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Theorem lautj 36049
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 2765 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 474 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐾 ∈ Lat)
4 simpr1 1248 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹𝐼)
53, 4jca 507 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐾 ∈ Lat ∧ 𝐹𝐼))
6 lautj.j . . . . 5 = (join‘𝐾)
71, 6latjcl 17317 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
873adant3r1 1233 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌) ∈ 𝐵)
9 lautj.i . . . 4 𝐼 = (LAut‘𝐾)
101, 9lautcl 36043 . . 3 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
115, 8, 10syl2anc 579 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) ∈ 𝐵)
12 simpr2 1250 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
131, 9lautcl 36043 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)
145, 12, 13syl2anc 579 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝐵)
15 simpr3 1252 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
161, 9lautcl 36043 . . . 4 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ 𝑌𝐵) → (𝐹𝑌) ∈ 𝐵)
175, 15, 16syl2anc 579 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝐵)
181, 6latjcl 17317 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
193, 14, 17, 18syl3anc 1490 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵)
201, 9laut1o 36041 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝐹𝐼) → 𝐹:𝐵1-1-onto𝐵)
21203ad2antr1 1239 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝐹:𝐵1-1-onto𝐵)
22 f1ocnvfv1 6724 . . . . 5 ((𝐹:𝐵1-1-onto𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
2321, 8, 22syl2anc 579 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌))) = (𝑋 𝑌))
241, 2, 6latlej1 17326 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
253, 14, 17, 24syl3anc 1490 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
26 f1ocnvfv2 6725 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2721, 19, 26syl2anc 579 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))) = ((𝐹𝑋) (𝐹𝑌)))
2825, 27breqtrrd 4837 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
29 f1ocnvdm 6732 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐵 ∧ ((𝐹𝑋) (𝐹𝑌)) ∈ 𝐵) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
3021, 19, 29syl2anc 579 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘((𝐹𝑋) (𝐹𝑌))) ∈ 𝐵)
311, 2, 9lautle 36040 . . . . . . 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 865 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3328, 32mpbird 248 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
341, 2, 6latlej2 17327 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝐹𝑋) ∈ 𝐵 ∧ (𝐹𝑌) ∈ 𝐵) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
353, 14, 17, 34syl3anc 1490 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
3635, 27breqtrrd 4837 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
371, 2, 9lautle 36040 . . . . . . 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 865 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝐹‘((𝐹𝑋) (𝐹𝑌))))))
3936, 38mpbird 248 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
401, 2, 6latjle12 17328 . . . . . 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 1491 . . . . 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 703 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋 𝑌)(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
4323, 42eqbrtrd 4831 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌))))
441, 2, 9lautcnvle 36045 . . . 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 865 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)) ↔ (𝐹‘(𝐹‘(𝑋 𝑌)))(le‘𝐾)(𝐹‘((𝐹𝑋) (𝐹𝑌)))))
4643, 45mpbird 248 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌))(le‘𝐾)((𝐹𝑋) (𝐹𝑌)))
471, 2, 6latlej1 17326 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
48473adant3r1 1233 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑋(le‘𝐾)(𝑋 𝑌))
491, 2, 9lautle 36040 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
505, 12, 8, 49syl12anc 865 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑋(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5148, 50mpbid 223 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑋)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
521, 2, 6latlej2 17327 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
53523adant3r1 1233 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → 𝑌(le‘𝐾)(𝑋 𝑌))
541, 2, 9lautle 36040 . . . . 5 (((𝐾 ∈ Lat ∧ 𝐹𝐼) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
555, 15, 8, 54syl12anc 865 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝑌(le‘𝐾)(𝑋 𝑌) ↔ (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌))))
5653, 55mpbid 223 . . 3 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹𝑌)(le‘𝐾)(𝐹‘(𝑋 𝑌)))
571, 2, 6latjle12 17328 . . . 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 1491 . . 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 703 . 2 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → ((𝐹𝑋) (𝐹𝑌))(le‘𝐾)(𝐹‘(𝑋 𝑌)))
601, 2, 3, 11, 19, 46, 59latasymd 17323 1 ((𝐾 ∈ Lat ∧ (𝐹𝐼𝑋𝐵𝑌𝐵)) → (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155   class class class wbr 4809  ccnv 5276  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  Basecbs 16130  lecple 16221  joincjn 17210  Latclat 17311  LAutclaut 35941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-map 8062  df-proset 17194  df-poset 17212  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-lat 17312  df-laut 35945
This theorem is referenced by:  ltrnj  36088
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