Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 31453 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 4813 | . 2 ⊢ (𝑋 ∈ 𝑈 → {𝑋} ⊆ 𝑈) | |
| 2 | lspsnss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 3 | lspsnss.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 4 | 2, 3 | lspssp 20884 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ {𝑋} ⊆ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
| 5 | 1, 4 | syl3an3 1162 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 {csn 4630 ‘cfv 6549 LModclmod 20755 LSubSpclss 20827 LSpanclspn 20867 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-lmod 20757 df-lss 20828 df-lsp 20868 |
| This theorem is referenced by: lspsnel3 20887 lspsnel6 20890 |
| Copyright terms: Public domain | W3C validator |