautset.i . . . 4 𝐼 = (LAutβ€˜πΎ)
41, 2, 3islaut 39556 . . 3 (𝐾 ∈ 𝑉 β†’ (𝐹 ∈ 𝐼 ↔ (𝐹:𝐡–1-1-onto→𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))))
54simplbda 499 . 2 ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))
6 breq1 5151 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ 𝑋 ≀ 𝑦))
7 fveq2 6897 . . . . 5 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
87breq1d 5158 . . . 4 (π‘₯ = 𝑋 β†’ ((πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦) ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘¦)))
96, 8bibi12d 345 . . 3 (π‘₯ = 𝑋 β†’ ((π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)) ↔ (𝑋 ≀ 𝑦 ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘¦))))
10 breq2 5152 . . . 4 (𝑦 = π‘Œ β†’ (𝑋 ≀ 𝑦 ↔ 𝑋 ≀ π‘Œ))
11 fveq2 6897 . . . . 5 (𝑦 = π‘Œ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘Œ))
1211breq2d 5160 . . . 4 (𝑦 = π‘Œ β†’ ((πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘¦) ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘Œ)))
1310, 12bibi12d 345 . . 3 (𝑦 = π‘Œ β†’ ((𝑋 ≀ 𝑦 ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘¦)) ↔ (𝑋 ≀ π‘Œ ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘Œ))))
149, 13rspc2v 3620 . 2 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)) β†’ (𝑋 ≀ π‘Œ ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘Œ))))
155, 14mpan9 506 1 (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋 ≀ π‘Œ ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   class class class wbr 5148  β€“1-1-ontoβ†’wf1o 6547  β€˜cfv 6548  Basecbs 17180  lecple 17240  LAutclaut 39458
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
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 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-laut 39462
This theorem is referenced by:  lautcnvle  39562  lautlt  39564  lautj  39566  lautm  39567  lauteq  39568  lautco  39570  ltrnle  39602
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