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Theorem lautm 35982
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 2765 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 474 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾 ∈ Lat)
4 simpr1 1248 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
53, 4jca 507 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐾 ∈ Lat ∧ 𝐹𝐼))
6 lautm.m . . . . 5 = (meet‘𝐾)
71, 6latmcl 17320 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1233 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautm.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 35975 . . 3 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
115, 8, 10syl2anc 579 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
12 simpr2 1250 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
131, 9lautcl 35975 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
145, 12, 13syl2anc 579 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
15 simpr3 1252 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
161, 9lautcl 35975 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 579 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latmcl 17320 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1490 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 2, 6latmle1 17344 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
21203adant3r1 1233 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑋)
221, 2, 9lautle 35972 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋)))
235, 8, 12, 22syl12anc 865 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋)))
2421, 23mpbid 223 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋))
251, 2, 6latmle2 17345 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
26253adant3r1 1233 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑌)
271, 2, 9lautle 35972 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)))
285, 8, 15, 27syl12anc 865 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)))
2926, 28mpbid 223 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌))
301, 2, 6latlem12 17346 . . . 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 1491 . . 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 703 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
331, 9laut1o 35973 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
34333ad2antr1 1239 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
35 f1ocnvfv2 6725 . . . 4 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
3634, 19, 35syl2anc 579 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
371, 2, 6latmle1 17344 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
383, 14, 17, 37syl3anc 1490 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
391, 2, 9lautcnvle 35977 . . . . . . . 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 865 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑋))))
4138, 40mpbid 223 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑋)))
42 f1ocnvfv1 6724 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑋𝐵) → (𝐹‘(𝐹𝑋)) = 𝑋)
4334, 12, 42syl2anc 579 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑋)) = 𝑋)
4441, 43breqtrd 4835 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑋)
451, 2, 6latmle2 17345 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
463, 14, 17, 45syl3anc 1490 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
471, 2, 9lautcnvle 35977 . . . . . . . 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 865 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑌))))
4946, 48mpbid 223 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑌)))
50 f1ocnvfv1 6724 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑌𝐵) → (𝐹‘(𝐹𝑌)) = 𝑌)
5134, 15, 50syl2anc 579 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑌)) = 𝑌)
5249, 51breqtrd 4835 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑌)
53 f1ocnvdm 6732 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
5434, 19, 53syl2anc 579 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
551, 2, 6latlem12 17346 . . . . . 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 1491 . . . . 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 703 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌))
581, 2, 9lautle 35972 . . . . 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 865 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌) ↔ (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))(le‘𝐾)(𝐹‘(𝑋 𝑌))))
6057, 59mpbid 223 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
6136, 60eqbrtrrd 4833 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
621, 2, 3, 11, 19, 32, 61latasymd 17325 1 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155   class class class wbr 4809  ccnv 5276  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  Basecbs 16132  lecple 16223  meetcmee 17213  Latclat 17313  LAutclaut 35873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-map 8062  df-proset 17196  df-poset 17214  df-lub 17242  df-glb 17243  df-join 17244  df-meet 17245  df-lat 17314  df-laut 35877
This theorem is referenced by:  ltrnm  36019
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