Theorem omllaw4 39225

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 39220	. . . . 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 39173	. . . 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 39173	. . . 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 2729	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
14		omllaw4.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
15	5, 12, 13, 14, 6	omllaw 39222	. . 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 39179	. . 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 39221	. . . . . 6 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
20	19	3ad2ant1 1133	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
21	5, 14	latmcl 18346	. . . . . . 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 39173	. . . . . 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 18346	. . . . 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 39175	. . . 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 18353	. . . . . 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 39219	. . . . . . 7 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
32	31	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ OL)
33	5, 13, 14, 6	oldmm2 39197	. . . . . 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 39174	. . . . . . . 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 7365	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∧ 𝑌))
38	37	oveq2d 7365	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌)))) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ 𝑌)))
39	30, 34, 38	3eqtr4d 2774	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌)) = (( ⊥ ‘𝑌)(join‘𝐾)(( ⊥ ‘𝑋) ∧ ( ⊥ ‘( ⊥ ‘𝑌)))))
40	39	eqeq2d 2740	. . 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 5092  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  lecple 17168  occoc 17169  joincjn 18217  meetcmee 18218  Latclat 18337  OPcops 39151  OLcol 39153  OMLcoml 39154

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-proset 18200  df-poset 18219  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-lat 18338  df-oposet 39155  df-ol 39157  df-oml 39158

This theorem is referenced by:  poml4N  39932  dihoml4c  41355