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Mirrors > Home > MPE Home > Th. List > lspprss | Structured version Visualization version GIF version |
Description: The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.) |
Ref | Expression |
---|---|
lspprss.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspprss.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspprss.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspprss.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lspprss.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
lspprss.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
Ref | Expression |
---|---|
lspprss | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspprss.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lspprss.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
3 | lspprss.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
4 | lspprss.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
5 | 3, 4 | prssd 4721 | . 2 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑈) |
6 | lspprss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | lspprss.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | 6, 7 | lspssp 19979 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ {𝑋, 𝑌} ⊆ 𝑈) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
9 | 1, 2, 5, 8 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 {cpr 4529 ‘cfv 6358 LModclmod 19853 LSubSpclss 19922 LSpanclspn 19962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-lmod 19855 df-lss 19923 df-lsp 19963 |
This theorem is referenced by: lsppratlem2 20139 dvh3dim2 39148 dvh3dim3N 39149 lclkrlem2n 39220 |
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