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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 4623 | . . 3 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
2 | 1 | fveq2i 6884 | . 2 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
3 | lsppr.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
4 | lsppr.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
5 | 4 | snssd 4804 | . . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
6 | lsppr.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
7 | 6 | snssd 4804 | . . . 4 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
8 | lsppr.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | lsppr.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | 8, 9 | lspun 20823 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
11 | 3, 5, 7, 10 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝑁‘((𝑁‘{𝑋}) ∪ (𝑁‘{𝑌})))) |
12 | eqid 2724 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
13 | 8, 12, 9 | lspsncl 20813 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
14 | 3, 4, 13 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑊)) |
15 | 8, 12, 9 | lspsncl 20813 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
16 | 3, 6, 15 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑊)) |
17 | eqid 2724 | . . . . 5 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
18 | 12, 9, 17 | lsmsp 20923 | . . . 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 1368 | . . 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 20921 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)))) |
25 | 24 | eqabdv 2859 | . . 3 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
26 | 11, 19, 25 | 3eqtr2d 2770 | . 2 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) |
27 | 2, 26 | eqtrid 2776 | 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 2098 {cab 2701 ∃wrex 3062 ∪ cun 3938 ⊆ wss 3940 {csn 4620 {cpr 4622 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 +gcplusg 17195 Scalarcsca 17198 ·𝑠 cvsca 17199 LSSumclsm 19543 LModclmod 20695 LSubSpclss 20767 LSpanclspn 20807 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-0g 17385 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19039 df-cntz 19222 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-ur 20076 df-ring 20129 df-lmod 20697 df-lss 20768 df-lsp 20808 |
This theorem is referenced by: lspprel 20931 |
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