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Theorem lautm 34881
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 2621 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 473 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾 ∈ Lat)
4 simpr1 1065 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
53, 4jca 554 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐾 ∈ Lat ∧ 𝐹𝐼))
6 lautm.m . . . . 5 = (meet‘𝐾)
71, 6latmcl 16976 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1271 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautm.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 34874 . . 3 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
115, 8, 10syl2anc 692 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
12 simpr2 1066 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
131, 9lautcl 34874 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
145, 12, 13syl2anc 692 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
15 simpr3 1067 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
161, 9lautcl 34874 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 692 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latmcl 16976 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1323 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 2, 6latmle1 17000 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑋)
21203adant3r1 1271 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑋)
221, 2, 9lautle 34871 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑋𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑋 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑋)))
235, 8, 12, 22syl12anc 1321 . . . 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 17001 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
26253adant3r1 1271 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)𝑌)
271, 2, 9lautle 34871 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝑋 𝑌) ∈ 𝐵𝑌𝐵)) → ((𝑋 𝑌)(le‘𝐾)𝑌 ↔ (𝐹‘(𝑋 𝑌))(le‘𝐾)(𝐹𝑌)))
285, 8, 15, 27syl12anc 1321 . . . 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 17002 . . . 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 1325 . . 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 955 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
331, 9laut1o 34872 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
34333ad2antr1 1224 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
35 f1ocnvfv2 6490 . . . 4 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
3634, 19, 35syl2anc 692 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
371, 2, 6latmle1 17000 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
383, 14, 17, 37syl3anc 1323 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑋))
391, 2, 9lautcnvle 34876 . . . . . . . 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 1321 . . . . . . 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 6489 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑋𝐵) → (𝐹‘(𝐹𝑋)) = 𝑋)
4334, 12, 42syl2anc 692 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑋)) = 𝑋)
4441, 43breqtrd 4641 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑋)
451, 2, 6latmle2 17001 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
463, 14, 17, 45syl3anc 1323 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹𝑌))
471, 2, 9lautcnvle 34876 . . . . . . . 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 1321 . . . . . . 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 6489 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵𝑌𝐵) → (𝐹‘(𝐹𝑌)) = 𝑌)
5134, 15, 50syl2anc 692 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹𝑌)) = 𝑌)
5249, 51breqtrd 4641 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)𝑌)
53 f1ocnvdm 6497 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
5434, 19, 53syl2anc 692 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
551, 2, 6latlem12 17002 . . . . . 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 1325 . . . . 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 955 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌)))(le‘𝐾)(𝑋 𝑌))
581, 2, 9lautle 34871 . . . . 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 1321 . . . 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 4639 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
621, 2, 3, 11, 19, 32, 61latasymd 16981 1 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4615  ccnv 5075  1-1-ontowf1o 5848  cfv 5849  (class class class)co 6607  Basecbs 15784  lecple 15872  meetcmee 16869  Latclat 16969  LAutclaut 34772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-map 7807  df-preset 16852  df-poset 16870  df-lub 16898  df-glb 16899  df-join 16900  df-meet 16901  df-lat 16970  df-laut 34776
This theorem is referenced by:  ltrnm  34918
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