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Theorem cmt2N 38425
Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 31111 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmt2.b 𝐡 = (Baseβ€˜πΎ)
cmt2.o βŠ₯ = (ocβ€˜πΎ)
cmt2.c 𝐢 = (cmβ€˜πΎ)
Assertion
Ref Expression
cmt2N ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ 𝑋𝐢( βŠ₯ β€˜π‘Œ)))
Proof of Theorem cmt2N
StepHypRef Expression
1 omllat 38417 . . . . . 6 (𝐾 ∈ OML β†’ 𝐾 ∈ Lat)
213ad2ant1 1131 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ Lat)
3 cmt2.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
4 eqid 2730 . . . . . . 7 (meetβ€˜πΎ) = (meetβ€˜πΎ)
53, 4latmcl 18399 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(meetβ€˜πΎ)π‘Œ) ∈ 𝐡)
61, 5syl3an1 1161 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(meetβ€˜πΎ)π‘Œ) ∈ 𝐡)
7 simp2 1135 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
8 omlop 38416 . . . . . . . 8 (𝐾 ∈ OML β†’ 𝐾 ∈ OP)
983ad2ant1 1131 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝐾 ∈ OP)
10 simp3 1136 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
11 cmt2.o . . . . . . . 8 βŠ₯ = (ocβ€˜πΎ)
123, 11opoccl 38369 . . . . . . 7 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
139, 10, 12syl2anc 582 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡)
143, 4latmcl 18399 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ (𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
152, 7, 13, 14syl3anc 1369 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ)) ∈ 𝐡)
16 eqid 2730 . . . . . 6 (joinβ€˜πΎ) = (joinβ€˜πΎ)
173, 16latjcom 18406 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋(meetβ€˜πΎ)π‘Œ) ∈ 𝐡 ∧ (𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ)) ∈ 𝐡) β†’ ((𝑋(meetβ€˜πΎ)π‘Œ)(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))) = ((𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)π‘Œ)))
182, 6, 15, 17syl3anc 1369 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋(meetβ€˜πΎ)π‘Œ)(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))) = ((𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)π‘Œ)))
193, 11opococ 38370 . . . . . . 7 ((𝐾 ∈ OP ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) = π‘Œ)
209, 10, 19syl2anc 582 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) = π‘Œ)
2120oveq2d 7429 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(meetβ€˜πΎ)( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))) = (𝑋(meetβ€˜πΎ)π‘Œ))
2221oveq2d 7429 . . . 4 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))) = ((𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)π‘Œ)))
2318, 22eqtr4d 2773 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋(meetβ€˜πΎ)π‘Œ)(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))) = ((𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)))))
2423eqeq2d 2741 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 = ((𝑋(meetβ€˜πΎ)π‘Œ)(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))) ↔ 𝑋 = ((𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))))
25 cmt2.c . . 3 𝐢 = (cmβ€˜πΎ)
263, 16, 4, 11, 25cmtvalN 38386 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ 𝑋 = ((𝑋(meetβ€˜πΎ)π‘Œ)(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ)))))
273, 16, 4, 11, 25cmtvalN 38386 . . 3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ ( βŠ₯ β€˜π‘Œ) ∈ 𝐡) β†’ (𝑋𝐢( βŠ₯ β€˜π‘Œ) ↔ 𝑋 = ((𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))))
2813, 27syld3an3 1407 . 2 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋𝐢( βŠ₯ β€˜π‘Œ) ↔ 𝑋 = ((𝑋(meetβ€˜πΎ)( βŠ₯ β€˜π‘Œ))(joinβ€˜πΎ)(𝑋(meetβ€˜πΎ)( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ))))))
2924, 26, 283bitr4d 310 1 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (π‘‹πΆπ‘Œ ↔ 𝑋𝐢( βŠ₯ β€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7413  Basecbs 17150  occoc 17211  joincjn 18270  meetcmee 18271  Latclat 18390  OPcops 38347  cmccmtN 38348  OMLcoml 38350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-lat 18391  df-oposet 38351  df-cmtN 38352  df-ol 38353  df-oml 38354
This theorem is referenced by:  cmt3N  38426  cmt4N  38427  omlfh1N  38433
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