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Mirrors > Home > MPE Home > Th. List > Mathboxes > opoc1 | Structured version Visualization version GIF version |
Description: Orthocomplement of orthoposet unit. (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 2818 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | opoc1.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
3 | 1, 2 | op0cl 36200 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
4 | opoc1.o | . . . . . 6 ⊢ ⊥ = (oc‘𝐾) | |
5 | 1, 4 | opoccl 36210 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 0 ∈ (Base‘𝐾)) → ( ⊥ ‘ 0 ) ∈ (Base‘𝐾)) |
6 | 3, 5 | mpdan 683 | . . . 4 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) ∈ (Base‘𝐾)) |
7 | eqid 2818 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | opoc1.u | . . . . 5 ⊢ 1 = (1.‘𝐾) | |
9 | 1, 7, 8 | ople1 36207 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘ 0 ) ∈ (Base‘𝐾)) → ( ⊥ ‘ 0 )(le‘𝐾) 1 ) |
10 | 6, 9 | mpdan 683 | . . 3 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 )(le‘𝐾) 1 ) |
11 | 1, 8 | op1cl 36201 | . . . 4 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
12 | 1, 7, 4 | oplecon1b 36217 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾) ∧ 0 ∈ (Base‘𝐾)) → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 0 )(le‘𝐾) 1 )) |
13 | 11, 3, 12 | mpd3an23 1454 | . . 3 ⊢ (𝐾 ∈ OP → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 0 )(le‘𝐾) 1 )) |
14 | 10, 13 | mpbird 258 | . 2 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 )(le‘𝐾) 0 ) |
15 | 1, 4 | opoccl 36210 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → ( ⊥ ‘ 1 ) ∈ (Base‘𝐾)) |
16 | 11, 15 | mpdan 683 | . . 3 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) ∈ (Base‘𝐾)) |
17 | 1, 7, 2 | ople0 36203 | . . 3 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘ 1 ) ∈ (Base‘𝐾)) → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 1 ) = 0 )) |
18 | 16, 17 | mpdan 683 | . 2 ⊢ (𝐾 ∈ OP → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 1 ) = 0 )) |
19 | 14, 18 | mpbid 233 | 1 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 lecple 16560 occoc 16561 0.cp0 17635 1.cp1 17636 OPcops 36188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-proset 17526 df-poset 17544 df-lub 17572 df-glb 17573 df-p0 17637 df-p1 17638 df-oposet 36192 |
This theorem is referenced by: opoc0 36219 olm11 36243 1cvrco 36488 1cvrjat 36491 pol1N 36926 doch1 38375 |
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