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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshplss | Structured version Visualization version GIF version |
Description: A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.) |
Ref | Expression |
---|---|
lshplss.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lshplss.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lshplss.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lshplss.u | ⊢ (𝜑 → 𝑈 ∈ 𝐻) |
Ref | Expression |
---|---|
lshplss | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshplss.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
2 | lshplss.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | eqid 2735 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2735 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | lshplss.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
6 | lshplss.h | . . . . 5 ⊢ 𝐻 = (LSHyp‘𝑊) | |
7 | 3, 4, 5, 6 | islshp 38961 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))) |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))) |
9 | 1, 8 | mpbid 232 | . 2 ⊢ (𝜑 → (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊))) |
10 | 9 | simp1d 1141 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 ∪ cun 3961 {csn 4631 ‘cfv 6563 Basecbs 17245 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 LSHypclsh 38957 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-lshyp 38959 |
This theorem is referenced by: lshpnel 38965 lshpnelb 38966 lshpne0 38968 lshpdisj 38969 lshpcmp 38970 lshpsmreu 39091 lshpkrlem1 39092 lshpkrlem5 39096 lshpkr 39099 dochshpncl 41367 dochshpsat 41437 lclkrlem2f 41495 |
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