Theorem omllaw4 40120

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 1154	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
2		omlop 40115	. . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
3	2	3ad2ant1 1151	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
4		simp3 1156	. . . 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 40068	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
8	3, 4, 7	syl2anc 596	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
9		simp2 1155	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
10	5, 6	opoccl 40068	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
11	3, 9, 10	syl2anc 596	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
12		omllaw4.l	. . . 4 ⊢ ≤ = (le‘𝐾)
13		eqid 2760	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
14		omllaw4.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
15	5, 12, 13, 14, 6	omllaw 40117	. . 3 ⊢ ((𝐾 ∈ OML ∧ ( ⊥ ‘𝑌) ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋) → ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
16	1, 8, 11, 15	syl3anc 1398	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋) → ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
17	5, 12, 6	oplecon3b 40074	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
18	2, 17	syl3an1 1181	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
19		omllat 40116	. . . . . 6 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
20	19	3ad2ant1 1151	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
21	5, 14	latmcl 18529	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵)
22	20, 11, 4, 21	syl3anc 1398	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵)
23	5, 6	opoccl 40068	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∈ 𝐵)
24	3, 22, 23	syl2anc 596	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∈ 𝐵)
25	5, 14	latmcl 18529	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) ∈ 𝐵)
26	20, 24, 4, 25	syl3anc 1398	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) ∈ 𝐵)
27	5, 6	opcon3b 40070	. . . 4 ⊢ ((𝐾 ∈ OP ∧ (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌))))
28	3, 26, 9, 27	syl3anc 1398	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌))))
29	5, 13	latjcom 18536	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
30	20, 22, 8, 29	syl3anc 1398	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
31		omlol 40114	. . . . . . 7 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
32	31	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
33	5, 13, 14, 6	oldmm2 40092	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)))
34	32, 22, 4, 33	syl3anc 1398	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)))
35	5, 6	opococ 40069	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
36	3, 4, 35	syl2anc 596	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
37	36	oveq2d 7430	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∧ 𝑌))
38	37	oveq2d 7430	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
39	30, 34, 38	3eqtr4d 2805	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌)))))
40	39	eqeq2d 2771	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) = ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) ↔ ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
41	28, 40	bitrd 282	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
42	16, 18, 41	3imtr4d 297	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  lecple 17350  occoc 17351  joincjn 18400  meetcmee 18401  Latclat 18520  OPcops 40046  OLcol 40048  OMLcoml 40049

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-proset 18383  df-poset 18402  df-lub 18433  df-glb 18434  df-join 18435  df-meet 18436  df-lat 18521  df-oposet 40050  df-ol 40052  df-oml 40053

This theorem is used by:  poml4N  40827  dihoml4c  42250