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Mirrors > Home > MPE Home > Th. List > Mathboxes > ople0 | Structured version Visualization version GIF version |
Description: An element less than or equal to zero equals zero. (chle0 29147 analog.) (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
op0le.b | ⊢ 𝐵 = (Base‘𝐾) |
op0le.l | ⊢ ≤ = (le‘𝐾) |
op0le.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
ople0 | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op0le.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | op0le.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | op0le.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | op0le 36202 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
5 | 4 | biantrud 532 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ (𝑋 ≤ 0 ∧ 0 ≤ 𝑋))) |
6 | opposet 36197 | . . . 4 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Poset) |
8 | simpr 485 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
9 | 1, 3 | op0cl 36200 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
11 | 1, 2 | posasymb 17550 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → ((𝑋 ≤ 0 ∧ 0 ≤ 𝑋) ↔ 𝑋 = 0 )) |
12 | 7, 8, 10, 11 | syl3anc 1363 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((𝑋 ≤ 0 ∧ 0 ≤ 𝑋) ↔ 𝑋 = 0 )) |
13 | 5, 12 | bitrd 280 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 lecple 16560 Posetcpo 17538 0.cp0 17635 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-glb 17573 df-p0 17637 df-oposet 36192 |
This theorem is referenced by: lub0N 36205 opoc1 36218 atlatmstc 36335 cvrat4 36459 lhpocnle 37032 cdleme22b 37357 tendoid 37789 tendoex 37991 |
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