Theorem omllaw2N 39704

Description: Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 31671 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
omllaw2N	⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌))

Proof of Theorem omllaw2N

Step	Hyp	Ref	Expression
1		omllaw.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		omllaw.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		omllaw.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		omllaw.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		omllaw.o	. . 3 ⊢ ⊥ = (oc‘𝐾)
6	1, 2, 3, 4, 5	omllaw 39703	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))
7		eqcom 2744	. . 3 ⊢ ((𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌 ↔ 𝑌 = (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)))
8		omllat 39702	. . . . . . 7 ⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
9	8	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat)
10		omlop 39701	. . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
11	1, 5	opoccl 39654	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
12	10, 11	sylan 581	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
13	12	3adant3 1133	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵)
14		simp3 1139	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
15	1, 4	latmcom 18420	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) = (𝑌 ∧ ( ⊥ ‘𝑋)))
16	9, 13, 14, 15	syl3anc 1374	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ∧ 𝑌) = (𝑌 ∧ ( ⊥ ‘𝑋)))
17	16	oveq2d 7376	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))
18	17	eqeq2d 2748	. . 3 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 = (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))
19	7, 18	bitrid 283	. 2 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌 ↔ 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋)))))
20	6, 19	sylibrd 259	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  lecple 17218  occoc 17219  joincjn 18268  meetcmee 18269  Latclat 18388  OPcops 39632  OMLcoml 39635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-glb 18302  df-meet 18304  df-lat 18389  df-oposet 39636  df-ol 39638  df-oml 39639

This theorem is referenced by:  omllaw5N  39707  cmtcomlemN  39708  cmtbr3N  39714