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Theorem cmtbr2N 35062
Description: Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 28795 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
cmtbr2N ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))

Proof of Theorem cmtbr2N
StepHypRef Expression
1 cmtbr2.b . . 3 𝐵 = (Base‘𝐾)
2 cmtbr2.o . . 3 = (oc‘𝐾)
3 cmtbr2.c . . 3 𝐶 = (cm‘𝐾)
41, 2, 3cmt4N 35061 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑋)𝐶( 𝑌)))
5 simp1 1130 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
6 omlop 35050 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ OP)
763ad2ant1 1127 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
8 simp2 1131 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
91, 2opoccl 35003 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
107, 8, 9syl2anc 573 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
11 simp3 1132 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
121, 2opoccl 35003 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
137, 11, 12syl2anc 573 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
14 cmtbr2.j . . . 4 = (join‘𝐾)
15 cmtbr2.m . . . 4 = (meet‘𝐾)
161, 14, 15, 2, 3cmtvalN 35020 . . 3 ((𝐾 ∈ OML ∧ ( 𝑋) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵) → (( 𝑋)𝐶( 𝑌) ↔ ( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌))))))
175, 10, 13, 16syl3anc 1476 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋)𝐶( 𝑌) ↔ ( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌))))))
18 eqcom 2778 . . . 4 (𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌))) ↔ ((𝑋 𝑌) (𝑋 ( 𝑌))) = 𝑋)
1918a1i 11 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌))) ↔ ((𝑋 𝑌) (𝑋 ( 𝑌))) = 𝑋))
20 omllat 35051 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ Lat)
21203ad2ant1 1127 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
221, 14latjcl 17259 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
2320, 22syl3an1 1166 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
241, 14latjcl 17259 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
2521, 8, 13, 24syl3anc 1476 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
261, 15latmcl 17260 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 ( 𝑌))) ∈ 𝐵)
2721, 23, 25, 26syl3anc 1476 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) (𝑋 ( 𝑌))) ∈ 𝐵)
281, 2opcon3b 35005 . . . 4 ((𝐾 ∈ OP ∧ ((𝑋 𝑌) (𝑋 ( 𝑌))) ∈ 𝐵𝑋𝐵) → (((𝑋 𝑌) (𝑋 ( 𝑌))) = 𝑋 ↔ ( 𝑋) = ( ‘((𝑋 𝑌) (𝑋 ( 𝑌))))))
297, 27, 8, 28syl3anc 1476 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((𝑋 𝑌) (𝑋 ( 𝑌))) = 𝑋 ↔ ( 𝑋) = ( ‘((𝑋 𝑌) (𝑋 ( 𝑌))))))
30 omlol 35049 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ OL)
31303ad2ant1 1127 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
321, 14, 15, 2oldmm1 35026 . . . . . 6 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) = (( ‘(𝑋 𝑌)) ( ‘(𝑋 ( 𝑌)))))
3331, 23, 25, 32syl3anc 1476 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) = (( ‘(𝑋 𝑌)) ( ‘(𝑋 ( 𝑌)))))
341, 14, 15, 2oldmj1 35030 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))
3530, 34syl3an1 1166 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))
361, 14, 15, 2oldmj1 35030 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) ( ‘( 𝑌))))
3731, 8, 13, 36syl3anc 1476 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) ( ‘( 𝑌))))
3835, 37oveq12d 6811 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(𝑋 𝑌)) ( ‘(𝑋 ( 𝑌)))) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌)))))
3933, 38eqtrd 2805 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌)))))
4039eqeq2d 2781 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) ↔ ( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌))))))
4119, 29, 403bitrrd 295 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌)))) ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
424, 17, 413bitrd 294 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  occoc 16157  joincjn 17152  meetcmee 17153  Latclat 17253  OPcops 34981  cmccmtN 34982  OLcol 34983  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 837  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:  cmtbr3N  35063
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