Theorem cmt2N 39268

Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 31574 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 39260	. . . . . 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 2735	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
5	3, 4	latmcl 18450	. . . . . 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 39259	. . . . . . . 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 39212	. . . . . . 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 18450	. . . . . 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 2735	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
17	3, 16	latjcom 18457	. . . . 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 39213	. . . . . . 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 7421	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)( ⊥ ‘( ⊥ ‘𝑌))) = (𝑋(meet‘𝐾)𝑌))
22	21	oveq2d 7421	. . . 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 2773	. . 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 2746	. 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 39229	. 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 39229	. . 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 1540   ∈ wcel 2108   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  occoc 17279  joincjn 18323  meetcmee 18324  Latclat 18441  OPcops 39190  cmccmtN 39191  OMLcoml 39193

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-lat 18442  df-oposet 39194  df-cmtN 39195  df-ol 39196  df-oml 39197

This theorem is referenced by:  cmt3N  39269  cmt4N  39270  omlfh1N  39276