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Theorem cmtbr2N 34858
Description: Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 28583 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 34857 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑋)𝐶( 𝑌)))
5 simp1 1081 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
6 omlop 34846 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ OP)
763ad2ant1 1102 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
8 simp2 1082 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
91, 2opoccl 34799 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
107, 8, 9syl2anc 694 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
11 simp3 1083 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
121, 2opoccl 34799 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
137, 11, 12syl2anc 694 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
14 cmtbr2.j . . . 4 = (join‘𝐾)
15 cmtbr2.m . . . 4 = (meet‘𝐾)
161, 14, 15, 2, 3cmtvalN 34816 . . 3 ((𝐾 ∈ OML ∧ ( 𝑋) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵) → (( 𝑋)𝐶( 𝑌) ↔ ( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌))))))
175, 10, 13, 16syl3anc 1366 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋)𝐶( 𝑌) ↔ ( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌))))))
18 eqcom 2658 . . . 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 34847 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ Lat)
21203ad2ant1 1102 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
221, 14latjcl 17098 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
2320, 22syl3an1 1399 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
241, 14latjcl 17098 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
2521, 8, 13, 24syl3anc 1366 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
261, 15latmcl 17099 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 ( 𝑌))) ∈ 𝐵)
2721, 23, 25, 26syl3anc 1366 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) (𝑋 ( 𝑌))) ∈ 𝐵)
281, 2opcon3b 34801 . . . 4 ((𝐾 ∈ OP ∧ ((𝑋 𝑌) (𝑋 ( 𝑌))) ∈ 𝐵𝑋𝐵) → (((𝑋 𝑌) (𝑋 ( 𝑌))) = 𝑋 ↔ ( 𝑋) = ( ‘((𝑋 𝑌) (𝑋 ( 𝑌))))))
297, 27, 8, 28syl3anc 1366 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((𝑋 𝑌) (𝑋 ( 𝑌))) = 𝑋 ↔ ( 𝑋) = ( ‘((𝑋 𝑌) (𝑋 ( 𝑌))))))
30 omlol 34845 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ OL)
31303ad2ant1 1102 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
321, 14, 15, 2oldmm1 34822 . . . . . 6 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) = (( ‘(𝑋 𝑌)) ( ‘(𝑋 ( 𝑌)))))
3331, 23, 25, 32syl3anc 1366 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) = (( ‘(𝑋 𝑌)) ( ‘(𝑋 ( 𝑌)))))
341, 14, 15, 2oldmj1 34826 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))
3530, 34syl3an1 1399 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))
361, 14, 15, 2oldmj1 34826 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) ( ‘( 𝑌))))
3731, 8, 13, 36syl3anc 1366 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) ( ‘( 𝑌))))
3835, 37oveq12d 6708 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(𝑋 𝑌)) ( ‘(𝑋 ( 𝑌)))) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌)))))
3933, 38eqtrd 2685 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌)))))
4039eqeq2d 2661 . . 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 1054   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  occoc 15996  joincjn 16991  meetcmee 16992  Latclat 17092  OPcops 34777  cmccmtN 34778  OLcol 34779  OMLcoml 34780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-lat 17093  df-oposet 34781  df-cmtN 34782  df-ol 34783  df-oml 34784
This theorem is referenced by:  cmtbr3N  34859
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