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Mirrors > Home > MPE Home > Th. List > Mathboxes > atlltn0 | Structured version Visualization version GIF version |
Description: A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.) |
Ref | Expression |
---|---|
atlltne0.b | ⊢ 𝐵 = (Base‘𝐾) |
atlltne0.s | ⊢ < = (lt‘𝐾) |
atlltne0.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
atlltn0 | ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ AtLat) | |
2 | atlltne0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | atlltne0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
4 | 2, 3 | atl0cl 39285 | . . . 4 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
6 | simpr 484 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2735 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | atlltne0.s | . . . 4 ⊢ < = (lt‘𝐾) | |
9 | 7, 8 | pltval 18390 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
10 | 1, 5, 6, 9 | syl3anc 1370 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
11 | necom 2992 | . . 3 ⊢ (𝑋 ≠ 0 ↔ 0 ≠ 𝑋) | |
12 | 2, 7, 3 | atl0le 39286 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
13 | 12 | biantrurd 532 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 ≠ 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
14 | 11, 13 | bitr2id 284 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋) ↔ 𝑋 ≠ 0 )) |
15 | 10, 14 | bitrd 279 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 lecple 17305 ltcplt 18366 0.cp0 18481 AtLatcal 39246 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-plt 18388 df-glb 18405 df-p0 18483 df-atl 39280 |
This theorem is referenced by: isat3 39289 |
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