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Mirrors > Home > MPE Home > Th. List > lspsnel | Structured version Visualization version GIF version |
Description: Member of span of the singleton of a vector. (elspansn 29349 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspsn.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lspsn.k | ⊢ 𝐾 = (Base‘𝐹) |
lspsn.v | ⊢ 𝑉 = (Base‘𝑊) |
lspsn.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lspsn.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspsnel | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑈 ∈ (𝑁‘{𝑋}) ↔ ∃𝑘 ∈ 𝐾 𝑈 = (𝑘 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsn.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | lspsn.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
3 | lspsn.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lspsn.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
5 | lspsn.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | 1, 2, 3, 4, 5 | lspsn 19767 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}) |
7 | 6 | eleq2d 2875 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑈 ∈ (𝑁‘{𝑋}) ↔ 𝑈 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)})) |
8 | id 22 | . . . . 5 ⊢ (𝑈 = (𝑘 · 𝑋) → 𝑈 = (𝑘 · 𝑋)) | |
9 | ovex 7168 | . . . . 5 ⊢ (𝑘 · 𝑋) ∈ V | |
10 | 8, 9 | eqeltrdi 2898 | . . . 4 ⊢ (𝑈 = (𝑘 · 𝑋) → 𝑈 ∈ V) |
11 | 10 | rexlimivw 3241 | . . 3 ⊢ (∃𝑘 ∈ 𝐾 𝑈 = (𝑘 · 𝑋) → 𝑈 ∈ V) |
12 | eqeq1 2802 | . . . 4 ⊢ (𝑣 = 𝑈 → (𝑣 = (𝑘 · 𝑋) ↔ 𝑈 = (𝑘 · 𝑋))) | |
13 | 12 | rexbidv 3256 | . . 3 ⊢ (𝑣 = 𝑈 → (∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋) ↔ ∃𝑘 ∈ 𝐾 𝑈 = (𝑘 · 𝑋))) |
14 | 11, 13 | elab3 3622 | . 2 ⊢ (𝑈 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ↔ ∃𝑘 ∈ 𝐾 𝑈 = (𝑘 · 𝑋)) |
15 | 7, 14 | syl6bb 290 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑈 ∈ (𝑁‘{𝑋}) ↔ ∃𝑘 ∈ 𝐾 𝑈 = (𝑘 · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 Vcvv 3441 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 LModclmod 19627 LSpanclspn 19736 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mgp 19233 df-ur 19245 df-ring 19292 df-lmod 19629 df-lss 19697 df-lsp 19737 |
This theorem is referenced by: lspsnss2 19770 lsmspsn 19849 lspsneleq 19880 lspsneq 19887 lspdisj 19890 rspsn 20020 rspsnel 30987 ccfldextdgrr 31145 lshpdisj 36283 lshpsmreu 36405 lkrlspeqN 36467 lcfl7lem 38795 lcfrvalsnN 38837 mapdpglem3 38971 hdmapglem7a 39223 |
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