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Theorem lautm 34398
 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 2610 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 472 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾 ∈ Lat)
4 simpr1 1060 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
53, 4jca 553 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐾 ∈ Lat ∧ 𝐹𝐼))
6 lautm.m . . . . 5 = (meet‘𝐾)
71, 6latmcl 16875 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1266 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautm.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 34391 . . 3 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
115, 8, 10syl2anc 691 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
12 simpr2 1061 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
131, 9lautcl 34391 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
145, 12, 13syl2anc 691 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
15 simpr3 1062 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
161, 9lautcl 34391 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 691 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latmcl 16875 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1318 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 2, 6latmle1 16899 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
21203adant3r1 1266 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑋)
221, 2, 9lautle 34388 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋)))
235, 8, 12, 22syl12anc 1316 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋)))
2421, 23mpbid 221 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋))
251, 2, 6latmle2 16900 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
26253adant3r1 1266 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑌)
271, 2, 9lautle 34388 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)))
285, 8, 15, 27syl12anc 1316 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)))
2926, 28mpbid 221 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌))
301, 2, 6latlem12 16901 . . . 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 1320 . . 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 958 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
331, 9laut1o 34389 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
34333ad2antr1 1219 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
35 f1ocnvfv2 6433 . . . 4 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
3634, 19, 35syl2anc 691 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
371, 2, 6latmle1 16899 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
383, 14, 17, 37syl3anc 1318 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
391, 2, 9lautcnvle 34393 . . . . . . . 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 1316 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑋))))
4138, 40mpbid 221 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑋)))
42 f1ocnvfv1 6432 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑋𝐵) → (𝐹‘(𝐹𝑋)) = 𝑋)
4334, 12, 42syl2anc 691 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑋)) = 𝑋)
4441, 43breqtrd 4609 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑋)
451, 2, 6latmle2 16900 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
463, 14, 17, 45syl3anc 1318 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
471, 2, 9lautcnvle 34393 . . . . . . . 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 1316 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌) ↔ (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑌))))
4946, 48mpbid 221 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝐹‘(𝐹𝑌)))
50 f1ocnvfv1 6432 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑌𝐵) → (𝐹‘(𝐹𝑌)) = 𝑌)
5134, 15, 50syl2anc 691 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑌)) = 𝑌)
5249, 51breqtrd 4609 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑌)
53 f1ocnvdm 6440 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
5434, 19, 53syl2anc 691 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
551, 2, 6latlem12 16901 . . . . . 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 1320 . . . . 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 958 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌))
581, 2, 9lautle 34388 . . . . 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 1316 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌) ↔ (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))(le‘𝐾)(𝐹‘(𝑋 𝑌))))
6057, 59mpbid 221 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
6136, 60eqbrtrrd 4607 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
621, 2, 3, 11, 19, 32, 61latasymd 16880 1 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ◡ccnv 5037  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  meetcmee 16768  Latclat 16868  LAutclaut 34289 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-preset 16751  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-lat 16869  df-laut 34293 This theorem is referenced by:  ltrnm  34435
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