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 2818 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | 1 | atex 36422 | . . . 4 ⊢ (𝐾 ∈ HL → (Atoms‘𝐾) ≠ ∅) |
3 | n0 4307 | . . . 4 ⊢ ((Atoms‘𝐾) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) | |
4 | 2, 3 | sylib 219 | . . 3 ⊢ (𝐾 ∈ HL → ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) |
5 | 4 | adantr 481 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) |
6 | eqid 2818 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | lplnn0.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
8 | 6, 1, 7 | lplnnleat 36558 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
9 | 8 | 3expa 1110 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
10 | hlop 36378 | . . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
11 | 10 | ad2antrr 722 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP) |
12 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 1 | atbase 36305 | . . . . . . 7 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾)) |
14 | 13 | adantl 482 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Base‘𝐾)) |
15 | lplnn0.z | . . . . . . 7 ⊢ 0 = (0.‘𝐾) | |
16 | 12, 6, 15 | op0le 36202 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑝) |
17 | 11, 14, 16 | syl2anc 584 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 0 (le‘𝐾)𝑝) |
18 | breq1 5060 | . . . . 5 ⊢ (𝑋 = 0 → (𝑋(le‘𝐾)𝑝 ↔ 0 (le‘𝐾)𝑝)) | |
19 | 17, 18 | syl5ibrcom 248 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 = 0 → 𝑋(le‘𝐾)𝑝)) |
20 | 19 | necon3bd 3027 | . . 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 1927 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 lecple 16560 0.cp0 17635 OPcops 36188 Atomscatm 36279 HLchlt 36366 LPlanesclpl 36508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 |
This theorem is referenced by: (None) |
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