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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 2798 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2798 | . . . . 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 36275 | . . . 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 235 | . 2 ⊢ (𝜑 → (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊))) |
10 | 9 | simp1d 1139 | 1 ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ∪ cun 3879 {csn 4525 ‘cfv 6324 Basecbs 16475 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 LSHypclsh 36271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-lshyp 36273 |
This theorem is referenced by: lshpnel 36279 lshpnelb 36280 lshpne0 36282 lshpdisj 36283 lshpcmp 36284 lshpsmreu 36405 lshpkrlem1 36406 lshpkrlem5 36410 lshpkr 36413 dochshpncl 38680 dochshpsat 38750 lclkrlem2f 38808 |
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