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Theorem cmtbr3N 37716
Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 30550 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 37711 . . . 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 37715 . . . . 5 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑋𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
873com23 1126 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
93, 8bitrd 278 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
10 oveq2 7365 . . . . . 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 37702 . . . . . . . . 9 (𝐾 ∈ OML → 𝐾 ∈ OL)
13123ad2ant1 1133 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
14 simp2 1137 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
15 omllat 37704 . . . . . . . . . 10 (𝐾 ∈ OML → 𝐾 ∈ Lat)
16153ad2ant1 1133 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
17 simp3 1138 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
181, 4latjcl 18328 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
1916, 17, 14, 18syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) ∈ 𝐵)
20 omlop 37703 . . . . . . . . . . 11 (𝐾 ∈ OML → 𝐾 ∈ OP)
21203ad2ant1 1133 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
221, 6opoccl 37656 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
2321, 14, 22syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
241, 4latjcl 18328 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
2516, 17, 23, 24syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
261, 5latmassOLD 37691 . . . . . . . 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 1372 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
281, 4latjcom 18336 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
2916, 17, 14, 28syl3anc 1371 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
3029oveq2d 7373 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = (𝑋 (𝑋 𝑌)))
311, 4, 5latabs2 18365 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3215, 31syl3an1 1163 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3330, 32eqtrd 2776 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = 𝑋)
341, 4latjcom 18336 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3516, 17, 23, 34syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3633, 35oveq12d 7375 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 (( 𝑋) 𝑌)))
3727, 36eqtr3d 2778 . . . . . 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 2777 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌))
4039ex 413 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋))) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
419, 40sylbid 239 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
42 simp1 1136 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
431, 6opoccl 37656 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
4421, 17, 43syl2anc 584 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
451, 5latmcl 18329 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4616, 14, 44, 45syl3anc 1371 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4742, 46, 143jca 1128 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ (𝑋 ( 𝑌)) ∈ 𝐵𝑋𝐵))
48 eqid 2736 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
491, 48, 5latmle1 18353 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
5016, 14, 44, 49syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
511, 48, 4, 5, 6omllaw2N 37706 . . . . . . . 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 37656 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
5421, 46, 53syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
551, 5latmcl 18329 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵𝑋𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
5616, 54, 14, 55syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
571, 4latjcom 18336 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋 ( 𝑌)) ∈ 𝐵 ∧ (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5816, 46, 56, 57syl3anc 1371 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5952, 58eqtr3d 2778 . . . . . 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 37681 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6212, 61syl3an1 1163 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6362oveq2d 7373 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (𝑋 (( 𝑋) 𝑌)))
641, 5latmcom 18352 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6516, 14, 54, 64syl3anc 1371 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6663, 65eqtr3d 2778 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6766eqeq1d 2738 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) ↔ (( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌)))
68 oveq1 7364 . . . . . . 7 ((( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
6967, 68syl6bi 252 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7069imp 407 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7160, 70eqtrd 2776 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7271ex 413 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
731, 4, 5, 6, 2cmtvalN 37673 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7472, 73sylibrd 258 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → 𝑋𝐶𝑌))
7541, 74impbid 211 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  occoc 17141  joincjn 18200  meetcmee 18201  Latclat 18320  OPcops 37634  cmccmtN 37635  OLcol 37636  OMLcoml 37637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-lat 18321  df-oposet 37638  df-cmtN 37639  df-ol 37640  df-oml 37641
This theorem is referenced by:  cmtbr4N  37717  omlfh1N  37720
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