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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 | β’ π = (Baseβπ) |
lsppr.a | β’ + = (+gβπ) |
lsppr.f | β’ πΉ = (Scalarβπ) |
lsppr.k | β’ πΎ = (BaseβπΉ) |
lsppr.t | β’ Β· = ( Β·π βπ) |
lsppr.n | β’ π = (LSpanβπ) |
lsppr.w | β’ (π β π β LMod) |
lsppr.x | β’ (π β π β π) |
lsppr.y | β’ (π β π β π) |
Ref | Expression |
---|---|
lsppr | β’ (π β (πβ{π, π}) = {π£ β£ βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4624 | . . 3 β’ {π, π} = ({π} βͺ {π}) | |
2 | 1 | fveq2i 6885 | . 2 β’ (πβ{π, π}) = (πβ({π} βͺ {π})) |
3 | lsppr.w | . . . 4 β’ (π β π β LMod) | |
4 | lsppr.x | . . . . 5 β’ (π β π β π) | |
5 | 4 | snssd 4805 | . . . 4 β’ (π β {π} β π) |
6 | lsppr.y | . . . . 5 β’ (π β π β π) | |
7 | 6 | snssd 4805 | . . . 4 β’ (π β {π} β π) |
8 | lsppr.v | . . . . 5 β’ π = (Baseβπ) | |
9 | lsppr.n | . . . . 5 β’ π = (LSpanβπ) | |
10 | 8, 9 | lspun 20826 | . . . 4 β’ ((π β LMod β§ {π} β π β§ {π} β π) β (πβ({π} βͺ {π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
11 | 3, 5, 7, 10 | syl3anc 1368 | . . 3 β’ (π β (πβ({π} βͺ {π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
12 | eqid 2724 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
13 | 8, 12, 9 | lspsncl 20816 | . . . . 5 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
14 | 3, 4, 13 | syl2anc 583 | . . . 4 β’ (π β (πβ{π}) β (LSubSpβπ)) |
15 | 8, 12, 9 | lspsncl 20816 | . . . . 5 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
16 | 3, 6, 15 | syl2anc 583 | . . . 4 β’ (π β (πβ{π}) β (LSubSpβπ)) |
17 | eqid 2724 | . . . . 5 β’ (LSSumβπ) = (LSSumβπ) | |
18 | 12, 9, 17 | lsmsp 20926 | . . . 4 β’ ((π β LMod β§ (πβ{π}) β (LSubSpβπ) β§ (πβ{π}) β (LSubSpβπ)) β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
19 | 3, 14, 16, 18 | syl3anc 1368 | . . 3 β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
20 | lsppr.a | . . . . 5 β’ + = (+gβπ) | |
21 | lsppr.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
22 | lsppr.k | . . . . 5 β’ πΎ = (BaseβπΉ) | |
23 | lsppr.t | . . . . 5 β’ Β· = ( Β·π βπ) | |
24 | 8, 20, 21, 22, 23, 17, 9, 3, 4, 6 | lsmspsn 20924 | . . . 4 β’ (π β (π£ β ((πβ{π})(LSSumβπ)(πβ{π})) β βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π)))) |
25 | 24 | eqabdv 2859 | . . 3 β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = {π£ β£ βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π))}) |
26 | 11, 19, 25 | 3eqtr2d 2770 | . 2 β’ (π β (πβ({π} βͺ {π})) = {π£ β£ βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π))}) |
27 | 2, 26 | eqtrid 2776 | 1 β’ (π β (πβ{π, π}) = {π£ β£ βπ β πΎ βπ β πΎ π£ = ((π Β· π) + (π Β· π))}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {cab 2701 βwrex 3062 βͺ cun 3939 β wss 3941 {csn 4621 {cpr 4623 βcfv 6534 (class class class)co 7402 Basecbs 17145 +gcplusg 17198 Scalarcsca 17201 Β·π cvsca 17202 LSSumclsm 19546 LModclmod 20698 LSubSpclss 20770 LSpanclspn 20810 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-cntz 19225 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20032 df-ur 20079 df-ring 20132 df-lmod 20700 df-lss 20771 df-lsp 20811 |
This theorem is referenced by: lspprel 20934 |
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