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Theorem omllaw3 37710
Description: Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 30381 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 7366 . . . . . 6 ((π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 β†’ (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))) = (𝑋(joinβ€˜πΎ) 0 ))
21adantl 483 . . . . 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 37705 . . . . . . . 8 (𝐾 ∈ OML β†’ 𝐾 ∈ OL)
4 omllaw3.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΎ)
5 eqid 2737 . . . . . . . . 9 (joinβ€˜πΎ) = (joinβ€˜πΎ)
6 omllaw3.z . . . . . . . . 9 0 = (0.β€˜πΎ)
74, 5, 6olj01 37690 . . . . . . . 8 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
83, 7sylan 581 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
983adant3 1133 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
109adantr 482 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 ) β†’ (𝑋(joinβ€˜πΎ) 0 ) = 𝑋)
112, 10eqtr2d 2778 . . . 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 715 . . 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 37708 . . . . 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 408 . . . 4 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ π‘Œ = (𝑋(joinβ€˜πΎ)(π‘Œ ∧ ( βŠ₯ β€˜π‘‹))))
1817adantrr 716 . . 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 2780 . 2 (((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑋 ≀ π‘Œ ∧ (π‘Œ ∧ ( βŠ₯ β€˜π‘‹)) = 0 )) β†’ 𝑋 = π‘Œ)
2019ex 414 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lecple 17141  occoc 17142  joincjn 18201  meetcmee 18202  0.cp0 18313  OLcol 37639  OMLcoml 37640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18185  df-poset 18203  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-lat 18322  df-oposet 37641  df-ol 37643  df-oml 37644
This theorem is referenced by:  omlfh1N  37723  atlatmstc  37784
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