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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpn0 | Structured version Visualization version GIF version |
Description: A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.) |
Ref | Expression |
---|---|
lhpne0.z | ⊢ 0 = (0.‘𝐾) |
lhpne0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpn0 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . . 4 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
2 | lhpne0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
3 | lhpne0.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 1, 2, 3 | lhp0lt 39465 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (lt‘𝐾)𝑊) |
5 | simpl 482 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ HL) | |
6 | hlop 38823 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
7 | eqid 2727 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | 7, 2 | op0cl 38645 | . . . . . 6 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝐾 ∈ HL → 0 ∈ (Base‘𝐾)) |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ (Base‘𝐾)) |
11 | simpr 484 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ 𝐻) | |
12 | 1 | pltne 18319 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 0 ∈ (Base‘𝐾) ∧ 𝑊 ∈ 𝐻) → ( 0 (lt‘𝐾)𝑊 → 0 ≠ 𝑊)) |
13 | 5, 10, 11, 12 | syl3anc 1369 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( 0 (lt‘𝐾)𝑊 → 0 ≠ 𝑊)) |
14 | 4, 13 | mpd 15 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ≠ 𝑊) |
15 | 14 | necomd 2991 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 ‘cfv 6542 Basecbs 17173 ltcplt 18293 0.cp0 18408 OPcops 38633 HLchlt 38811 LHypclh 39446 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-oposet 38637 df-ol 38639 df-oml 38640 df-covers 38727 df-ats 38728 df-atl 38759 df-cvlat 38783 df-hlat 38812 df-lhyp 39450 |
This theorem is referenced by: lhpexle 39467 |
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