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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 2823 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | 1 | atex 36544 | . . . 4 ⊢ (𝐾 ∈ HL → (Atoms‘𝐾) ≠ ∅) |
3 | n0 4312 | . . . 4 ⊢ ((Atoms‘𝐾) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) | |
4 | 2, 3 | sylib 220 | . . 3 ⊢ (𝐾 ∈ HL → ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) |
5 | 4 | adantr 483 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) |
6 | eqid 2823 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | llnn0.n | . . . . 5 ⊢ 𝑁 = (LLines‘𝐾) | |
8 | 6, 1, 7 | llnnleat 36651 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
9 | 8 | 3expa 1114 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
10 | hlop 36500 | . . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
11 | 10 | ad2antrr 724 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP) |
12 | eqid 2823 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 1 | atbase 36427 | . . . . . . 7 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾)) |
14 | 13 | adantl 484 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Base‘𝐾)) |
15 | llnn0.z | . . . . . . 7 ⊢ 0 = (0.‘𝐾) | |
16 | 12, 6, 15 | op0le 36324 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑝) |
17 | 11, 14, 16 | syl2anc 586 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 0 (le‘𝐾)𝑝) |
18 | breq1 5071 | . . . . 5 ⊢ (𝑋 = 0 → (𝑋(le‘𝐾)𝑝 ↔ 0 (le‘𝐾)𝑝)) | |
19 | 17, 18 | syl5ibrcom 249 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 = 0 → 𝑋(le‘𝐾)𝑝)) |
20 | 19 | necon3bd 3032 | . . 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 1936 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 lecple 16574 0.cp0 17649 OPcops 36310 Atomscatm 36401 HLchlt 36488 LLinesclln 36629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 |
This theorem is referenced by: 2llnm3N 36707 cdleme22b 37479 |
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