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Theorem cmtbr4N 36570
 Description: Alternate definition for the commutes relation. (cmbr4i 29394 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 36569 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
7 omllat 36557 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ Lat)
8 cmtbr4.l . . . . . 6 = (le‘𝐾)
91, 8, 3latmle2 17682 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
107, 9syl3an1 1160 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
11 breq1 5034 . . . 4 ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 ↔ (𝑋 𝑌) 𝑌))
1210, 11syl5ibrcom 250 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → (𝑋 (( 𝑋) 𝑌)) 𝑌))
1373ad2ant1 1130 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
14 simp2 1134 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
15 omlop 36556 . . . . . . . . . . . 12 (𝐾 ∈ OML → 𝐾 ∈ OP)
16153ad2ant1 1130 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
171, 4opoccl 36509 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
1816, 14, 17syl2anc 587 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
19 simp3 1135 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
201, 2latjcl 17656 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
2113, 18, 19, 20syl3anc 1368 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
221, 8, 3latmle1 17681 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵) → (𝑋 (( 𝑋) 𝑌)) 𝑋)
2313, 14, 21, 22syl3anc 1368 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) 𝑋)
2423anim1i 617 . . . . . . 7 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌) → ((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌))
2524ex 416 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → ((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌)))
261, 3latmcl 17657 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵) → (𝑋 (( 𝑋) 𝑌)) ∈ 𝐵)
2713, 14, 21, 26syl3anc 1368 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) ∈ 𝐵)
281, 8, 3latlem12 17683 . . . . . . 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 1369 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌) ↔ (𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌)))
3025, 29sylibd 242 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → (𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌)))
311, 8, 2latlej2 17666 . . . . . . 7 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → 𝑌 (( 𝑋) 𝑌))
3213, 18, 19, 31syl3anc 1368 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌 (( 𝑋) 𝑌))
331, 8, 3latmlem2 17687 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑌𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵𝑋𝐵)) → (𝑌 (( 𝑋) 𝑌) → (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌))))
3413, 19, 21, 14, 33syl13anc 1369 . . . . . 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 530 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → ((𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌) ∧ (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌)))))
371, 3latmcl 17657 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
387, 37syl3an1 1160 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
391, 8latasymb 17659 . . . . 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 1368 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌) ∧ (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌))) ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
4136, 40sylibd 242 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
4212, 41impbid 215 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) ↔ (𝑋 (( 𝑋) 𝑌)) 𝑌))
436, 42bitrd 282 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5031  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  lecple 16567  occoc 16568  joincjn 17549  meetcmee 17550  Latclat 17650  OPcops 36487  cmccmtN 36488  OMLcoml 36490 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-proset 17533  df-poset 17551  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-lat 17651  df-oposet 36491  df-cmtN 36492  df-ol 36493  df-oml 36494 This theorem is referenced by:  lecmtN  36571
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