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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > op1le | Structured version Visualization version GIF version |
Description: If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 28856 analog.) (Contributed by NM, 5-Dec-2011.) |
Ref | Expression |
---|---|
ople1.b | ⊢ 𝐵 = (Base‘𝐾) |
ople1.l | ⊢ ≤ = (le‘𝐾) |
ople1.u | ⊢ 1 = (1.‘𝐾) |
Ref | Expression |
---|---|
op1le | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 1 ≤ 𝑋 ↔ 𝑋 = 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ople1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | ople1.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | ople1.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
4 | 1, 2, 3 | ople1 35265 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 1 ) |
5 | 4 | biantrurd 530 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 1 ≤ 𝑋 ↔ (𝑋 ≤ 1 ∧ 1 ≤ 𝑋))) |
6 | opposet 35255 | . . . 4 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | |
7 | 6 | adantr 474 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Poset) |
8 | simpr 479 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
9 | 1, 3 | op1cl 35259 | . . . 4 ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵) |
10 | 9 | adantr 474 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 1 ∈ 𝐵) |
11 | 1, 2 | posasymb 17304 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵) → ((𝑋 ≤ 1 ∧ 1 ≤ 𝑋) ↔ 𝑋 = 1 )) |
12 | 7, 8, 10, 11 | syl3anc 1496 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((𝑋 ≤ 1 ∧ 1 ≤ 𝑋) ↔ 𝑋 = 1 )) |
13 | 5, 12 | bitrd 271 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 1 ≤ 𝑋 ↔ 𝑋 = 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4872 ‘cfv 6122 Basecbs 16221 lecple 16311 Posetcpo 17292 1.cp1 17390 OPcops 35246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-proset 17280 df-poset 17298 df-lub 17326 df-p1 17392 df-oposet 35250 |
This theorem is referenced by: glb0N 35267 lhpj1 36096 |
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