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Theorem cmtbr4N 37717
Description: Alternate definition for the commutes relation. (cmbr4i 30543 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 37716 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
7 omllat 37704 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ Lat)
8 cmtbr4.l . . . . . 6 = (le‘𝐾)
91, 8, 3latmle2 18354 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
107, 9syl3an1 1163 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
11 breq1 5108 . . . 4 ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 ↔ (𝑋 𝑌) 𝑌))
1210, 11syl5ibrcom 246 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) → (𝑋 (( 𝑋) 𝑌)) 𝑌))
1373ad2ant1 1133 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
14 simp2 1137 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
15 omlop 37703 . . . . . . . . . . . 12 (𝐾 ∈ OML → 𝐾 ∈ OP)
16153ad2ant1 1133 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
171, 4opoccl 37656 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
1816, 14, 17syl2anc 584 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
19 simp3 1138 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
201, 2latjcl 18328 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
2113, 18, 19, 20syl3anc 1371 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
221, 8, 3latmle1 18353 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵) → (𝑋 (( 𝑋) 𝑌)) 𝑋)
2313, 14, 21, 22syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) 𝑋)
2423anim1i 615 . . . . . . 7 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌) → ((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌))
2524ex 413 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → ((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌)))
261, 3latmcl 18329 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵) → (𝑋 (( 𝑋) 𝑌)) ∈ 𝐵)
2713, 14, 21, 26syl3anc 1371 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) ∈ 𝐵)
281, 8, 3latlem12 18355 . . . . . . 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 1372 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌) ↔ (𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌)))
3025, 29sylibd 238 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → (𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌)))
311, 8, 2latlej2 18338 . . . . . . 7 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → 𝑌 (( 𝑋) 𝑌))
3213, 18, 19, 31syl3anc 1371 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌 (( 𝑋) 𝑌))
331, 8, 3latmlem2 18359 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑌𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵𝑋𝐵)) → (𝑌 (( 𝑋) 𝑌) → (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌))))
3413, 19, 21, 14, 33syl13anc 1372 . . . . . 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 527 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → ((𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌) ∧ (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌)))))
371, 3latmcl 18329 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
387, 37syl3an1 1163 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
391, 8latasymb 18331 . . . . 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 1371 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌) ∧ (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌))) ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
4136, 40sylibd 238 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
4212, 41impbid 211 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌) ↔ (𝑋 (( 𝑋) 𝑌)) 𝑌))
436, 42bitrd 278 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) 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  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:  lecmtN  37718
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