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Theorem lautj 39458
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 2724 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 482 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾 ∈ Lat)
4 simpr1 1191 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
53, 4jca 511 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐾 ∈ Lat ∧ 𝐹𝐼))
6 lautj.j . . . . 5 = (join‘𝐾)
71, 6latjcl 18396 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1179 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautj.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 39452 . . 3 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
115, 8, 10syl2anc 583 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
12 simpr2 1192 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
131, 9lautcl 39452 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
145, 12, 13syl2anc 583 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
15 simpr3 1193 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
161, 9lautcl 39452 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 583 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latjcl 18396 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1368 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 9laut1o 39450 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
21203ad2antr1 1185 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
22 f1ocnvfv1 7267 . . . . 5 ((𝐹:𝐵1-1-onto𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
2321, 8, 22syl2anc 583 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
241, 2, 6latlej1 18405 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
253, 14, 17, 24syl3anc 1368 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
26 f1ocnvfv2 7268 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2721, 19, 26syl2anc 583 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2825, 27breqtrrd 5167 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
29 f1ocnvdm 7276 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
3021, 19, 29syl2anc 583 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
311, 2, 9lautle 39449 . . . . . . 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 834 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3328, 32mpbird 257 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
341, 2, 6latlej2 18406 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
353, 14, 17, 34syl3anc 1368 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
3635, 27breqtrrd 5167 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
371, 2, 9lautle 39449 . . . . . . 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 834 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3936, 38mpbird 257 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
401, 2, 6latjle12 18407 . . . . . 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 1369 . . . . 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 709 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
4323, 42eqbrtrd 5161 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
441, 2, 9lautcnvle 39454 . . . 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 834 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)) ↔ (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
4643, 45mpbird 257 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
471, 2, 6latlej1 18405 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
48473adant3r1 1179 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝑋 𝑌))
491, 2, 9lautle 39449 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
505, 12, 8, 49syl12anc 834 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5148, 50mpbid 231 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
521, 2, 6latlej2 18406 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
53523adant3r1 1179 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝑋 𝑌))
541, 2, 9lautle 39449 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
555, 15, 8, 54syl12anc 834 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5653, 55mpbid 231 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
571, 2, 6latjle12 18407 . . . 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 1369 . . 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 709 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
601, 2, 3, 11, 19, 46, 59latasymd 18402 1 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5139  ccnv 5666  1-1-ontowf1o 6533  cfv 6534  (class class class)co 7402  Basecbs 17145  lecple 17205  joincjn 18268  Latclat 18388  LAutclaut 39350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-proset 18252  df-poset 18270  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-lat 18389  df-laut 39354
This theorem is referenced by:  ltrnj  39497
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