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Theorem cmtbr3N 33358
Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 27653 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtbr2.b 𝐵 = (Base‘𝐾)
cmtbr2.j = (join‘𝐾)
cmtbr2.m = (meet‘𝐾)
cmtbr2.o = (oc‘𝐾)
cmtbr2.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
cmtbr3N ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))

Proof of Theorem cmtbr3N
StepHypRef Expression
1 cmtbr2.b . . . . 5 𝐵 = (Base‘𝐾)
2 cmtbr2.c . . . . 5 𝐶 = (cm‘𝐾)
31, 2cmtcomN 33353 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))
4 cmtbr2.j . . . . . 6 = (join‘𝐾)
5 cmtbr2.m . . . . . 6 = (meet‘𝐾)
6 cmtbr2.o . . . . . 6 = (oc‘𝐾)
71, 4, 5, 6, 2cmtbr2N 33357 . . . . 5 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑋𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
873com23 1262 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
93, 8bitrd 266 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
10 oveq2 6531 . . . . . 6 (𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋))) → (𝑋 𝑌) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
1110adantl 480 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 𝑌) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
12 omlol 33344 . . . . . . . . 9 (𝐾 ∈ OML → 𝐾 ∈ OL)
13123ad2ant1 1074 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
14 simp2 1054 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
15 omllat 33346 . . . . . . . . . 10 (𝐾 ∈ OML → 𝐾 ∈ Lat)
16153ad2ant1 1074 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
17 simp3 1055 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
181, 4latjcl 16816 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
1916, 17, 14, 18syl3anc 1317 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) ∈ 𝐵)
20 omlop 33345 . . . . . . . . . . 11 (𝐾 ∈ OML → 𝐾 ∈ OP)
21203ad2ant1 1074 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
221, 6opoccl 33298 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
2321, 14, 22syl2anc 690 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
241, 4latjcl 16816 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
2516, 17, 23, 24syl3anc 1317 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
261, 5latmassOLD 33333 . . . . . . . 8 ((𝐾 ∈ OL ∧ (𝑋𝐵 ∧ (𝑌 𝑋) ∈ 𝐵 ∧ (𝑌 ( 𝑋)) ∈ 𝐵)) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
2713, 14, 19, 25, 26syl13anc 1319 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
281, 4latjcom 16824 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
2916, 17, 14, 28syl3anc 1317 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
3029oveq2d 6539 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = (𝑋 (𝑋 𝑌)))
311, 4, 5latabs2 16853 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3215, 31syl3an1 1350 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3330, 32eqtrd 2639 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = 𝑋)
341, 4latjcom 16824 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3516, 17, 23, 34syl3anc 1317 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3633, 35oveq12d 6541 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 (( 𝑋) 𝑌)))
3727, 36eqtr3d 2641 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))) = (𝑋 (( 𝑋) 𝑌)))
3837adantr 479 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))) = (𝑋 (( 𝑋) 𝑌)))
3911, 38eqtr2d 2640 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌))
4039ex 448 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋))) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
419, 40sylbid 228 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
42 simp1 1053 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
431, 6opoccl 33298 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
4421, 17, 43syl2anc 690 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
451, 5latmcl 16817 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4616, 14, 44, 45syl3anc 1317 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4742, 46, 143jca 1234 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ (𝑋 ( 𝑌)) ∈ 𝐵𝑋𝐵))
48 eqid 2605 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
491, 48, 5latmle1 16841 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
5016, 14, 44, 49syl3anc 1317 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
511, 48, 4, 5, 6omllaw2N 33348 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋 ( 𝑌)) ∈ 𝐵𝑋𝐵) → ((𝑋 ( 𝑌))(le‘𝐾)𝑋 → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = 𝑋))
5247, 50, 51sylc 62 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = 𝑋)
531, 6opoccl 33298 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
5421, 46, 53syl2anc 690 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
551, 5latmcl 16817 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵𝑋𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
5616, 54, 14, 55syl3anc 1317 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
571, 4latjcom 16824 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋 ( 𝑌)) ∈ 𝐵 ∧ (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5816, 46, 56, 57syl3anc 1317 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5952, 58eqtr3d 2641 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋 = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
6059adantr 479 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → 𝑋 = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
611, 4, 5, 6oldmm3N 33323 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6212, 61syl3an1 1350 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6362oveq2d 6539 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (𝑋 (( 𝑋) 𝑌)))
641, 5latmcom 16840 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6516, 14, 54, 64syl3anc 1317 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6663, 65eqtr3d 2641 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6766eqeq1d 2607 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) ↔ (( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌)))
68 oveq1 6530 . . . . . . 7 ((( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
6967, 68syl6bi 241 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7069imp 443 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7160, 70eqtrd 2639 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7271ex 448 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
731, 4, 5, 6, 2cmtvalN 33315 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7472, 73sylibrd 247 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → 𝑋𝐶𝑌))
7541, 74impbid 200 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  lecple 15717  occoc 15718  joincjn 16709  meetcmee 16710  Latclat 16810  OPcops 33276  cmccmtN 33277  OLcol 33278  OMLcoml 33279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-preset 16693  df-poset 16711  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-lat 16811  df-oposet 33280  df-cmtN 33281  df-ol 33282  df-oml 33283
This theorem is referenced by:  cmtbr4N  33359  omlfh1N  33362
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