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Theorem olm11 38699
Description: The meet of an ortholattice element with one equals itself. (chm1i 31279 analog.) (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
olm1.b 𝐡 = (Baseβ€˜πΎ)
olm1.m ∧ = (meetβ€˜πΎ)
olm1.u 1 = (1.β€˜πΎ)
Assertion
Ref Expression
olm11 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∧ 1 ) = 𝑋)

Proof of Theorem olm11
StepHypRef Expression
1 olop 38686 . . . . . . 7 (𝐾 ∈ OL β†’ 𝐾 ∈ OP)
21adantr 480 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ 𝐾 ∈ OP)
3 eqid 2728 . . . . . . 7 (0.β€˜πΎ) = (0.β€˜πΎ)
4 olm1.u . . . . . . 7 1 = (1.β€˜πΎ)
5 eqid 2728 . . . . . . 7 (ocβ€˜πΎ) = (ocβ€˜πΎ)
63, 4, 5opoc1 38674 . . . . . 6 (𝐾 ∈ OP β†’ ((ocβ€˜πΎ)β€˜ 1 ) = (0.β€˜πΎ))
72, 6syl 17 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜ 1 ) = (0.β€˜πΎ))
87oveq2d 7436 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ (((ocβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜ 1 )) = (((ocβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)(0.β€˜πΎ)))
9 olm1.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
109, 5opoccl 38666 . . . . . 6 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
111, 10sylan 579 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡)
12 eqid 2728 . . . . . 6 (joinβ€˜πΎ) = (joinβ€˜πΎ)
139, 12, 3olj01 38697 . . . . 5 ((𝐾 ∈ OL ∧ ((ocβ€˜πΎ)β€˜π‘‹) ∈ 𝐡) β†’ (((ocβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)(0.β€˜πΎ)) = ((ocβ€˜πΎ)β€˜π‘‹))
1411, 13syldan 590 . . . 4 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ (((ocβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)(0.β€˜πΎ)) = ((ocβ€˜πΎ)β€˜π‘‹))
158, 14eqtrd 2768 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ (((ocβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜ 1 )) = ((ocβ€˜πΎ)β€˜π‘‹))
1615fveq2d 6901 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜ 1 ))) = ((ocβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘‹)))
179, 4op1cl 38657 . . . 4 (𝐾 ∈ OP β†’ 1 ∈ 𝐡)
182, 17syl 17 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ 1 ∈ 𝐡)
19 olm1.m . . . 4 ∧ = (meetβ€˜πΎ)
209, 12, 19, 5oldmj4 38696 . . 3 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡 ∧ 1 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜ 1 ))) = (𝑋 ∧ 1 ))
2118, 20mpd3an3 1459 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜(((ocβ€˜πΎ)β€˜π‘‹)(joinβ€˜πΎ)((ocβ€˜πΎ)β€˜ 1 ))) = (𝑋 ∧ 1 ))
229, 5opococ 38667 . . 3 ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘‹)) = 𝑋)
231, 22sylan 579 . 2 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ ((ocβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜π‘‹)) = 𝑋)
2416, 21, 233eqtr3d 2776 1 ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∧ 1 ) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  occoc 17241  joincjn 18303  meetcmee 18304  0.cp0 18415  1.cp1 18416  OPcops 38644  OLcol 38646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-proset 18287  df-poset 18305  df-lub 18338  df-glb 18339  df-join 18340  df-meet 18341  df-p0 18417  df-p1 18418  df-lat 18424  df-oposet 38648  df-ol 38650
This theorem is referenced by:  olm12  38700  lhpmcvr3  39498  trljat1  39639  trljat2  39640  cdlemc1  39664  cdlemc6  39669  cdleme0cp  39687  cdleme0cq  39688  cdleme1  39700  cdleme4  39711  cdleme5  39713  cdleme8  39723  cdleme9  39726  cdleme10  39727  cdleme20c  39784  cdleme20j  39791  cdleme22e  39817  cdleme22eALTN  39818  cdleme30a  39851  cdleme35b  39923  cdleme35e  39926  cdleme42a  39944  trlcoabs2N  40195  trlcolem  40199  cdlemi1  40291  cdlemk4  40307  dia2dimlem1  40537  cdlemn10  40679  dihglbcpreN  40773
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