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Theorem cmtbr2N 38579
Description: Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 31273 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtbr2.b 𝐡 = (Baseβ€˜πΎ)
cmtbr2.j ∨ = (joinβ€˜πΎ)
cmtbr2.m ∧ = (meetβ€˜πΎ)
cmtbr2.o βŠ₯ = (ocβ€˜πΎ)
cmtbr2.c 𝐢 = (cmβ€˜πΎ)
Assertion
Ref Expression
cmtbr2N ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ 𝑋 = ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))))
Proof of Theorem cmtbr2N
StepHypRef Expression
1 cmtbr2.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 cmtbr2.o . . 3 βŠ₯ = (ocβ€˜πΎ)
3 cmtbr2.c . . 3 𝐢 = (cmβ€˜πΎ)
41, 2, 3cmt4N 38578 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ ( βŠ₯ β€˜π‘‹)𝐢( βŠ₯ β€˜π‘Œ)))
5 simp1 1133 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OML)
6 omlop 38567 . . . . 5 (𝐾 ∈ OML β†’ 𝐾 ∈ OP)
763ad2ant1 1130 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
8 simp2 1134 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
91, 2opoccl 38520 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
107, 8, 9syl2anc 583 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
11 simp3 1135 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
121, 2opoccl 38520 . . . 4 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
137, 11, 12syl2anc 583 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
14 cmtbr2.j . . . 4 ∨ = (joinβ€˜πΎ)
15 cmtbr2.m . . . 4 ∧ = (meetβ€˜πΎ)
161, 14, 15, 2, 3cmtvalN 38537 . . 3 ((𝐾 ∈ OML ∧ ( βŠ₯ β€˜π‘‹) ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹)𝐢( βŠ₯ β€˜π‘Œ) ↔ ( βŠ₯ β€˜π‘‹) = ((( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))))
175, 10, 13, 16syl3anc 1368 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹)𝐢( βŠ₯ β€˜π‘Œ) ↔ ( βŠ₯ β€˜π‘‹) = ((( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))))
18 eqcom 2731 . . . 4 (𝑋 = ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) ↔ ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) = 𝑋)
1918a1i 11 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 = ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) ↔ ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) = 𝑋))
20 omllat 38568 . . . . . 6 (𝐾 ∈ OML β†’ 𝐾 ∈ Lat)
21203ad2ant1 1130 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
221, 14latjcl 18391 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) ∈ 𝐡)
2320, 22syl3an1 1160 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) ∈ 𝐡)
241, 14latjcl 18391 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
2521, 8, 13, 24syl3anc 1368 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
261, 15latmcl 18392 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 ∨ π‘Œ) ∈ 𝐡 ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡) β†’ ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡)
2721, 23, 25, 26syl3anc 1368 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡)
281, 2opcon3b 38522 . . . 4 ((𝐾 ∈ OP ∧ ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) = 𝑋 ↔ ( βŠ₯ β€˜π‘‹) = ( βŠ₯ β€˜((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))))))
297, 27, 8, 28syl3anc 1368 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) = 𝑋 ↔ ( βŠ₯ β€˜π‘‹) = ( βŠ₯ β€˜((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))))))
30 omlol 38566 . . . . . . 7 (𝐾 ∈ OML β†’ 𝐾 ∈ OL)
31303ad2ant1 1130 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OL)
321, 14, 15, 2oldmm1 38543 . . . . . 6 ((𝐾 ∈ OL ∧ (𝑋 ∨ π‘Œ) ∈ 𝐡 ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡) β†’ ( βŠ₯ β€˜((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))) = (( βŠ₯ β€˜(𝑋 ∨ π‘Œ)) ∨ ( βŠ₯ β€˜(𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))))
3331, 23, 25, 32syl3anc 1368 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))) = (( βŠ₯ β€˜(𝑋 ∨ π‘Œ)) ∨ ( βŠ₯ β€˜(𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))))
341, 14, 15, 2oldmj1 38547 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∨ π‘Œ)) = (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜π‘Œ)))
3530, 34syl3an1 1160 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∨ π‘Œ)) = (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜π‘Œ)))
361, 14, 15, 2oldmj1 38547 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) = (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))
3731, 8, 13, 36syl3anc 1368 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) = (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))
3835, 37oveq12d 7419 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜(𝑋 ∨ π‘Œ)) ∨ ( βŠ₯ β€˜(𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))) = ((( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))))
3933, 38eqtrd 2764 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))) = ((( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))))
4039eqeq2d 2735 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) = ( βŠ₯ β€˜((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))) ↔ ( βŠ₯ β€˜π‘‹) = ((( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))))
4119, 29, 403bitrrd 306 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (( βŠ₯ β€˜π‘‹) = ((( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜π‘Œ)) ∨ (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))) ↔ 𝑋 = ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))))
424, 17, 413bitrd 305 1 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ 𝑋 = ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5138  β€˜cfv 6533  (class class class)co 7401  Basecbs 17140  occoc 17201  joincjn 18263  meetcmee 18264  Latclat 18383  OPcops 38498  cmccmtN 38499  OLcol 38500  OMLcoml 38501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-proset 18247  df-poset 18265  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-lat 18384  df-oposet 38502  df-cmtN 38503  df-ol 38504  df-oml 38505
This theorem is referenced by:  cmtbr3N  38580
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