Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 37477 | . 2 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 ) |
5 | eqid 2736 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | 5, 2 | op1cl 37460 | . . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
7 | 5, 1 | op0cl 37459 | . . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
8 | 5, 3 | opcon1b 37473 | . . 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 231 | 1 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ‘cfv 6479 Basecbs 17009 occoc 17067 0.cp0 18238 1.cp1 18239 OPcops 37447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-proset 18110 df-poset 18128 df-lub 18161 df-glb 18162 df-p0 18240 df-p1 18241 df-oposet 37451 |
This theorem is referenced by: 1cvrjat 37751 doch0 39634 |
Copyright terms: Public domain | W3C validator |