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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 2727 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2727 | . . . . 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 38388 | . . . 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 231 | . 2 ⊢ (𝜑 → (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊))) |
10 | 9 | simp1d 1140 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∃wrex 3065 ∪ cun 3942 {csn 4624 ‘cfv 6542 Basecbs 17171 LModclmod 20732 LSubSpclss 20804 LSpanclspn 20844 LSHypclsh 38384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-lshyp 38386 |
This theorem is referenced by: lshpnel 38392 lshpnelb 38393 lshpne0 38395 lshpdisj 38396 lshpcmp 38397 lshpsmreu 38518 lshpkrlem1 38519 lshpkrlem5 38523 lshpkr 38526 dochshpncl 40794 dochshpsat 40864 lclkrlem2f 40922 |
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