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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 475 | . . 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 35884 | . . . 4 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
5 | 4 | adantr 473 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
6 | simpr 477 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2772 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | atlltne0.s | . . . 4 ⊢ < = (lt‘𝐾) | |
9 | 7, 8 | pltval 17422 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
10 | 1, 5, 6, 9 | syl3anc 1351 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
11 | necom 3014 | . . 3 ⊢ (𝑋 ≠ 0 ↔ 0 ≠ 𝑋) | |
12 | 2, 7, 3 | atl0le 35885 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
13 | 12 | biantrurd 525 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 ≠ 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
14 | 11, 13 | syl5rbb 276 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋) ↔ 𝑋 ≠ 0 )) |
15 | 10, 14 | bitrd 271 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 class class class wbr 4923 ‘cfv 6182 Basecbs 16333 lecple 16422 ltcplt 17403 0.cp0 17499 AtLatcal 35845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-plt 17420 df-glb 17437 df-p0 17501 df-atl 35879 |
This theorem is referenced by: isat3 35888 |
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