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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 2728 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | 1 | atex 38874 | . . . 4 ⊢ (𝐾 ∈ HL → (Atoms‘𝐾) ≠ ∅) |
3 | n0 4343 | . . . 4 ⊢ ((Atoms‘𝐾) ≠ ∅ ↔ ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) | |
4 | 2, 3 | sylib 217 | . . 3 ⊢ (𝐾 ∈ HL → ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) |
5 | 4 | adantr 480 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ∃𝑝 𝑝 ∈ (Atoms‘𝐾)) |
6 | eqid 2728 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
7 | lplnn0.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
8 | 6, 1, 7 | lplnnleat 39010 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
9 | 8 | 3expa 1116 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ¬ 𝑋(le‘𝐾)𝑝) |
10 | hlop 38829 | . . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
11 | 10 | ad2antrr 725 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ OP) |
12 | eqid 2728 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | 12, 1 | atbase 38756 | . . . . . . 7 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾)) |
14 | 13 | adantl 481 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Base‘𝐾)) |
15 | lplnn0.z | . . . . . . 7 ⊢ 0 = (0.‘𝐾) | |
16 | 12, 6, 15 | op0le 38653 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑝) |
17 | 11, 14, 16 | syl2anc 583 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 0 (le‘𝐾)𝑝) |
18 | breq1 5146 | . . . . 5 ⊢ (𝑋 = 0 → (𝑋(le‘𝐾)𝑝 ↔ 0 (le‘𝐾)𝑝)) | |
19 | 17, 18 | syl5ibrcom 246 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑋 = 0 → 𝑋(le‘𝐾)𝑝)) |
20 | 19 | necon3bd 2950 | . . 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 1931 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2936 ∅c0 4319 class class class wbr 5143 ‘cfv 6543 Basecbs 17174 lecple 17234 0.cp0 18409 OPcops 38639 Atomscatm 38730 HLchlt 38817 LPlanesclpl 38960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-oposet 38643 df-ol 38645 df-oml 38646 df-covers 38733 df-ats 38734 df-atl 38765 df-cvlat 38789 df-hlat 38818 df-llines 38966 df-lplanes 38967 |
This theorem is referenced by: (None) |
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