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Theorem lautj 38556
Description: Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)
Hypotheses
Ref Expression
lautj.b 𝐵 = (Base‘𝐾)
lautj.j = (join‘𝐾)
lautj.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautj ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))

Proof of Theorem lautj
StepHypRef Expression
1 lautj.b . 2 𝐵 = (Base‘𝐾)
2 eqid 2736 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 483 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾 ∈ Lat)
4 simpr1 1194 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
53, 4jca 512 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐾 ∈ Lat ∧ 𝐹𝐼))
6 lautj.j . . . . 5 = (join‘𝐾)
71, 6latjcl 18328 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1182 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautj.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 38550 . . 3 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
115, 8, 10syl2anc 584 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
12 simpr2 1195 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
131, 9lautcl 38550 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
145, 12, 13syl2anc 584 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
15 simpr3 1196 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
161, 9lautcl 38550 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 584 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latjcl 18328 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1371 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 9laut1o 38548 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
21203ad2antr1 1188 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
22 f1ocnvfv1 7222 . . . . 5 ((𝐹:𝐵1-1-onto𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
2321, 8, 22syl2anc 584 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
241, 2, 6latlej1 18337 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
253, 14, 17, 24syl3anc 1371 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
26 f1ocnvfv2 7223 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2721, 19, 26syl2anc 584 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2825, 27breqtrrd 5133 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
29 f1ocnvdm 7231 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
3021, 19, 29syl2anc 584 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
311, 2, 9lautle 38547 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋𝐵 ∧ (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)) → (𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
325, 12, 30, 31syl12anc 835 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3328, 32mpbird 256 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
341, 2, 6latlej2 18338 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
353, 14, 17, 34syl3anc 1371 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
3635, 27breqtrrd 5133 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
371, 2, 9lautle 38547 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑌𝐵 ∧ (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)) → (𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
385, 15, 30, 37syl12anc 835 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3936, 38mpbird 256 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
401, 2, 6latjle12 18339 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵 ∧ (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)) → ((𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ∧ 𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))) ↔ (𝑋 𝑌)(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
413, 12, 15, 30, 40syl13anc 1372 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ∧ 𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))) ↔ (𝑋 𝑌)(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
4233, 39, 41mpbi2and 710 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
4323, 42eqbrtrd 5127 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
441, 2, 9lautcnvle 38552 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝐹‘(𝑋 𝑌)) ∈ 𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)) → ((𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)) ↔ (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
455, 11, 19, 44syl12anc 835 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)) ↔ (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
4643, 45mpbird 256 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
471, 2, 6latlej1 18337 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
48473adant3r1 1182 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝑋 𝑌))
491, 2, 9lautle 38547 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
505, 12, 8, 49syl12anc 835 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5148, 50mpbid 231 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
521, 2, 6latlej2 18338 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
53523adant3r1 1182 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝑋 𝑌))
541, 2, 9lautle 38547 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
555, 15, 8, 54syl12anc 835 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5653, 55mpbid 231 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
571, 2, 6latjle12 18339 . . . 4 ((𝐾 ∈ Lat ∧ ((𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵 ∧ (𝐹‘(𝑋 𝑌)) ∈ 𝐵)) → (((𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌)) ∧ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))) ↔ ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌))))
583, 14, 17, 11, 57syl13anc 1372 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌)) ∧ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))) ↔ ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5951, 56, 58mpbi2and 710 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
601, 2, 3, 11, 19, 46, 59latasymd 18334 1 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5105  ccnv 5632  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  Latclat 18320  LAutclaut 38448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-proset 18184  df-poset 18202  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-lat 18321  df-laut 38452
This theorem is referenced by:  ltrnj  38595
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