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Theorem lautle 38576
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 38575 . . 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 501 . 2 ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))
6 breq1 5113 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ 𝑋 ≀ 𝑦))
7 fveq2 6847 . . . . 5 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
87breq1d 5120 . . . 4 (π‘₯ = 𝑋 β†’ ((πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦) ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘¦)))
96, 8bibi12d 346 . . 3 (π‘₯ = 𝑋 β†’ ((π‘₯ ≀ 𝑦 ↔ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)) ↔ (𝑋 ≀ 𝑦 ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘¦))))
10 breq2 5114 . . . 4 (𝑦 = π‘Œ β†’ (𝑋 ≀ 𝑦 ↔ 𝑋 ≀ π‘Œ))
11 fveq2 6847 . . . . 5 (𝑦 = π‘Œ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘Œ))
1211breq2d 5122 . . . 4 (𝑦 = π‘Œ β†’ ((πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘¦) ↔ (πΉβ€˜π‘‹) ≀ (πΉβ€˜π‘Œ)))
1310, 12bibi12d 346 . . 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 3593 . 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 508 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   class class class wbr 5110  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  Basecbs 17090  lecple 17147  LAutclaut 38477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-laut 38481
This theorem is referenced by:  lautcnvle  38581  lautlt  38583  lautj  38585  lautm  38586  lauteq  38587  lautco  38589  ltrnle  38621
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