Theorem omllaw4 39909

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 1152	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OML)
2		omlop 39904	. . . . 5 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
3	2	3ad2ant1 1149	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OP)
4		simp3 1154	. . . 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 39857	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
8	3, 4, 7	syl2anc 595	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
9		simp2 1153	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
10	5, 6	opoccl 39857	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
11	3, 9, 10	syl2anc 595	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
12		omllaw4.l	. . . 4 ⊢ ≤ = (le‘𝐾)
13		eqid 2769	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
14		omllaw4.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
15	5, 12, 13, 14, 6	omllaw 39906	. . 3 ⊢ ((𝐾 ∈ OML ∧ ( ⊥ ‘𝑌) ∈ 𝐵 ∧ ( ⊥ ‘𝑋) ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋) → ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
16	1, 8, 11, 15	syl3anc 1396	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋) → ( ⊥ ‘𝑋) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))))))
17	5, 12, 6	oplecon3b 39863	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
18	2, 17	syl3an1 1179	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
19		omllat 39905	. . . . . 6 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
20	19	3ad2ant1 1149	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
21	5, 14	latmcl 18495	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵)
22	20, 11, 4, 21	syl3anc 1396	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵)
23	5, 6	opoccl 39857	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∈ 𝐵)
24	3, 22, 23	syl2anc 595	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∈ 𝐵)
25	5, 14	latmcl 18495	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) ∈ 𝐵)
26	20, 24, 4, 25	syl3anc 1396	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) ∈ 𝐵)
27	5, 6	opcon3b 39859	. . . 4 ⊢ ((𝐾 ∈ OP ∧ (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌))))
28	3, 26, 9, 27	syl3anc 1396	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋 ↔ ( ⊥ ‘𝑋) = ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌))))
29	5, 13	latjcom 18502	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
30	20, 22, 8, 29	syl3anc 1396	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
31		omlol 39903	. . . . . . 7 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
32	31	3ad2ant1 1149	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
33	5, 13, 14, 6	oldmm2 39881	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (( ⊥ ‘𝑋) ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)))
34	32, 22, 4, 33	syl3anc 1396	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = ((( ⊥ ‘𝑋) ∧ 𝑌)(join‘𝐾)( ⊥ ‘𝑌)))
35	5, 6	opococ 39858	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
36	3, 4, 35	syl2anc 595	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
37	36	oveq2d 7427	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∧ 𝑌))
38	37	oveq2d 7427	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
39	30, 34, 38	3eqtr4d 2814	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌)))))
40	39	eqeq2d 2780	. . 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  lecple 17316  occoc 17317  joincjn 18366  meetcmee 18367  Latclat 18486  OPcops 39835  OLcol 39837  OMLcoml 39838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-proset 18349  df-poset 18368  df-lub 18399  df-glb 18400  df-join 18401  df-meet 18402  df-lat 18487  df-oposet 39839  df-ol 39841  df-oml 39842

This theorem is referenced by:  poml4N  40616  dihoml4c  42039