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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opoc1 | Structured version Visualization version GIF version |
Description: Orthocomplement of orthoposet unity. (Contributed by NM, 24-Jan-2012.) |
Ref | Expression |
---|---|
opoc1.z | ⊢ 0 = (0.‘𝐾) |
opoc1.u | ⊢ 1 = (1.‘𝐾) |
opoc1.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
opoc1 | ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | opoc1.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
3 | 1, 2 | op0cl 38893 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
4 | opoc1.o | . . . . . 6 ⊢ ⊥ = (oc‘𝐾) | |
5 | 1, 4 | opoccl 38903 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 0 ∈ (Base‘𝐾)) → ( ⊥ ‘ 0 ) ∈ (Base‘𝐾)) |
6 | 3, 5 | mpdan 685 | . . . 4 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) ∈ (Base‘𝐾)) |
7 | eqid 2726 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | opoc1.u | . . . . 5 ⊢ 1 = (1.‘𝐾) | |
9 | 1, 7, 8 | ople1 38900 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘ 0 ) ∈ (Base‘𝐾)) → ( ⊥ ‘ 0 )(le‘𝐾) 1 ) |
10 | 6, 9 | mpdan 685 | . . 3 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 )(le‘𝐾) 1 ) |
11 | 1, 8 | op1cl 38894 | . . . 4 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
12 | 1, 7, 4 | oplecon1b 38910 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾) ∧ 0 ∈ (Base‘𝐾)) → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 0 )(le‘𝐾) 1 )) |
13 | 11, 3, 12 | mpd3an23 1460 | . . 3 ⊢ (𝐾 ∈ OP → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 0 )(le‘𝐾) 1 )) |
14 | 10, 13 | mpbird 256 | . 2 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 )(le‘𝐾) 0 ) |
15 | 1, 4 | opoccl 38903 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → ( ⊥ ‘ 1 ) ∈ (Base‘𝐾)) |
16 | 11, 15 | mpdan 685 | . . 3 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) ∈ (Base‘𝐾)) |
17 | 1, 7, 2 | ople0 38896 | . . 3 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘ 1 ) ∈ (Base‘𝐾)) → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 1 ) = 0 )) |
18 | 16, 17 | mpdan 685 | . 2 ⊢ (𝐾 ∈ OP → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 1 ) = 0 )) |
19 | 14, 18 | mpbid 231 | 1 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 class class class wbr 5144 ‘cfv 6544 Basecbs 17206 lecple 17266 occoc 17267 0.cp0 18441 1.cp1 18442 OPcops 38881 |
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 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-proset 18313 df-poset 18331 df-lub 18364 df-glb 18365 df-p0 18443 df-p1 18444 df-oposet 38885 |
This theorem is referenced by: opoc0 38912 olm11 38936 1cvrco 39182 1cvrjat 39185 pol1N 39620 doch1 41069 |
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