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Theorem lautm 35902
Description: Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
Hypotheses
Ref Expression
lautm.b 𝐵 = (Base‘𝐾)
lautm.m = (meet‘𝐾)
lautm.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautm ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))

Proof of Theorem lautm
StepHypRef Expression
1 lautm.b . 2 𝐵 = (Base‘𝐾)
2 eqid 2771 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 468 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾 ∈ Lat)
4 simpr1 1233 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
53, 4jca 501 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐾 ∈ Lat ∧ 𝐹𝐼))
6 lautm.m . . . . 5 = (meet‘𝐾)
71, 6latmcl 17260 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1197 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautm.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 35895 . . 3 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
115, 8, 10syl2anc 573 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
12 simpr2 1235 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
131, 9lautcl 35895 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
145, 12, 13syl2anc 573 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
15 simpr3 1237 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
161, 9lautcl 35895 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 573 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latmcl 17260 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1476 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 2, 6latmle1 17284 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
21203adant3r1 1197 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑋)
221, 2, 9lautle 35892 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋)))
235, 8, 12, 22syl12anc 1474 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋)))
2421, 23mpbid 222 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋))
251, 2, 6latmle2 17285 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
26253adant3r1 1197 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑌)
271, 2, 9lautle 35892 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)))
285, 8, 15, 27syl12anc 1474 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)))
2926, 28mpbid 222 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌))
301, 2, 6latlem12 17286 . . . 4 ((𝐾 ∈ Lat ∧ ((𝐹‘(𝑋 𝑌)) ∈ 𝐵 ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵)) → (((𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋) ∧ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)) ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌))))
313, 11, 14, 17, 30syl13anc 1478 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋) ∧ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)) ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌))))
3224, 29, 31mpbi2and 691 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
331, 9laut1o 35893 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
34333ad2antr1 1203 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
35 f1ocnvfv2 6676 . . . 4 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
3634, 19, 35syl2anc 573 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
371, 2, 6latmle1 17284 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
383, 14, 17, 37syl3anc 1476 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
391, 2, 9lautcnvle 35897 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵 ∧ (𝐹𝑋) ∈ 𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑋))))
405, 19, 14, 39syl12anc 1474 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑋))))
4138, 40mpbid 222 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑋)))
42 f1ocnvfv1 6675 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑋𝐵) → (𝐹‘(𝐹𝑋)) = 𝑋)
4334, 12, 42syl2anc 573 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑋)) = 𝑋)
4441, 43breqtrd 4812 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑋)
451, 2, 6latmle2 17285 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
463, 14, 17, 45syl3anc 1476 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
471, 2, 9lautcnvle 35897 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑌))))
485, 19, 17, 47syl12anc 1474 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑌))))
4946, 48mpbid 222 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑌)))
50 f1ocnvfv1 6675 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑌𝐵) → (𝐹‘(𝐹𝑌)) = 𝑌)
5134, 15, 50syl2anc 573 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑌)) = 𝑌)
5249, 51breqtrd 4812 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑌)
53 f1ocnvdm 6683 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
5434, 19, 53syl2anc 573 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
551, 2, 6latlem12 17286 . . . . . 6 ((𝐾 ∈ Lat ∧ ((𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑋 ∧ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑌) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌)))
563, 54, 12, 15, 55syl13anc 1478 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑋 ∧ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑌) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌)))
5744, 52, 56mpbi2and 691 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌))
581, 2, 9lautle 35892 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → ((𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌) ↔ (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))(le‘𝐾)(𝐹‘(𝑋 𝑌))))
595, 54, 8, 58syl12anc 1474 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌) ↔ (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))(le‘𝐾)(𝐹‘(𝑋 𝑌))))
6057, 59mpbid 222 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
6136, 60eqbrtrrd 4810 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
621, 2, 3, 11, 19, 32, 61latasymd 17265 1 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  ccnv 5248  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  meetcmee 17153  Latclat 17253  LAutclaut 35793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-preset 17136  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254  df-laut 35797
This theorem is referenced by:  ltrnm  35939
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