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Theorem cmtbr3N 33559
 Description: Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 27851 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 33554 . . . 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 33558 . . . . 5 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑋𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
873com23 1263 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
93, 8bitrd 267 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))))
10 oveq2 6557 . . . . . 6 (𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋))) → (𝑋 𝑌) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
1110adantl 481 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 𝑌) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
12 omlol 33545 . . . . . . . . 9 (𝐾 ∈ OML → 𝐾 ∈ OL)
13123ad2ant1 1075 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
14 simp2 1055 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
15 omllat 33547 . . . . . . . . . 10 (𝐾 ∈ OML → 𝐾 ∈ Lat)
16153ad2ant1 1075 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
17 simp3 1056 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
181, 4latjcl 16874 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
1916, 17, 14, 18syl3anc 1318 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) ∈ 𝐵)
20 omlop 33546 . . . . . . . . . . 11 (𝐾 ∈ OML → 𝐾 ∈ OP)
21203ad2ant1 1075 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
221, 6opoccl 33499 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
2321, 14, 22syl2anc 691 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
241, 4latjcl 16874 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
2516, 17, 23, 24syl3anc 1318 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) ∈ 𝐵)
261, 5latmassOLD 33534 . . . . . . . 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 1320 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))))
281, 4latjcom 16882 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
2916, 17, 14, 28syl3anc 1318 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = (𝑋 𝑌))
3029oveq2d 6565 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = (𝑋 (𝑋 𝑌)))
311, 4, 5latabs2 16911 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3215, 31syl3an1 1351 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
3330, 32eqtrd 2644 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑌 𝑋)) = 𝑋)
341, 4latjcom 16882 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ ( 𝑋) ∈ 𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3516, 17, 23, 34syl3anc 1318 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 ( 𝑋)) = (( 𝑋) 𝑌))
3633, 35oveq12d 6567 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (𝑌 𝑋)) (𝑌 ( 𝑋))) = (𝑋 (( 𝑋) 𝑌)))
3727, 36eqtr3d 2646 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))) = (𝑋 (( 𝑋) 𝑌)))
3837adantr 480 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 ((𝑌 𝑋) (𝑌 ( 𝑋)))) = (𝑋 (( 𝑋) 𝑌)))
3911, 38eqtr2d 2645 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋)))) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌))
4039ex 449 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 = ((𝑌 𝑋) (𝑌 ( 𝑋))) → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
419, 40sylbid 229 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
42 simp1 1054 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
431, 6opoccl 33499 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
4421, 17, 43syl2anc 691 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
451, 5latmcl 16875 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4616, 14, 44, 45syl3anc 1318 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
4742, 46, 143jca 1235 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ (𝑋 ( 𝑌)) ∈ 𝐵𝑋𝐵))
48 eqid 2610 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
491, 48, 5latmle1 16899 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
5016, 14, 44, 49syl3anc 1318 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌))(le‘𝐾)𝑋)
511, 48, 4, 5, 6omllaw2N 33549 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋 ( 𝑌)) ∈ 𝐵𝑋𝐵) → ((𝑋 ( 𝑌))(le‘𝐾)𝑋 → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = 𝑋))
5247, 50, 51sylc 63 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = 𝑋)
531, 6opoccl 33499 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
5421, 46, 53syl2anc 691 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) ∈ 𝐵)
551, 5latmcl 16875 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵𝑋𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
5616, 54, 14, 55syl3anc 1318 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵)
571, 4latjcom 16882 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑋 ( 𝑌)) ∈ 𝐵 ∧ (( ‘(𝑋 ( 𝑌))) 𝑋) ∈ 𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5816, 46, 56, 57syl3anc 1318 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 ( 𝑌)) (( ‘(𝑋 ( 𝑌))) 𝑋)) = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
5952, 58eqtr3d 2646 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋 = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
6059adantr 480 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → 𝑋 = ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))))
611, 4, 5, 6oldmm3N 33524 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6212, 61syl3an1 1351 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
6362oveq2d 6565 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (𝑋 (( 𝑋) 𝑌)))
641, 5latmcom 16898 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( ‘(𝑋 ( 𝑌))) ∈ 𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6516, 14, 54, 64syl3anc 1318 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( ‘(𝑋 ( 𝑌)))) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6663, 65eqtr3d 2646 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) = (( ‘(𝑋 ( 𝑌))) 𝑋))
6766eqeq1d 2612 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) ↔ (( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌)))
68 oveq1 6556 . . . . . . 7 ((( ‘(𝑋 ( 𝑌))) 𝑋) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
6967, 68syl6bi 242 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7069imp 444 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → ((( ‘(𝑋 ( 𝑌))) 𝑋) (𝑋 ( 𝑌))) = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7160, 70eqtrd 2644 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌))))
7271ex 449 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
731, 4, 5, 6, 2cmtvalN 33516 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
7472, 73sylibrd 248 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → 𝑋𝐶𝑌))
7541, 74impbid 201 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  occoc 15776  joincjn 16767  meetcmee 16768  Latclat 16868  OPcops 33477  cmccmtN 33478  OLcol 33479  OMLcoml 33480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-lat 16869  df-oposet 33481  df-cmtN 33482  df-ol 33483  df-oml 33484 This theorem is referenced by:  cmtbr4N  33560  omlfh1N  33563
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