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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnn0N | Structured version Visualization version GIF version |
Description: A lattice plane is nonzero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lplnn0.z | ⊢ 0 = (0.‘𝐾) |
lplnn0.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
Ref | Expression |
---|---|
lplnn0N | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → 𝑋 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | 1 | atex 36702 | . . . 4 ⊢ (𝐾 ∈ HL → (Atoms‘𝐾) ≠ ∅) |
3 | n0 4260 | . . . 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 ∈ 𝑃) → ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) |
6 | eqid 2798 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | lplnn0.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
8 | 6, 1, 7 | lplnnleat 36838 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
9 | 8 | 3expa 1115 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
10 | hlop 36658 | . . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
11 | 10 | ad2antrr 725 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP) |
12 | eqid 2798 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 1 | atbase 36585 | . . . . . . 7 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾)) |
14 | 13 | adantl 485 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Base‘𝐾)) |
15 | lplnn0.z | . . . . . . 7 ⊢ 0 = (0.‘𝐾) | |
16 | 12, 6, 15 | op0le 36482 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑝) |
17 | 11, 14, 16 | syl2anc 587 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 0 (le‘𝐾)𝑝) |
18 | breq1 5033 | . . . . 5 ⊢ (𝑋 = 0 → (𝑋(le‘𝐾)𝑝 ↔ 0 (le‘𝐾)𝑝)) | |
19 | 17, 18 | syl5ibrcom 250 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 = 0 → 𝑋(le‘𝐾)𝑝)) |
20 | 19 | necon3bd 3001 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑋(le‘𝐾)𝑝 → 𝑋 ≠ 0 )) |
21 | 9, 20 | mpd 15 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋 ≠ 0 ) |
22 | 5, 21 | exlimddv 1936 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 lecple 16564 0.cp0 17639 OPcops 36468 Atomscatm 36559 HLchlt 36646 LPlanesclpl 36788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 |
This theorem is referenced by: (None) |
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