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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opoc0 | Structured version Visualization version GIF version |
Description: Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.) |
Ref | Expression |
---|---|
opoc1.z | ⊢ 0 = (0.‘𝐾) |
opoc1.u | ⊢ 1 = (1.‘𝐾) |
opoc1.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
opoc0 | ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opoc1.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
2 | opoc1.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
3 | opoc1.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
4 | 1, 2, 3 | opoc1 39184 | . 2 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 ) |
5 | eqid 2735 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 5, 2 | op1cl 39167 | . . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
7 | 5, 1 | op0cl 39166 | . . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
8 | 5, 3 | opcon1b 39180 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾) ∧ 0 ∈ (Base‘𝐾)) → (( ⊥ ‘ 1 ) = 0 ↔ ( ⊥ ‘ 0 ) = 1 )) |
9 | 6, 7, 8 | mpd3an23 1462 | . 2 ⊢ (𝐾 ∈ OP → (( ⊥ ‘ 1 ) = 0 ↔ ( ⊥ ‘ 0 ) = 1 )) |
10 | 4, 9 | mpbid 232 | 1 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 Basecbs 17245 occoc 17306 0.cp0 18481 1.cp1 18482 OPcops 39154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-proset 18352 df-poset 18371 df-lub 18404 df-glb 18405 df-p0 18483 df-p1 18484 df-oposet 39158 |
This theorem is referenced by: 1cvrjat 39458 doch0 41341 |
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