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Theorem cmtbr4N 34057
Description: Alternate definition for the commutes relation. (cmbr4i 28330 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtbr4.b 𝐵 = (Base‘𝐾)
cmtbr4.l = (le‘𝐾)
cmtbr4.j = (join‘𝐾)
cmtbr4.m = (meet‘𝐾)
cmtbr4.o = (oc‘𝐾)
cmtbr4.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
cmtbr4N ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) 𝑌))

Proof of Theorem cmtbr4N
StepHypRef Expression
1 cmtbr4.b . . 3 𝐵 = (Base‘𝐾)
2 cmtbr4.j . . 3 = (join‘𝐾)
3 cmtbr4.m . . 3 = (meet‘𝐾)
4 cmtbr4.o . . 3 = (oc‘𝐾)
5 cmtbr4.c . . 3 𝐶 = (cm‘𝐾)
61, 2, 3, 4, 5cmtbr3N 34056 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
7 omllat 34044 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ Lat)
8 cmtbr4.l . . . . . 6 = (le‘𝐾)
91, 8, 3latmle2 17009 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
107, 9syl3an1 1356 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
11 breq1 4621 . . . 4 ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 ↔ (𝑋 𝑌) 𝑌))
1210, 11syl5ibrcom 237 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → (𝑋 (( 𝑋) 𝑌)) 𝑌))
1373ad2ant1 1080 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
14 simp2 1060 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
15 omlop 34043 . . . . . . . . . . . 12 (𝐾 ∈ OML → 𝐾 ∈ OP)
16153ad2ant1 1080 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
171, 4opoccl 33996 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
1816, 14, 17syl2anc 692 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
19 simp3 1061 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
201, 2latjcl 16983 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
2113, 18, 19, 20syl3anc 1323 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
221, 8, 3latmle1 17008 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵) → (𝑋 (( 𝑋) 𝑌)) 𝑋)
2313, 14, 21, 22syl3anc 1323 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) 𝑋)
2423anim1i 591 . . . . . . 7 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌) → ((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌))
2524ex 450 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → ((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌)))
261, 3latmcl 16984 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵) → (𝑋 (( 𝑋) 𝑌)) ∈ 𝐵)
2713, 14, 21, 26syl3anc 1323 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) ∈ 𝐵)
281, 8, 3latlem12 17010 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑋 (( 𝑋) 𝑌)) ∈ 𝐵𝑋𝐵𝑌𝐵)) → (((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌) ↔ (𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌)))
2913, 27, 14, 19, 28syl13anc 1325 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌) ↔ (𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌)))
3025, 29sylibd 229 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → (𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌)))
311, 8, 2latlej2 16993 . . . . . . 7 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → 𝑌 (( 𝑋) 𝑌))
3213, 18, 19, 31syl3anc 1323 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌 (( 𝑋) 𝑌))
331, 8, 3latmlem2 17014 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑌𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵𝑋𝐵)) → (𝑌 (( 𝑋) 𝑌) → (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌))))
3413, 19, 21, 14, 33syl13anc 1325 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌 (( 𝑋) 𝑌) → (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌))))
3532, 34mpd 15 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌)))
3630, 35jctird 566 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → ((𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌) ∧ (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌)))))
371, 3latmcl 16984 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
387, 37syl3an1 1356 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
391, 8latasymb 16986 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 (( 𝑋) 𝑌)) ∈ 𝐵 ∧ (𝑋 𝑌) ∈ 𝐵) → (((𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌) ∧ (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌))) ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
4013, 27, 38, 39syl3anc 1323 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌) ∧ (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌))) ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
4136, 40sylibd 229 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
4212, 41impbid 202 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) ↔ (𝑋 (( 𝑋) 𝑌)) 𝑌))
436, 42bitrd 268 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  occoc 15881  joincjn 16876  meetcmee 16877  Latclat 16977  OPcops 33974  cmccmtN 33975  OMLcoml 33977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16860  df-poset 16878  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-lat 16978  df-oposet 33978  df-cmtN 33979  df-ol 33980  df-oml 33981
This theorem is referenced by:  lecmtN  34058
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