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Theorem opoc1 38674
Description: Orthocomplement of orthoposet unity. (Contributed by NM, 24-Jan-2012.)
Hypotheses
Ref Expression
opoc1.z 0 = (0.β€˜πΎ)
opoc1.u 1 = (1.β€˜πΎ)
opoc1.o βŠ₯ = (ocβ€˜πΎ)
Assertion
Ref Expression
opoc1 (𝐾 ∈ OP β†’ ( βŠ₯ β€˜ 1 ) = 0 )

Proof of Theorem opoc1
StepHypRef Expression
1 eqid 2728 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 opoc1.z . . . . . 6 0 = (0.β€˜πΎ)
31, 2op0cl 38656 . . . . 5 (𝐾 ∈ OP β†’ 0 ∈ (Baseβ€˜πΎ))
4 opoc1.o . . . . . 6 βŠ₯ = (ocβ€˜πΎ)
51, 4opoccl 38666 . . . . 5 ((𝐾 ∈ OP ∧ 0 ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜ 0 ) ∈ (Baseβ€˜πΎ))
63, 5mpdan 686 . . . 4 (𝐾 ∈ OP β†’ ( βŠ₯ β€˜ 0 ) ∈ (Baseβ€˜πΎ))
7 eqid 2728 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
8 opoc1.u . . . . 5 1 = (1.β€˜πΎ)
91, 7, 8ople1 38663 . . . 4 ((𝐾 ∈ OP ∧ ( βŠ₯ β€˜ 0 ) ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜ 0 )(leβ€˜πΎ) 1 )
106, 9mpdan 686 . . 3 (𝐾 ∈ OP β†’ ( βŠ₯ β€˜ 0 )(leβ€˜πΎ) 1 )
111, 8op1cl 38657 . . . 4 (𝐾 ∈ OP β†’ 1 ∈ (Baseβ€˜πΎ))
121, 7, 4oplecon1b 38673 . . . 4 ((𝐾 ∈ OP ∧ 1 ∈ (Baseβ€˜πΎ) ∧ 0 ∈ (Baseβ€˜πΎ)) β†’ (( βŠ₯ β€˜ 1 )(leβ€˜πΎ) 0 ↔ ( βŠ₯ β€˜ 0 )(leβ€˜πΎ) 1 ))
1311, 3, 12mpd3an23 1460 . . 3 (𝐾 ∈ OP β†’ (( βŠ₯ β€˜ 1 )(leβ€˜πΎ) 0 ↔ ( βŠ₯ β€˜ 0 )(leβ€˜πΎ) 1 ))
1410, 13mpbird 257 . 2 (𝐾 ∈ OP β†’ ( βŠ₯ β€˜ 1 )(leβ€˜πΎ) 0 )
151, 4opoccl 38666 . . . 4 ((𝐾 ∈ OP ∧ 1 ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜ 1 ) ∈ (Baseβ€˜πΎ))
1611, 15mpdan 686 . . 3 (𝐾 ∈ OP β†’ ( βŠ₯ β€˜ 1 ) ∈ (Baseβ€˜πΎ))
171, 7, 2ople0 38659 . . 3 ((𝐾 ∈ OP ∧ ( βŠ₯ β€˜ 1 ) ∈ (Baseβ€˜πΎ)) β†’ (( βŠ₯ β€˜ 1 )(leβ€˜πΎ) 0 ↔ ( βŠ₯ β€˜ 1 ) = 0 ))
1816, 17mpdan 686 . 2 (𝐾 ∈ OP β†’ (( βŠ₯ β€˜ 1 )(leβ€˜πΎ) 0 ↔ ( βŠ₯ β€˜ 1 ) = 0 ))
1914, 18mpbid 231 1 (𝐾 ∈ OP β†’ ( βŠ₯ β€˜ 1 ) = 0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099   class class class wbr 5148  β€˜cfv 6548  Basecbs 17179  lecple 17239  occoc 17240  0.cp0 18414  1.cp1 18415  OPcops 38644
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
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-proset 18286  df-poset 18304  df-lub 18337  df-glb 18338  df-p0 18416  df-p1 18417  df-oposet 38648
This theorem is referenced by:  opoc0  38675  olm11  38699  1cvrco  38945  1cvrjat  38948  pol1N  39383  doch1  40832
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