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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 4586 | . 2 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑈) |
6 | lspprss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | lspprss.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | 6, 7 | lspssp 19394 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ {𝑋, 𝑌} ⊆ 𝑈) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
9 | 1, 2, 5, 8 | syl3anc 1439 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 {cpr 4400 ‘cfv 6137 LModclmod 19266 LSubSpclss 19335 LSpanclspn 19377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-0g 16499 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-grp 17823 df-lmod 19268 df-lss 19336 df-lsp 19378 |
This theorem is referenced by: lsppratlem2 19556 dvh3dim2 37611 dvh3dim3N 37612 lclkrlem2n 37683 |
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