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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 4780 | . 2 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑈) |
6 | lspprss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | lspprss.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | 6, 7 | lspssp 20434 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ {𝑋, 𝑌} ⊆ 𝑈) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
9 | 1, 2, 5, 8 | syl3anc 1371 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 {cpr 4586 ‘cfv 6493 LModclmod 20307 LSubSpclss 20377 LSpanclspn 20417 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-0g 17315 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-grp 18743 df-lmod 20309 df-lss 20378 df-lsp 20418 |
This theorem is referenced by: lsppratlem2 20594 dvh3dim2 39878 dvh3dim3N 39879 lclkrlem2n 39950 |
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