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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpne | Structured version Visualization version GIF version |
Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.) |
Ref | Expression |
---|---|
lshpne.v | ⊢ 𝑉 = (Base‘𝑊) |
lshpne.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lshpne.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lshpne.u | ⊢ (𝜑 → 𝑈 ∈ 𝐻) |
Ref | Expression |
---|---|
lshpne | ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpne.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
2 | lshpne.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | lshpne.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2821 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | eqid 2821 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
6 | lshpne.h | . . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊) | |
7 | 3, 4, 5, 6 | islshp 36114 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉))) |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉))) |
9 | 1, 8 | mpbid 234 | . 2 ⊢ (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉)) |
10 | 9 | simp2d 1139 | 1 ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ∪ cun 3933 {csn 4566 ‘cfv 6354 Basecbs 16482 LModclmod 19633 LSubSpclss 19702 LSpanclspn 19742 LSHypclsh 36110 |
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-sep 5202 ax-nul 5209 ax-pr 5329 |
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-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-lshyp 36112 |
This theorem is referenced by: lshpnel 36118 lshpcmp 36123 lkrshp3 36241 lkrshp4 36243 dochshpncl 38519 dochlkr 38520 dochkrshp 38521 dochsatshpb 38587 |
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