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 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 2821 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | opoc1.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
3 | 1, 2 | op0cl 36314 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
4 | opoc1.o | . . . . . 6 ⊢ ⊥ = (oc‘𝐾) | |
5 | 1, 4 | opoccl 36324 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 0 ∈ (Base‘𝐾)) → ( ⊥ ‘ 0 ) ∈ (Base‘𝐾)) |
6 | 3, 5 | mpdan 685 | . . . 4 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) ∈ (Base‘𝐾)) |
7 | eqid 2821 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | opoc1.u | . . . . 5 ⊢ 1 = (1.‘𝐾) | |
9 | 1, 7, 8 | ople1 36321 | . . . 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 36315 | . . . 4 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
12 | 1, 7, 4 | oplecon1b 36331 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾) ∧ 0 ∈ (Base‘𝐾)) → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 0 )(le‘𝐾) 1 )) |
13 | 11, 3, 12 | mpd3an23 1459 | . . 3 ⊢ (𝐾 ∈ OP → (( ⊥ ‘ 1 )(le‘𝐾) 0 ↔ ( ⊥ ‘ 0 )(le‘𝐾) 1 )) |
14 | 10, 13 | mpbird 259 | . 2 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 )(le‘𝐾) 0 ) |
15 | 1, 4 | opoccl 36324 | . . . 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 36317 | . . 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 234 | 1 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 Basecbs 16477 lecple 16566 occoc 16567 0.cp0 17641 1.cp1 17642 OPcops 36302 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-proset 17532 df-poset 17550 df-lub 17578 df-glb 17579 df-p0 17643 df-p1 17644 df-oposet 36306 |
This theorem is referenced by: opoc0 36333 olm11 36357 1cvrco 36602 1cvrjat 36605 pol1N 37040 doch1 38489 |
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