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Theorem omllaw3 38581
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 31123 analog.) (Contributed by NM, 19-Oct-2011.)
Hypotheses
Ref Expression
omllaw3.b 𝐡 = (Baseβ€˜πΎ)
omllaw3.l ≀ = (leβ€˜πΎ)
omllaw3.m ∧ = (meetβ€˜πΎ)
omllaw3.o βŠ₯ = (ocβ€˜πΎ)
omllaw3.z 0 = (0.β€˜πΎ)
Assertion
Ref Expression
omllaw3 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 ) β†’ 𝑋 = π‘Œ))
Proof of Theorem omllaw3
StepHypRef Expression
1 oveq2 7420 . . . . . 6 ((π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 β†’ (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))) = (𝑋(joinβ€˜πΎ) 0 ))
21adantl 481 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 ) β†’ (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))) = (𝑋(joinβ€˜πΎ) 0 ))
3 omlol 38576 . . . . . . . 8 (𝐾 ∈ OML β†’ 𝐾 ∈ OL)
4 omllaw3.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΎ)
5 eqid 2731 . . . . . . . . 9 (joinβ€˜πΎ) = (joinβ€˜πΎ)
6 omllaw3.z . . . . . . . . 9 0 = (0.β€˜πΎ)
74, 5, 6olj01 38561 . . . . . . . 8 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
83, 7sylan 579 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
983adant3 1131 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
109adantr 480 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 ) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
112, 10eqtr2d 2772 . . . 4 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 ) β†’ 𝑋 = (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))
1211adantrl 713 . . 3 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑋 ≀ π‘Œ ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 )) β†’ 𝑋 = (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))
13 omllaw3.l . . . . . 6 ≀ = (leβ€˜πΎ)
14 omllaw3.m . . . . . 6 ∧ = (meetβ€˜πΎ)
15 omllaw3.o . . . . . 6 βŠ₯ = (ocβ€˜πΎ)
164, 13, 5, 14, 15omllaw 38579 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ≀ π‘Œ β†’ π‘Œ = (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹)))))
1716imp 406 . . . 4 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ π‘Œ = (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))
1817adantrr 714 . . 3 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑋 ≀ π‘Œ ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 )) β†’ π‘Œ = (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))
1912, 18eqtr4d 2774 . 2 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑋 ≀ π‘Œ ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 )) β†’ 𝑋 = π‘Œ)
2019ex 412 1 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 ≀ π‘Œ ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 ) β†’ 𝑋 = π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211  occoc 17212  joincjn 18274  meetcmee 18275  0.cp0 18386  OLcol 38510  OMLcoml 38511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18258  df-poset 18276  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-oposet 38512  df-ol 38514  df-oml 38515
This theorem is referenced by:  omlfh1N  38594  atlatmstc  38655
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