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Theorem cmtbr2N 38061
Description: Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 30827 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 38060 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ ( βŠ₯ β€˜π‘‹)𝐢( βŠ₯ β€˜π‘Œ)))
5 simp1 1137 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OML)
6 omlop 38049 . . . . 5 (𝐾 ∈ OML β†’ 𝐾 ∈ OP)
763ad2ant1 1134 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
8 simp2 1138 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
91, 2opoccl 38002 . . . 4 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
107, 8, 9syl2anc 585 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝐡)
11 simp3 1139 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
121, 2opoccl 38002 . . . 4 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
137, 11, 12syl2anc 585 . . 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 38019 . . 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 1372 . 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 2740 . . . 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 38050 . . . . . 6 (𝐾 ∈ OML β†’ 𝐾 ∈ Lat)
21203ad2ant1 1134 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
221, 14latjcl 18388 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) ∈ 𝐡)
2320, 22syl3an1 1164 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) ∈ 𝐡)
241, 14latjcl 18388 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
2521, 8, 13, 24syl3anc 1372 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
261, 15latmcl 18389 . . . . 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 1372 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ∨ π‘Œ) ∧ (𝑋 ∨ ( βŠ₯ β€˜π‘Œ))) ∈ 𝐡)
281, 2opcon3b 38004 . . . 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 1372 . . 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 38048 . . . . . . 7 (𝐾 ∈ OML β†’ 𝐾 ∈ OL)
31303ad2ant1 1134 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OL)
321, 14, 15, 2oldmm1 38025 . . . . . 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 1372 . . . . 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 38029 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜(𝑋 ∨ π‘Œ)) = (( βŠ₯ β€˜π‘‹) ∧ ( βŠ₯ β€˜π‘Œ)))
3530, 34syl3an1 1164 . . . . . 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 38029 . . . . . . 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 1372 . . . . . 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 7422 . . . . 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 2773 . . . 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 2744 . . 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 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7404  Basecbs 17140  occoc 17201  joincjn 18260  meetcmee 18261  Latclat 18380  OPcops 37980  cmccmtN 37981  OLcol 37982  OMLcoml 37983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-proset 18244  df-poset 18262  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-lat 18381  df-oposet 37984  df-cmtN 37985  df-ol 37986  df-oml 37987
This theorem is referenced by:  cmtbr3N  38062
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