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Mirrors > Home > MPE Home > Th. List > Mathboxes > atlle0 | Structured version Visualization version GIF version |
Description: An element less than or equal to zero equals zero. (chle0 29214 analog.) (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
atl0le.b | ⊢ 𝐵 = (Base‘𝐾) |
atl0le.l | ⊢ ≤ = (le‘𝐾) |
atl0le.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
atlle0 | ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atl0le.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | atl0le.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | atl0le.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | atl0le 36434 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
5 | 4 | biantrud 534 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ (𝑋 ≤ 0 ∧ 0 ≤ 𝑋))) |
6 | atlpos 36431 | . . . 4 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Poset) |
8 | simpr 487 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
9 | 1, 3 | atl0cl 36433 | . . . 4 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
11 | 1, 2 | posasymb 17556 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → ((𝑋 ≤ 0 ∧ 0 ≤ 𝑋) ↔ 𝑋 = 0 )) |
12 | 7, 8, 10, 11 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ((𝑋 ≤ 0 ∧ 0 ≤ 𝑋) ↔ 𝑋 = 0 )) |
13 | 5, 12 | bitrd 281 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 Basecbs 16477 lecple 16566 Posetcpo 17544 0.cp0 17641 AtLatcal 36394 |
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-proset 17532 df-poset 17550 df-glb 17579 df-p0 17643 df-lat 17650 df-atl 36428 |
This theorem is referenced by: dia0 38182 |
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