Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lspprel | Structured version Visualization version GIF version |
Description: Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.) |
Ref | Expression |
---|---|
lsppr.v | ⊢ 𝑉 = (Base‘𝑊) |
lsppr.a | ⊢ + = (+g‘𝑊) |
lsppr.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lsppr.k | ⊢ 𝐾 = (Base‘𝐹) |
lsppr.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lsppr.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lsppr.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsppr.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lsppr.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lspprel | ⊢ (𝜑 → (𝑍 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsppr.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lsppr.a | . . . 4 ⊢ + = (+g‘𝑊) | |
3 | lsppr.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | lsppr.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
5 | lsppr.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
6 | lsppr.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
7 | lsppr.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
8 | lsppr.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
9 | lsppr.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | lsppr 20344 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
11 | 10 | eleq2d 2824 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑁‘{𝑋, 𝑌}) ↔ 𝑍 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))})) |
12 | id 22 | . . . . . 6 ⊢ (𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)) → 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))) | |
13 | ovex 7302 | . . . . . 6 ⊢ ((𝑘 · 𝑋) + (𝑙 · 𝑌)) ∈ V | |
14 | 12, 13 | eqeltrdi 2847 | . . . . 5 ⊢ (𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)) → 𝑍 ∈ V) |
15 | 14 | rexlimivw 3210 | . . . 4 ⊢ (∃𝑙 ∈ 𝐾 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)) → 𝑍 ∈ V) |
16 | 15 | rexlimivw 3210 | . . 3 ⊢ (∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)) → 𝑍 ∈ V) |
17 | eqeq1 2742 | . . . 4 ⊢ (𝑣 = 𝑍 → (𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)) ↔ 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)))) | |
18 | 17 | 2rexbidv 3228 | . . 3 ⊢ (𝑣 = 𝑍 → (∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)) ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)))) |
19 | 16, 18 | elab3 3618 | . 2 ⊢ (𝑍 ∈ {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))} ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))) |
20 | 11, 19 | bitrdi 287 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 {cab 2715 ∃wrex 3065 Vcvv 3431 {cpr 4565 ‘cfv 6428 (class class class)co 7269 Basecbs 16901 +gcplusg 16951 Scalarcsca 16954 ·𝑠 cvsca 16955 LModclmod 20112 LSpanclspn 20222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8487 df-en 8723 df-dom 8724 df-sdom 8725 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-nn 11963 df-2 12025 df-sets 16854 df-slot 16872 df-ndx 16884 df-base 16902 df-ress 16931 df-plusg 16964 df-0g 17141 df-mgm 18315 df-sgrp 18364 df-mnd 18375 df-submnd 18420 df-grp 18569 df-minusg 18570 df-sbg 18571 df-subg 18741 df-cntz 18912 df-lsm 19230 df-cmn 19377 df-abl 19378 df-mgp 19710 df-ur 19727 df-ring 19774 df-lmod 20114 df-lss 20183 df-lsp 20223 |
This theorem is referenced by: lspfixed 20379 lspexch 20380 ccfldextdgrr 31729 |
Copyright terms: Public domain | W3C validator |