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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 33507 | . 2 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 ) |
5 | eqid 2610 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 5, 2 | op1cl 33490 | . . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
7 | 5, 1 | op0cl 33489 | . . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
8 | 5, 3 | opcon1b 33503 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾) ∧ 0 ∈ (Base‘𝐾)) → (( ⊥ ‘ 1 ) = 0 ↔ ( ⊥ ‘ 0 ) = 1 )) |
9 | 6, 7, 8 | mpd3an23 1418 | . 2 ⊢ (𝐾 ∈ OP → (( ⊥ ‘ 1 ) = 0 ↔ ( ⊥ ‘ 0 ) = 1 )) |
10 | 4, 9 | mpbid 221 | 1 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 Basecbs 15695 occoc 15776 0.cp0 16860 1.cp1 16861 OPcops 33477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-preset 16751 df-poset 16769 df-lub 16797 df-glb 16798 df-p0 16862 df-p1 16863 df-oposet 33481 |
This theorem is referenced by: 1cvrjat 33779 doch0 35665 |
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