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Mirrors > Home > MPE Home > Th. List > lspsnss | Structured version Visualization version GIF version |
Description: The span of the singleton of a subspace member is included in the subspace. (spansnss 30511 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.) |
Ref | Expression |
---|---|
lspsnss.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lspsnss.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspsnss | ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4768 | . 2 ⊢ (𝑋 ∈ 𝑈 → {𝑋} ⊆ 𝑈) | |
2 | lspsnss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lspsnss.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 2, 3 | lspssp 20447 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ {𝑋} ⊆ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
5 | 1, 4 | syl3an3 1165 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 {csn 4586 ‘cfv 6496 LModclmod 20320 LSubSpclss 20390 LSpanclspn 20430 |
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 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 |
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 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-0g 17322 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-grp 18750 df-lmod 20322 df-lss 20391 df-lsp 20431 |
This theorem is referenced by: lspsnel3 20450 lspsnel6 20453 |
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