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Theorem omllaw3 38103
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 30676 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 7413 . . . . . 6 ((π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 β†’ (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))) = (𝑋(joinβ€˜πΎ) 0 ))
21adantl 482 . . . . 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 38098 . . . . . . . 8 (𝐾 ∈ OML β†’ 𝐾 ∈ OL)
4 omllaw3.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΎ)
5 eqid 2732 . . . . . . . . 9 (joinβ€˜πΎ) = (joinβ€˜πΎ)
6 omllaw3.z . . . . . . . . 9 0 = (0.β€˜πΎ)
74, 5, 6olj01 38083 . . . . . . . 8 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
83, 7sylan 580 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
983adant3 1132 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
109adantr 481 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 ) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
112, 10eqtr2d 2773 . . . 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 714 . . 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 38101 . . . . 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 407 . . . 4 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ π‘Œ = (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))
1817adantrr 715 . . 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 2775 . 2 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑋 ≀ π‘Œ ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 )) β†’ 𝑋 = π‘Œ)
2019ex 413 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  occoc 17201  joincjn 18260  meetcmee 18261  0.cp0 18372  OLcol 38032  OMLcoml 38033
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-oposet 38034  df-ol 38036  df-oml 38037
This theorem is referenced by:  omlfh1N  38116  atlatmstc  38177
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