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Theorem lautj 37719
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 2738 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 486 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾 ∈ Lat)
4 simpr1 1195 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
53, 4jca 515 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐾 ∈ Lat ∧ 𝐹𝐼))
6 lautj.j . . . . 5 = (join‘𝐾)
71, 6latjcl 17770 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1183 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautj.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 37713 . . 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 1196 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
131, 9lautcl 37713 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
145, 12, 13syl2anc 587 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
15 simpr3 1197 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
161, 9lautcl 37713 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 587 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latjcl 17770 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1372 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 9laut1o 37711 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
21203ad2antr1 1189 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
22 f1ocnvfv1 7038 . . . . 5 ((𝐹:𝐵1-1-onto𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
2321, 8, 22syl2anc 587 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
241, 2, 6latlej1 17779 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
253, 14, 17, 24syl3anc 1372 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
26 f1ocnvfv2 7039 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2721, 19, 26syl2anc 587 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2825, 27breqtrrd 5055 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
29 f1ocnvdm 7046 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
3021, 19, 29syl2anc 587 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
311, 2, 9lautle 37710 . . . . . . 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 836 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3328, 32mpbird 260 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
341, 2, 6latlej2 17780 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
353, 14, 17, 34syl3anc 1372 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
3635, 27breqtrrd 5055 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
371, 2, 9lautle 37710 . . . . . . 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 836 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3936, 38mpbird 260 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
401, 2, 6latjle12 17781 . . . . . 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 1373 . . . . 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 712 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
4323, 42eqbrtrd 5049 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
441, 2, 9lautcnvle 37715 . . . 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 836 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)) ↔ (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
4643, 45mpbird 260 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
471, 2, 6latlej1 17779 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
48473adant3r1 1183 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝑋 𝑌))
491, 2, 9lautle 37710 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
505, 12, 8, 49syl12anc 836 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5148, 50mpbid 235 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
521, 2, 6latlej2 17780 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
53523adant3r1 1183 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝑋 𝑌))
541, 2, 9lautle 37710 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
555, 15, 8, 54syl12anc 836 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5653, 55mpbid 235 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
571, 2, 6latjle12 17781 . . . 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 1373 . . 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 712 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
601, 2, 3, 11, 19, 46, 59latasymd 17776 1 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113   class class class wbr 5027  ccnv 5518  1-1-ontowf1o 6332  cfv 6333  (class class class)co 7164  Basecbs 16579  lecple 16668  joincjn 17663  Latclat 17764  LAutclaut 37611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-map 8432  df-proset 17647  df-poset 17665  df-lub 17693  df-glb 17694  df-join 17695  df-meet 17696  df-lat 17765  df-laut 37615
This theorem is referenced by:  ltrnj  37758
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