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Theorem cmtbr2N 33361
Description: Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 27645 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 33360 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑋)𝐶( 𝑌)))
5 simp1 1053 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
6 omlop 33349 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ OP)
763ad2ant1 1074 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
8 simp2 1054 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
91, 2opoccl 33302 . . . 4 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)
107, 8, 9syl2anc 690 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑋) ∈ 𝐵)
11 simp3 1055 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
121, 2opoccl 33302 . . . 4 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
137, 11, 12syl2anc 690 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
14 cmtbr2.j . . . 4 = (join‘𝐾)
15 cmtbr2.m . . . 4 = (meet‘𝐾)
161, 14, 15, 2, 3cmtvalN 33319 . . 3 ((𝐾 ∈ OML ∧ ( 𝑋) ∈ 𝐵 ∧ ( 𝑌) ∈ 𝐵) → (( 𝑋)𝐶( 𝑌) ↔ ( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌))))))
175, 10, 13, 16syl3anc 1317 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋)𝐶( 𝑌) ↔ ( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌))))))
18 eqcom 2616 . . . 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 33350 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ Lat)
21203ad2ant1 1074 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
221, 14latjcl 16820 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
2320, 22syl3an1 1350 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
241, 14latjcl 16820 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
2521, 8, 13, 24syl3anc 1317 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 ( 𝑌)) ∈ 𝐵)
261, 15latmcl 16821 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 ( 𝑌))) ∈ 𝐵)
2721, 23, 25, 26syl3anc 1317 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) (𝑋 ( 𝑌))) ∈ 𝐵)
281, 2opcon3b 33304 . . . 4 ((𝐾 ∈ OP ∧ ((𝑋 𝑌) (𝑋 ( 𝑌))) ∈ 𝐵𝑋𝐵) → (((𝑋 𝑌) (𝑋 ( 𝑌))) = 𝑋 ↔ ( 𝑋) = ( ‘((𝑋 𝑌) (𝑋 ( 𝑌))))))
297, 27, 8, 28syl3anc 1317 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((𝑋 𝑌) (𝑋 ( 𝑌))) = 𝑋 ↔ ( 𝑋) = ( ‘((𝑋 𝑌) (𝑋 ( 𝑌))))))
30 omlol 33348 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ OL)
31303ad2ant1 1074 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
321, 14, 15, 2oldmm1 33325 . . . . . 6 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 ( 𝑌)) ∈ 𝐵) → ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) = (( ‘(𝑋 𝑌)) ( ‘(𝑋 ( 𝑌)))))
3331, 23, 25, 32syl3anc 1317 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) = (( ‘(𝑋 𝑌)) ( ‘(𝑋 ( 𝑌)))))
341, 14, 15, 2oldmj1 33329 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))
3530, 34syl3an1 1350 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))
361, 14, 15, 2oldmj1 33329 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) ( ‘( 𝑌))))
3731, 8, 13, 36syl3anc 1317 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) ( ‘( 𝑌))))
3835, 37oveq12d 6545 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( ‘(𝑋 𝑌)) ( ‘(𝑋 ( 𝑌)))) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌)))))
3933, 38eqtrd 2643 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌)))))
4039eqeq2d 2619 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = ( ‘((𝑋 𝑌) (𝑋 ( 𝑌)))) ↔ ( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌))))))
4119, 29, 403bitrrd 293 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = ((( 𝑋) ( 𝑌)) (( 𝑋) ( ‘( 𝑌)))) ↔ 𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
424, 17, 413bitrd 292 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  w3a 1030   = wceq 1474  wcel 1976   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  occoc 15722  joincjn 16713  meetcmee 16714  Latclat 16814  OPcops 33280  cmccmtN 33281  OLcol 33282  OMLcoml 33283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16697  df-poset 16715  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-lat 16815  df-oposet 33284  df-cmtN 33285  df-ol 33286  df-oml 33287
This theorem is referenced by:  cmtbr3N  33362
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