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Theorem cmt2N 33351
Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 27642 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 33343 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ Lat)
213ad2ant1 1074 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
3 cmt2.b . . . . . . 7 𝐵 = (Base‘𝐾)
4 eqid 2609 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
53, 4latmcl 16821 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
61, 5syl3an1 1350 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
7 simp2 1054 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
8 omlop 33342 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OP)
983ad2ant1 1074 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
10 simp3 1055 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
11 cmt2.o . . . . . . . 8 = (oc‘𝐾)
123, 11opoccl 33295 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( 𝑌) ∈ 𝐵)
139, 10, 12syl2anc 690 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( 𝑌) ∈ 𝐵)
143, 4latmcl 16821 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋(meet‘𝐾)( 𝑌)) ∈ 𝐵)
152, 7, 13, 14syl3anc 1317 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)( 𝑌)) ∈ 𝐵)
16 eqid 2609 . . . . . 6 (join‘𝐾) = (join‘𝐾)
173, 16latjcom 16828 . . . . 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 1317 . . . 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 33296 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
209, 10, 19syl2anc 690 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ( ‘( 𝑌)) = 𝑌)
2120oveq2d 6543 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)( ‘( 𝑌))) = (𝑋(meet‘𝐾)𝑌))
2221oveq2d 6543 . . . 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 2646 . . 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 2619 . 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 33312 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)( 𝑌)))))
273, 16, 4, 11, 25cmtvalN 33312 . . 3 ((𝐾 ∈ OML ∧ 𝑋𝐵 ∧ ( 𝑌) ∈ 𝐵) → (𝑋𝐶( 𝑌) ↔ 𝑋 = ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ‘( 𝑌))))))
2813, 27syld3an3 1362 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶( 𝑌) ↔ 𝑋 = ((𝑋(meet‘𝐾)( 𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ‘( 𝑌))))))
2924, 26, 283bitr4d 298 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶( 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 33273  cmccmtN 33274  OMLcoml 33276
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 2033  ax-13 2233  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-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-lat 16815  df-oposet 33277  df-cmtN 33278  df-ol 33279  df-oml 33280
This theorem is referenced by:  cmt3N  33352  cmt4N  33353  omlfh1N  33359
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