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Theorem lautj 34894
Description: Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)
Hypotheses
Ref Expression
lautj.b 𝐵 = (Base‘𝐾)
lautj.j = (join‘𝐾)
lautj.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautj ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))

Proof of Theorem lautj
StepHypRef Expression
1 lautj.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 lautj.j . . . . 5 = (join‘𝐾)
71, 6latjcl 16983 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1271 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautj.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 34888 . . 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 34888 . . . 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 34888 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 692 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latjcl 16983 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1323 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 9laut1o 34886 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
21203ad2antr1 1224 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
22 f1ocnvfv1 6492 . . . . 5 ((𝐹:𝐵1-1-onto𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
2321, 8, 22syl2anc 692 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
241, 2, 6latlej1 16992 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
253, 14, 17, 24syl3anc 1323 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
26 f1ocnvfv2 6493 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2721, 19, 26syl2anc 692 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2825, 27breqtrrd 4646 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
29 f1ocnvdm 6500 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
3021, 19, 29syl2anc 692 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
311, 2, 9lautle 34885 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋𝐵 ∧ (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)) → (𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
325, 12, 30, 31syl12anc 1321 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3328, 32mpbird 247 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
341, 2, 6latlej2 16993 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
353, 14, 17, 34syl3anc 1323 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
3635, 27breqtrrd 4646 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
371, 2, 9lautle 34885 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑌𝐵 ∧ (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)) → (𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
385, 15, 30, 37syl12anc 1321 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3936, 38mpbird 247 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
401, 2, 6latjle12 16994 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵 ∧ (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)) → ((𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ∧ 𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))) ↔ (𝑋 𝑌)(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
413, 12, 15, 30, 40syl13anc 1325 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ∧ 𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))) ↔ (𝑋 𝑌)(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
4233, 39, 41mpbi2and 955 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
4323, 42eqbrtrd 4640 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
441, 2, 9lautcnvle 34890 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ ((𝐹‘(𝑋 𝑌)) ∈ 𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)) → ((𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)) ↔ (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
455, 11, 19, 44syl12anc 1321 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)) ↔ (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
4643, 45mpbird 247 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
471, 2, 6latlej1 16992 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
48473adant3r1 1271 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝑋 𝑌))
491, 2, 9lautle 34885 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
505, 12, 8, 49syl12anc 1321 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5148, 50mpbid 222 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
521, 2, 6latlej2 16993 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
53523adant3r1 1271 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝑋 𝑌))
541, 2, 9lautle 34885 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
555, 15, 8, 54syl12anc 1321 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5653, 55mpbid 222 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
571, 2, 6latjle12 16994 . . . 4 ((𝐾 ∈ Lat ∧ ((𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵 ∧ (𝐹‘(𝑋 𝑌)) ∈ 𝐵)) → (((𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌)) ∧ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))) ↔ ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌))))
583, 14, 17, 11, 57syl13anc 1325 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (((𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌)) ∧ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))) ↔ ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5951, 56, 58mpbi2and 955 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
601, 2, 3, 11, 19, 46, 59latasymd 16989 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 4618  ccnv 5078  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  joincjn 16876  Latclat 16977  LAutclaut 34786
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-preset 16860  df-poset 16878  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-lat 16978  df-laut 34790
This theorem is referenced by:  ltrnj  34933
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