Theorem cmt2N 39232

Description: Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 31622 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 39224	. . . . . 6 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
2	1	3ad2ant1 1132	. . . . 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 18498	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
6	1, 5	syl3an1 1162	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
7		simp2 1136	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
8		omlop 39223	. . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
9	8	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
10		simp3 1137	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
11		cmt2.o	. . . . . . . 8 ⊢ ⊥ = (oc‘𝐾)
12	3, 11	opoccl 39176	. . . . . . 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 18498	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋(meet‘𝐾)( ⊥ ‘𝑌)) ∈ 𝐵)
15	2, 7, 13, 14	syl3anc 1370	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)( ⊥ ‘𝑌)) ∈ 𝐵)
16		eqid 2735	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
17	3, 16	latjcom 18505	. . . . 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 1370	. . . 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 39177	. . . . . . 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 7447	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(meet‘𝐾)( ⊥ ‘( ⊥ ‘𝑌))) = (𝑋(meet‘𝐾)𝑌))
22	21	oveq2d 7447	. . . 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 2778	. . 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 39193	. 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 39193	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋𝐶( ⊥ ‘𝑌) ↔ 𝑋 = ((𝑋(meet‘𝐾)( ⊥ ‘𝑌))(join‘𝐾)(𝑋(meet‘𝐾)( ⊥ ‘( ⊥ ‘𝑌))))))
28	13, 27	syld3an3 1408	. 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 1537   ∈ wcel 2106   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  occoc 17306  joincjn 18369  meetcmee 18370  Latclat 18489  OPcops 39154  cmccmtN 39155  OMLcoml 39157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-lat 18490  df-oposet 39158  df-cmtN 39159  df-ol 39160  df-oml 39161

This theorem is referenced by:  cmt3N  39233  cmt4N  39234  omlfh1N  39240