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Theorem cmtbr4N 35064
Description: Alternate definition for the commutes relation. (cmbr4i 28800 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 35063 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
7 omllat 35051 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ Lat)
8 cmtbr4.l . . . . . 6 = (le‘𝐾)
91, 8, 3latmle2 17285 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
107, 9syl3an1 1166 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) 𝑌)
11 breq1 4789 . . . 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 1127 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
14 simp2 1131 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
15 omlop 35050 . . . . . . . . . . . 12 (𝐾 ∈ OML → 𝐾 ∈ OP)
16153ad2ant1 1127 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
171, 4opoccl 35003 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
1816, 14, 17syl2anc 565 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
19 simp3 1132 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
201, 2latjcl 17259 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
2113, 18, 19, 20syl3anc 1476 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌) ∈ 𝐵)
221, 8, 3latmle1 17284 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵) → (𝑋 (( 𝑋) 𝑌)) 𝑋)
2313, 14, 21, 22syl3anc 1476 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) 𝑋)
2423anim1i 594 . . . . . . 7 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌) → ((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌))
2524ex 397 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → ((𝑋 (( 𝑋) 𝑌)) 𝑋 ∧ (𝑋 (( 𝑋) 𝑌)) 𝑌)))
261, 3latmcl 17260 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵) → (𝑋 (( 𝑋) 𝑌)) ∈ 𝐵)
2713, 14, 21, 26syl3anc 1476 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) 𝑌)) ∈ 𝐵)
281, 8, 3latlem12 17286 . . . . . . 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 1478 . . . . . 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 17269 . . . . . . 7 ((𝐾 ∈ Lat ∧ ( 𝑋) ∈ 𝐵𝑌𝐵) → 𝑌 (( 𝑋) 𝑌))
3213, 18, 19, 31syl3anc 1476 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌 (( 𝑋) 𝑌))
331, 8, 3latmlem2 17290 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑌𝐵 ∧ (( 𝑋) 𝑌) ∈ 𝐵𝑋𝐵)) → (𝑌 (( 𝑋) 𝑌) → (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌))))
3413, 19, 21, 14, 33syl13anc 1478 . . . . . 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 510 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 (( 𝑋) 𝑌)) 𝑌 → ((𝑋 (( 𝑋) 𝑌)) (𝑋 𝑌) ∧ (𝑋 𝑌) (𝑋 (( 𝑋) 𝑌)))))
371, 3latmcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
387, 37syl3an1 1166 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
391, 8latasymb 17262 . . . . 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 1476 . . . 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 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  occoc 16157  joincjn 17152  meetcmee 17153  Latclat 17253  OPcops 34981  cmccmtN 34982  OMLcoml 34984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254  df-oposet 34985  df-cmtN 34986  df-ol 34987  df-oml 34988
This theorem is referenced by:  lecmtN  35065
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