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Theorem cmt2N 35059
Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 28792 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmt2.b 𝐵 = (Base‘𝐾)
cmt2.o = (oc‘𝐾)
cmt2.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
cmt2N ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶( 𝑌)))

Proof of Theorem cmt2N
StepHypRef Expression
1 omllat 35051 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ Lat)
213ad2ant1 1127 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
3 cmt2.b . . . . . . 7 𝐵 = (Base‘𝐾)
4 eqid 2771 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
53, 4latmcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
61, 5syl3an1 1166 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
7 simp2 1131 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
8 omlop 35050 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OP)
983ad2ant1 1127 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
10 simp3 1132 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
11 cmt2.o . . . . . . . 8 = (oc‘𝐾)
123, 11opoccl 35003 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
139, 10, 12syl2anc 573 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
143, 4latmcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋(meet‘𝐾)( 𝑌)) ∈ 𝐵)
152, 7, 13, 14syl3anc 1476 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)( 𝑌)) ∈ 𝐵)
16 eqid 2771 . . . . . 6 (join‘𝐾) = (join‘𝐾)
173, 16latjcom 17267 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵 ∧ (𝑋(meet‘𝐾)( 𝑌)) ∈ 𝐵) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)( 𝑌))) = ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)𝑌)))
182, 6, 15, 17syl3anc 1476 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)( 𝑌))) = ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)𝑌)))
193, 11opococ 35004 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
209, 10, 19syl2anc 573 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
2120oveq2d 6809 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)( ‘( 𝑌))) = (𝑋(meet‘𝐾)𝑌))
2221oveq2d 6809 . . . 4 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ‘( 𝑌)))) = ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)𝑌)))
2318, 22eqtr4d 2808 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)( 𝑌))) = ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ‘( 𝑌)))))
2423eqeq2d 2781 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)( 𝑌))) ↔ 𝑋 = ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ‘( 𝑌))))))
25 cmt2.c . . 3 𝐶 = (cm‘𝐾)
263, 16, 4, 11, 25cmtvalN 35020 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)( 𝑌)))))
273, 16, 4, 11, 25cmtvalN 35020 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋𝐶( 𝑌) ↔ 𝑋 = ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ‘( 𝑌))))))
2813, 27syld3an3 1515 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶( 𝑌) ↔ 𝑋 = ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ‘( 𝑌))))))
2924, 26, 283bitr4d 300 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶( 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  occoc 16157  joincjn 17152  meetcmee 17153  Latclat 17253  OPcops 34981  cmccmtN 34982  OMLcoml 34984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-lat 17254  df-oposet 34985  df-cmtN 34986  df-ol 34987  df-oml 34988
This theorem is referenced by:  cmt3N  35060  cmt4N  35061  omlfh1N  35067
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