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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 2725 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | eqid 2725 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
6 | lshpne.h | . . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊) | |
7 | 3, 4, 5, 6 | islshp 38601 | . . . 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 231 | . 2 ⊢ (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉)) |
10 | 9 | simp2d 1140 | 1 ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 ∪ cun 3942 {csn 4630 ‘cfv 6549 Basecbs 17199 LModclmod 20772 LSubSpclss 20844 LSpanclspn 20884 LSHypclsh 38597 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-lshyp 38599 |
This theorem is referenced by: lshpnel 38605 lshpcmp 38610 lkrshp3 38728 lkrshp4 38730 dochshpncl 41007 dochlkr 41008 dochkrshp 41009 dochsatshpb 41075 |
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