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Mirrors > Home > MPE Home > Th. List > lsppr | Structured version Visualization version GIF version |
Description: Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.) |
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 |
---|---|
lsppr | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4569 | . . 3 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
2 | 1 | fveq2i 6672 | . 2 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
3 | lsppr.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
4 | lsppr.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
5 | 4 | snssd 4741 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
6 | lsppr.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
7 | 6 | snssd 4741 | . . . 4 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
8 | lsppr.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | lsppr.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | 8, 9 | lspun 19758 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
11 | 3, 5, 7, 10 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
12 | eqid 2821 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
13 | 8, 12, 9 | lspsncl 19748 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
14 | 3, 4, 13 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
15 | 8, 12, 9 | lspsncl 19748 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
16 | 3, 6, 15 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | eqid 2821 | . . . . 5 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
18 | 12, 9, 17 | lsmsp 19857 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
19 | 3, 14, 16, 18 | syl3anc 1367 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
20 | lsppr.a | . . . . 5 ⊢ + = (+g‘𝑊) | |
21 | lsppr.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
22 | lsppr.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
23 | lsppr.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
24 | 8, 20, 21, 22, 23, 17, 9, 3, 4, 6 | lsmspsn 19855 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)))) |
25 | 24 | abbi2dv 2950 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
26 | 11, 19, 25 | 3eqtr2d 2862 | . 2 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
27 | 2, 26 | syl5eq 2868 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 ∪ cun 3933 ⊆ wss 3935 {csn 4566 {cpr 4568 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 Scalarcsca 16567 ·𝑠 cvsca 16568 LSSumclsm 18758 LModclmod 19633 LSubSpclss 19702 LSpanclspn 19742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-subg 18275 df-cntz 18446 df-lsm 18760 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-lmod 19635 df-lss 19703 df-lsp 19743 |
This theorem is referenced by: lspprel 19865 |
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