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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnn0 | Structured version Visualization version GIF version |
Description: A lattice line is nonzero. (Contributed by NM, 15-Jul-2012.) |
Ref | Expression |
---|---|
llnn0.z | ⊢ 0 = (0.‘𝐾) |
llnn0.n | ⊢ 𝑁 = (LLines‘𝐾) |
Ref | Expression |
---|---|
llnn0 | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | 1 | atex 37170 | . . . 4 ⊢ (𝐾 ∈ HL → (Atoms‘𝐾) ≠ ∅) |
3 | n0 4270 | . . . 4 ⊢ ((Atoms‘𝐾) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) | |
4 | 2, 3 | sylib 221 | . . 3 ⊢ (𝐾 ∈ HL → ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) |
5 | 4 | adantr 484 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) |
6 | eqid 2738 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | llnn0.n | . . . . 5 ⊢ 𝑁 = (LLines‘𝐾) | |
8 | 6, 1, 7 | llnnleat 37277 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
9 | 8 | 3expa 1120 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
10 | hlop 37126 | . . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
11 | 10 | ad2antrr 726 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP) |
12 | eqid 2738 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 1 | atbase 37053 | . . . . . . 7 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾)) |
14 | 13 | adantl 485 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Base‘𝐾)) |
15 | llnn0.z | . . . . . . 7 ⊢ 0 = (0.‘𝐾) | |
16 | 12, 6, 15 | op0le 36950 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑝) |
17 | 11, 14, 16 | syl2anc 587 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 0 (le‘𝐾)𝑝) |
18 | breq1 5065 | . . . . 5 ⊢ (𝑋 = 0 → (𝑋(le‘𝐾)𝑝 ↔ 0 (le‘𝐾)𝑝)) | |
19 | 17, 18 | syl5ibrcom 250 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 = 0 → 𝑋(le‘𝐾)𝑝)) |
20 | 19 | necon3bd 2955 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑋(le‘𝐾)𝑝 → 𝑋 ≠ 0 )) |
21 | 9, 20 | mpd 15 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋 ≠ 0 ) |
22 | 5, 21 | exlimddv 1943 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2111 ≠ wne 2941 ∅c0 4246 class class class wbr 5062 ‘cfv 6389 Basecbs 16773 lecple 16822 0.cp0 17942 OPcops 36936 Atomscatm 37027 HLchlt 37114 LLinesclln 37255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-id 5464 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-riota 7179 df-ov 7225 df-oprab 7226 df-proset 17815 df-poset 17833 df-plt 17849 df-lub 17865 df-glb 17866 df-join 17867 df-meet 17868 df-p0 17944 df-p1 17945 df-lat 17951 df-clat 18018 df-oposet 36940 df-ol 36942 df-oml 36943 df-covers 37030 df-ats 37031 df-atl 37062 df-cvlat 37086 df-hlat 37115 df-llines 37262 |
This theorem is referenced by: 2llnm3N 37333 cdleme22b 38105 |
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