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Theorem cmtbr3N 34360
Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 28437 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 34355 . . . 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 34359 . . . . 5 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑋𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
873com23 1269 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
93, 8bitrd 268 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
10 oveq2 6643 . . . . . 6 (𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋))) → (𝑋 𝑌) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
1110adantl 482 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 𝑌) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
12 omlol 34346 . . . . . . . . 9 (𝐾 ∈ OML → 𝐾 ∈ OL)
13123ad2ant1 1080 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
14 simp2 1060 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
15 omllat 34348 . . . . . . . . . 10 (𝐾 ∈ OML → 𝐾 ∈ Lat)
16153ad2ant1 1080 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
17 simp3 1061 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
181, 4latjcl 17032 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
1916, 17, 14, 18syl3anc 1324 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) ∈ 𝐵)
20 omlop 34347 . . . . . . . . . . 11 (𝐾 ∈ OML → 𝐾 ∈ OP)
21203ad2ant1 1080 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
221, 6opoccl 34300 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
2321, 14, 22syl2anc 692 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
241, 4latjcl 17032 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
2516, 17, 23, 24syl3anc 1324 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
261, 5latmassOLD 34335 . . . . . . . 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 1326 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
281, 4latjcom 17040 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
2916, 17, 14, 28syl3anc 1324 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
3029oveq2d 6651 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = (𝑋 (𝑋 𝑌)))
311, 4, 5latabs2 17069 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3215, 31syl3an1 1357 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3330, 32eqtrd 2654 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = 𝑋)
341, 4latjcom 17040 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3516, 17, 23, 34syl3anc 1324 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3633, 35oveq12d 6653 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 (( 𝑋) 𝑌)))
3727, 36eqtr3d 2656 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))) = (𝑋 (( 𝑋) 𝑌)))
3837adantr 481 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))) = (𝑋 (( 𝑋) 𝑌)))
3911, 38eqtr2d 2655 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌))
4039ex 450 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋))) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
419, 40sylbid 230 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
42 simp1 1059 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
431, 6opoccl 34300 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
4421, 17, 43syl2anc 692 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
451, 5latmcl 17033 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4616, 14, 44, 45syl3anc 1324 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4742, 46, 143jca 1240 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ (𝑋 ( 𝑌)) ∈ 𝐵𝑋𝐵))
48 eqid 2620 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
491, 48, 5latmle1 17057 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
5016, 14, 44, 49syl3anc 1324 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
511, 48, 4, 5, 6omllaw2N 34350 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋 ( 𝑌)) ∈ 𝐵𝑋𝐵) → ((𝑋 ( 𝑌))(le‘𝐾)𝑋 → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = 𝑋))
5247, 50, 51sylc 65 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = 𝑋)
531, 6opoccl 34300 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
5421, 46, 53syl2anc 692 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
551, 5latmcl 17033 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵𝑋𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
5616, 54, 14, 55syl3anc 1324 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
571, 4latjcom 17040 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋 ( 𝑌)) ∈ 𝐵 ∧ (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5816, 46, 56, 57syl3anc 1324 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5952, 58eqtr3d 2656 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋 = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
6059adantr 481 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → 𝑋 = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
611, 4, 5, 6oldmm3N 34325 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6212, 61syl3an1 1357 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6362oveq2d 6651 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (𝑋 (( 𝑋) 𝑌)))
641, 5latmcom 17056 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6516, 14, 54, 64syl3anc 1324 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6663, 65eqtr3d 2656 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6766eqeq1d 2622 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) ↔ (( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌)))
68 oveq1 6642 . . . . . . 7 ((( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
6967, 68syl6bi 243 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7069imp 445 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7160, 70eqtrd 2654 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7271ex 450 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
731, 4, 5, 6, 2cmtvalN 34317 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7472, 73sylibrd 249 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → 𝑋𝐶𝑌))
7541, 74impbid 202 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  occoc 15930  joincjn 16925  meetcmee 16926  Latclat 17026  OPcops 34278  cmccmtN 34279  OLcol 34280  OMLcoml 34281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-preset 16909  df-poset 16927  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-lat 17027  df-oposet 34282  df-cmtN 34283  df-ol 34284  df-oml 34285
This theorem is referenced by:  cmtbr4N  34361  omlfh1N  34364
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