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Mathbox for Norm Megill |
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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 28642 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 34995 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
5 | 4 | biantrud 521 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ (𝑋 ≤ 0 ∧ 0 ≤ 𝑋))) |
6 | opposet 34990 | . . . 4 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | |
7 | 6 | adantr 466 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Poset) |
8 | simpr 471 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
9 | 1, 3 | op0cl 34993 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
10 | 9 | adantr 466 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
11 | 1, 2 | posasymb 17160 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → ((𝑋 ≤ 0 ∧ 0 ≤ 𝑋) ↔ 𝑋 = 0 )) |
12 | 7, 8, 10, 11 | syl3anc 1476 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((𝑋 ≤ 0 ∧ 0 ≤ 𝑋) ↔ 𝑋 = 0 )) |
13 | 5, 12 | bitrd 268 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ‘cfv 6031 Basecbs 16064 lecple 16156 Posetcpo 17148 0.cp0 17245 OPcops 34981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-preset 17136 df-poset 17154 df-glb 17183 df-p0 17247 df-oposet 34985 |
This theorem is referenced by: lub0N 34998 opoc1 35011 atlatmstc 35128 cvrat4 35251 lhpocnle 35824 cdleme22b 36150 tendoid 36582 tendoex 36784 |
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