Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 2821 | . . . 4 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
2 | lhpne0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
3 | lhpne0.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 1, 2, 3 | lhp0lt 37133 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (lt‘𝐾)𝑊) |
5 | simpl 485 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ HL) | |
6 | hlop 36492 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
7 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | 7, 2 | op0cl 36314 | . . . . . 6 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝐾 ∈ HL → 0 ∈ (Base‘𝐾)) |
10 | 9 | adantr 483 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ (Base‘𝐾)) |
11 | simpr 487 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ 𝐻) | |
12 | 1 | pltne 17566 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 0 ∈ (Base‘𝐾) ∧ 𝑊 ∈ 𝐻) → ( 0 (lt‘𝐾)𝑊 → 0 ≠ 𝑊)) |
13 | 5, 10, 11, 12 | syl3anc 1367 | . . 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 3071 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑊 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 Basecbs 16477 ltcplt 17545 0.cp0 17641 OPcops 36302 HLchlt 36480 LHypclh 37114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-lhyp 37118 |
This theorem is referenced by: lhpexle 37135 |
Copyright terms: Public domain | W3C validator |