Theorem omllaw4 39269

Description: Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)

Hypotheses

Ref	Expression
omllaw4.b	⊢ 𝐵 = (Base‘𝐾)
omllaw4.l	⊢ ≤ = (le‘𝐾)
omllaw4.m	⊢ ∧ = (meet‘𝐾)
omllaw4.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
omllaw4	⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋))

Proof of Theorem omllaw4

Step	Hyp	Ref	Expression
1		simp1 1136	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
2		omlop 39264	. . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
4		simp3 1138	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
5		omllaw4.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
6		omllaw4.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
7	5, 6	opoccl 39217	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
8	3, 4, 7	syl2anc 584	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
9		simp2 1137	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
10	5, 6	opoccl 39217	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
11	3, 9, 10	syl2anc 584	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
12		omllaw4.l	. . . 4 ⊢ ≤ = (le‘𝐾)
13		eqid 2736	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
14		omllaw4.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
15	5, 12, 13, 14, 6	omllaw 39266	. . 3 ⊢ ((𝐾 ∈ OML ∧ ( ⊥ ‘𝑌) ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋) → ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
16	1, 8, 11, 15	syl3anc 1373	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋) → ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
17	5, 12, 6	oplecon3b 39223	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
18	2, 17	syl3an1 1163	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
19		omllat 39265	. . . . . 6 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
20	19	3ad2ant1 1133	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
21	5, 14	latmcl 18455	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵)
22	20, 11, 4, 21	syl3anc 1373	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵)
23	5, 6	opoccl 39217	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∈ 𝐵)
24	3, 22, 23	syl2anc 584	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∈ 𝐵)
25	5, 14	latmcl 18455	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) ∈ 𝐵)
26	20, 24, 4, 25	syl3anc 1373	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) ∈ 𝐵)
27	5, 6	opcon3b 39219	. . . 4 ⊢ ((𝐾 ∈ OP ∧ (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌))))
28	3, 26, 9, 27	syl3anc 1373	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌))))
29	5, 13	latjcom 18462	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
30	20, 22, 8, 29	syl3anc 1373	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
31		omlol 39263	. . . . . . 7 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
32	31	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
33	5, 13, 14, 6	oldmm2 39241	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)))
34	32, 22, 4, 33	syl3anc 1373	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)))
35	5, 6	opococ 39218	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
36	3, 4, 35	syl2anc 584	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
37	36	oveq2d 7426	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∧ 𝑌))
38	37	oveq2d 7426	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
39	30, 34, 38	3eqtr4d 2781	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌)))))
40	39	eqeq2d 2747	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) ↔ ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
41	28, 40	bitrd 279	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
42	16, 18, 41	3imtr4d 294	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  lecple 17283  occoc 17284  joincjn 18328  meetcmee 18329  Latclat 18446  OPcops 39195  OLcol 39197  OMLcoml 39198

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-proset 18311  df-poset 18330  df-lub 18361  df-glb 18362  df-join 18363  df-meet 18364  df-lat 18447  df-oposet 39199  df-ol 39201  df-oml 39202

This theorem is referenced by:  poml4N  39977  dihoml4c  41400