Theorem omllaw3 39245

Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 31372 analog.) (Contributed by NM, 19-Oct-2011.)

Hypotheses

Ref	Expression
omllaw3.b	⊢ 𝐵 = (Base‘𝐾)
omllaw3.l	⊢ ≤ = (le‘𝐾)
omllaw3.m	⊢ ∧ = (meet‘𝐾)
omllaw3.o	⊢ ⊥ = (oc‘𝐾)
omllaw3.z	⊢ 0 = (0.‘𝐾)

Assertion

Ref	Expression
omllaw3	⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = 𝑌))

Proof of Theorem omllaw3

Step	Hyp	Ref	Expression
1		oveq2 7398	. . . . . 6 ⊢ ((𝑌 ∧ ( ⊥ ‘𝑋)) = 0 → (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))) = (𝑋(join‘𝐾) 0 ))
2	1	adantl 481	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))) = (𝑋(join‘𝐾) 0 ))
3		omlol 39240	. . . . . . . 8 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
4		omllaw3.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5		eqid 2730	. . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾)
6		omllaw3.z	. . . . . . . . 9 ⊢ 0 = (0.‘𝐾)
7	4, 5, 6	olj01 39225	. . . . . . . 8 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
8	3, 7	sylan 580	. . . . . . 7 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
9	8	3adant3 1132	. . . . . 6 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(join‘𝐾) 0 ) = 𝑋)
10	9	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → (𝑋(join‘𝐾) 0 ) = 𝑋)
11	2, 10	eqtr2d 2766	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))))
12	11	adantrl 716	. . 3 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑋 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))))
13		omllaw3.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
14		omllaw3.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
15		omllaw3.o	. . . . . 6 ⊢ ⊥ = (oc‘𝐾)
16	4, 13, 5, 14, 15	omllaw 39243	. . . . 5 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋)))))
17	16	imp 406	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))))
18	17	adantrr 717	. . 3 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑌 = (𝑋(join‘𝐾)(𝑌 ∧ ( ⊥ ‘𝑋))))
19	12, 18	eqtr4d 2768	. 2 ⊢ (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 )) → 𝑋 = 𝑌)
20	19	ex 412	1 ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = 𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  occoc 17235  joincjn 18279  meetcmee 18280  0.cp0 18389  OLcol 39174  OMLcoml 39175

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-proset 18262  df-poset 18281  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-lat 18398  df-oposet 39176  df-ol 39178  df-oml 39179

This theorem is referenced by:  omlfh1N  39258  atlatmstc  39319