Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lautm Structured version   Visualization version   GIF version

Theorem lautm 37845
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 2737 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 486 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾 ∈ Lat)
4 simpr1 1196 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
53, 4jca 515 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐾 ∈ Lat ∧ 𝐹𝐼))
6 lautm.m . . . . 5 = (meet‘𝐾)
71, 6latmcl 17946 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1184 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautm.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 37838 . . 3 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
115, 8, 10syl2anc 587 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
12 simpr2 1197 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
131, 9lautcl 37838 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
145, 12, 13syl2anc 587 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
15 simpr3 1198 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
161, 9lautcl 37838 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 587 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latmcl 17946 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1373 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 2, 6latmle1 17970 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
21203adant3r1 1184 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑋)
221, 2, 9lautle 37835 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋)))
235, 8, 12, 22syl12anc 837 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋)))
2421, 23mpbid 235 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋))
251, 2, 6latmle2 17971 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
26253adant3r1 1184 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑌)
271, 2, 9lautle 37835 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)))
285, 8, 15, 27syl12anc 837 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)))
2926, 28mpbid 235 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌))
301, 2, 6latlem12 17972 . . . 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 1374 . . 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 712 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
331, 9laut1o 37836 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
34333ad2antr1 1190 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
35 f1ocnvfv2 7088 . . . 4 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
3634, 19, 35syl2anc 587 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
371, 2, 6latmle1 17970 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
383, 14, 17, 37syl3anc 1373 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
391, 2, 9lautcnvle 37840 . . . . . . . 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 837 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑋))))
4138, 40mpbid 235 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑋)))
42 f1ocnvfv1 7087 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑋𝐵) → (𝐹‘(𝐹𝑋)) = 𝑋)
4334, 12, 42syl2anc 587 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑋)) = 𝑋)
4441, 43breqtrd 5079 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑋)
451, 2, 6latmle2 17971 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
463, 14, 17, 45syl3anc 1373 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
471, 2, 9lautcnvle 37840 . . . . . . . 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 837 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑌))))
4946, 48mpbid 235 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑌)))
50 f1ocnvfv1 7087 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑌𝐵) → (𝐹‘(𝐹𝑌)) = 𝑌)
5134, 15, 50syl2anc 587 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑌)) = 𝑌)
5249, 51breqtrd 5079 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑌)
53 f1ocnvdm 7095 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
5434, 19, 53syl2anc 587 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
551, 2, 6latlem12 17972 . . . . . 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 1374 . . . . 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 712 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌))
581, 2, 9lautle 37835 . . . . 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 837 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌) ↔ (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))(le‘𝐾)(𝐹‘(𝑋 𝑌))))
6057, 59mpbid 235 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
6136, 60eqbrtrrd 5077 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
621, 2, 3, 11, 19, 32, 61latasymd 17951 1 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110   class class class wbr 5053  ccnv 5550  1-1-ontowf1o 6379  cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  meetcmee 17819  Latclat 17937  LAutclaut 37736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-proset 17802  df-poset 17820  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-lat 17938  df-laut 37740
This theorem is referenced by:  ltrnm  37882
  Copyright terms: Public domain W3C validator