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Theorem cmtbr3N 38062
Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 30839 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 38057 . . . 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 38061 . . . . 5 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑋𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
873com23 1127 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
93, 8bitrd 279 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
10 oveq2 7412 . . . . . 6 (𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋))) → (𝑋 𝑌) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
1110adantl 483 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 𝑌) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
12 omlol 38048 . . . . . . . . 9 (𝐾 ∈ OML → 𝐾 ∈ OL)
13123ad2ant1 1134 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
14 simp2 1138 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
15 omllat 38050 . . . . . . . . . 10 (𝐾 ∈ OML → 𝐾 ∈ Lat)
16153ad2ant1 1134 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
17 simp3 1139 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
181, 4latjcl 18388 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
1916, 17, 14, 18syl3anc 1372 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) ∈ 𝐵)
20 omlop 38049 . . . . . . . . . . 11 (𝐾 ∈ OML → 𝐾 ∈ OP)
21203ad2ant1 1134 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
221, 6opoccl 38002 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
2321, 14, 22syl2anc 585 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
241, 4latjcl 18388 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
2516, 17, 23, 24syl3anc 1372 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
261, 5latmassOLD 38037 . . . . . . . 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 1373 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
281, 4latjcom 18396 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
2916, 17, 14, 28syl3anc 1372 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
3029oveq2d 7420 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = (𝑋 (𝑋 𝑌)))
311, 4, 5latabs2 18425 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3215, 31syl3an1 1164 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3330, 32eqtrd 2773 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = 𝑋)
341, 4latjcom 18396 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3516, 17, 23, 34syl3anc 1372 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3633, 35oveq12d 7422 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 (( 𝑋) 𝑌)))
3727, 36eqtr3d 2775 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))) = (𝑋 (( 𝑋) 𝑌)))
3837adantr 482 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))) = (𝑋 (( 𝑋) 𝑌)))
3911, 38eqtr2d 2774 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌))
4039ex 414 . . 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 1137 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
431, 6opoccl 38002 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
4421, 17, 43syl2anc 585 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
451, 5latmcl 18389 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4616, 14, 44, 45syl3anc 1372 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4742, 46, 143jca 1129 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ (𝑋 ( 𝑌)) ∈ 𝐵𝑋𝐵))
48 eqid 2733 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
491, 48, 5latmle1 18413 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
5016, 14, 44, 49syl3anc 1372 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
511, 48, 4, 5, 6omllaw2N 38052 . . . . . . . 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 38002 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
5421, 46, 53syl2anc 585 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
551, 5latmcl 18389 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵𝑋𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
5616, 54, 14, 55syl3anc 1372 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
571, 4latjcom 18396 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋 ( 𝑌)) ∈ 𝐵 ∧ (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5816, 46, 56, 57syl3anc 1372 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5952, 58eqtr3d 2775 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋 = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
6059adantr 482 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → 𝑋 = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
611, 4, 5, 6oldmm3N 38027 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6212, 61syl3an1 1164 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6362oveq2d 7420 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (𝑋 (( 𝑋) 𝑌)))
641, 5latmcom 18412 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6516, 14, 54, 64syl3anc 1372 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6663, 65eqtr3d 2775 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6766eqeq1d 2735 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) ↔ (( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌)))
68 oveq1 7411 . . . . . . 7 ((( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
6967, 68syl6bi 253 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7069imp 408 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7160, 70eqtrd 2773 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7271ex 414 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
731, 4, 5, 6, 2cmtvalN 38019 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7472, 73sylibrd 259 . 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 397  w3a 1088   = wceq 1542  wcel 2107   class class class wbr 5147  cfv 6540  (class class class)co 7404  Basecbs 17140  lecple 17200  occoc 17201  joincjn 18260  meetcmee 18261  Latclat 18380  OPcops 37980  cmccmtN 37981  OLcol 37982  OMLcoml 37983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-proset 18244  df-poset 18262  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-lat 18381  df-oposet 37984  df-cmtN 37985  df-ol 37986  df-oml 37987
This theorem is referenced by:  cmtbr4N  38063  omlfh1N  38066
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