Theorem cmt2N 39289

Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 31565 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋𝐶( ⊥ ‘𝑌)))

Proof of Theorem cmt2N

Step	Hyp	Ref	Expression
1		omllat 39281	. . . . . 6 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
2	1	3ad2ant1 1133	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
3		cmt2.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
4		eqid 2731	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
5	3, 4	latmcl 18341	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
6	1, 5	syl3an1 1163	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
7		simp2 1137	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
8		omlop 39280	. . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
9	8	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
10		simp3 1138	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
11		cmt2.o	. . . . . . . 8 ⊢ ⊥ = (oc‘𝐾)
12	3, 11	opoccl 39233	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
13	9, 10, 12	syl2anc 584	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
14	3, 4	latmcl 18341	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋(meet‘𝐾)( ⊥ ‘𝑌)) ∈ 𝐵)
15	2, 7, 13, 14	syl3anc 1373	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)( ⊥ ‘𝑌)) ∈ 𝐵)
16		eqid 2731	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
17	3, 16	latjcom 18348	. . . . 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)))
18	2, 6, 15, 17	syl3anc 1373	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)( ⊥ ‘𝑌))) = ((𝑋(meet‘𝐾)( ⊥ ‘𝑌))(join‘𝐾)(𝑋(meet‘𝐾)𝑌)))
19	3, 11	opococ 39234	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
20	9, 10, 19	syl2anc 584	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
21	20	oveq2d 7357	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)( ⊥ ‘( ⊥ ‘𝑌))) = (𝑋(meet‘𝐾)𝑌))
22	21	oveq2d 7357	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(meet‘𝐾)( ⊥ ‘𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ⊥ ‘( ⊥ ‘𝑌)))) = ((𝑋(meet‘𝐾)( ⊥ ‘𝑌))(join‘𝐾)(𝑋(meet‘𝐾)𝑌)))
23	18, 22	eqtr4d 2769	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)( ⊥ ‘𝑌))) = ((𝑋(meet‘𝐾)( ⊥ ‘𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ⊥ ‘( ⊥ ‘𝑌)))))
24	23	eqeq2d 2742	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (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)
26	3, 16, 4, 11, 25	cmtvalN 39250	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)( ⊥ ‘𝑌)))))
27	3, 16, 4, 11, 25	cmtvalN 39250	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋𝐶( ⊥ ‘𝑌) ↔ 𝑋 = ((𝑋(meet‘𝐾)( ⊥ ‘𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ⊥ ‘( ⊥ ‘𝑌))))))
28	13, 27	syld3an3 1411	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶( ⊥ ‘𝑌) ↔ 𝑋 = ((𝑋(meet‘𝐾)( ⊥ ‘𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ⊥ ‘( ⊥ ‘𝑌))))))
29	24, 26, 28	3bitr4d 311	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋𝐶( ⊥ ‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   class class class wbr 5086  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  occoc 17164  joincjn 18212  meetcmee 18213  Latclat 18332  OPcops 39211  cmccmtN 39212  OMLcoml 39214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-lub 18245  df-glb 18246  df-join 18247  df-meet 18248  df-lat 18333  df-oposet 39215  df-cmtN 39216  df-ol 39217  df-oml 39218

This theorem is referenced by:  cmt3N  39290  cmt4N  39291  omlfh1N  39297