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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 4747 | . 2 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑈) |
6 | lspprss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | lspprss.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
8 | 6, 7 | lspssp 19689 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ {𝑋, 𝑌} ⊆ 𝑈) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
9 | 1, 2, 5, 8 | syl3anc 1363 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 {cpr 4559 ‘cfv 6348 LModclmod 19563 LSubSpclss 19632 LSpanclspn 19672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-lmod 19565 df-lss 19633 df-lsp 19673 |
This theorem is referenced by: lsppratlem2 19849 dvh3dim2 38464 dvh3dim3N 38465 lclkrlem2n 38536 |
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