![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > lspsnid | GIF version |
Description: A vector belongs to the span of its singleton. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspsnid.v | ⊢ 𝑉 = (Base‘𝑊) |
lspsnid.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspsnid | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 3748 | . . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑋} ⊆ 𝑉) | |
2 | lspsnid.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lspsnid.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | 2, 3 | lspssid 13589 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉) → {𝑋} ⊆ (𝑁‘{𝑋})) |
5 | 1, 4 | sylan2 286 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑋} ⊆ (𝑁‘{𝑋})) |
6 | snssg 3738 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ (𝑁‘{𝑋}) ↔ {𝑋} ⊆ (𝑁‘{𝑋}))) | |
7 | 6 | adantl 277 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ (𝑁‘{𝑋}) ↔ {𝑋} ⊆ (𝑁‘{𝑋}))) |
8 | 5, 7 | mpbird 167 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 ⊆ wss 3141 {csn 3604 ‘cfv 5228 Basecbs 12476 LModclmod 13476 LSpanclspn 13575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-cnex 7916 ax-resscn 7917 ax-1re 7919 ax-addrcl 7922 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-5 8995 df-6 8996 df-ndx 12479 df-slot 12480 df-base 12482 df-plusg 12564 df-mulr 12565 df-sca 12567 df-vsca 12568 df-0g 12725 df-mgm 12794 df-sgrp 12827 df-mnd 12840 df-grp 12902 df-lmod 13478 df-lssm 13542 df-lsp 13576 |
This theorem is referenced by: lspsnel6 13597 lssats2 13603 lspsneli 13604 lspsn 13605 lspsneq0 13615 |
Copyright terms: Public domain | W3C validator |