| Mathbox for Norm Megill |
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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 2737 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
| 5 | eqid 2737 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
| 6 | lshpne.h | . . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊) | |
| 7 | 3, 4, 5, 6 | islshp 38980 | . . . 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 232 | . 2 ⊢ (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉)) |
| 10 | 9 | simp2d 1144 | 1 ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∪ cun 3949 {csn 4626 ‘cfv 6561 Basecbs 17247 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 LSHypclsh 38976 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-lshyp 38978 |
| This theorem is referenced by: lshpnel 38984 lshpcmp 38989 lkrshp3 39107 lkrshp4 39109 dochshpncl 41386 dochlkr 41387 dochkrshp 41388 dochsatshpb 41454 |
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