Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lsmpr | Structured version Visualization version GIF version |
Description: The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.) |
Ref | Expression |
---|---|
lsmpr.v | β’ π = (Baseβπ) |
lsmpr.n | β’ π = (LSpanβπ) |
lsmpr.p | β’ β = (LSSumβπ) |
lsmpr.w | β’ (π β π β LMod) |
lsmpr.x | β’ (π β π β π) |
lsmpr.y | β’ (π β π β π) |
Ref | Expression |
---|---|
lsmpr | β’ (π β (πβ{π, π}) = ((πβ{π}) β (πβ{π}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsmpr.w | . . 3 β’ (π β π β LMod) | |
2 | lsmpr.x | . . . 4 β’ (π β π β π) | |
3 | 2 | snssd 4768 | . . 3 β’ (π β {π} β π) |
4 | lsmpr.y | . . . 4 β’ (π β π β π) | |
5 | 4 | snssd 4768 | . . 3 β’ (π β {π} β π) |
6 | lsmpr.v | . . . 4 β’ π = (Baseβπ) | |
7 | lsmpr.n | . . . 4 β’ π = (LSpanβπ) | |
8 | 6, 7 | lspun 20371 | . . 3 β’ ((π β LMod β§ {π} β π β§ {π} β π) β (πβ({π} βͺ {π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
9 | 1, 3, 5, 8 | syl3anc 1372 | . 2 β’ (π β (πβ({π} βͺ {π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
10 | df-pr 4588 | . . . 4 β’ {π, π} = ({π} βͺ {π}) | |
11 | 10 | fveq2i 6841 | . . 3 β’ (πβ{π, π}) = (πβ({π} βͺ {π})) |
12 | 11 | a1i 11 | . 2 β’ (π β (πβ{π, π}) = (πβ({π} βͺ {π}))) |
13 | eqid 2738 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
14 | 6, 13, 7 | lspsncl 20361 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
15 | 1, 2, 14 | syl2anc 585 | . . 3 β’ (π β (πβ{π}) β (LSubSpβπ)) |
16 | 6, 13, 7 | lspsncl 20361 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
17 | 1, 4, 16 | syl2anc 585 | . . 3 β’ (π β (πβ{π}) β (LSubSpβπ)) |
18 | lsmpr.p | . . . 4 β’ β = (LSSumβπ) | |
19 | 13, 7, 18 | lsmsp 20470 | . . 3 β’ ((π β LMod β§ (πβ{π}) β (LSubSpβπ) β§ (πβ{π}) β (LSubSpβπ)) β ((πβ{π}) β (πβ{π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
20 | 1, 15, 17, 19 | syl3anc 1372 | . 2 β’ (π β ((πβ{π}) β (πβ{π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
21 | 9, 12, 20 | 3eqtr4d 2788 | 1 β’ (π β (πβ{π, π}) = ((πβ{π}) β (πβ{π}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βͺ cun 3907 β wss 3909 {csn 4585 {cpr 4587 βcfv 6492 (class class class)co 7350 Basecbs 17018 LSSumclsm 19345 LModclmod 20245 LSubSpclss 20315 LSpanclspn 20355 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-0g 17258 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-submnd 18537 df-grp 18686 df-minusg 18687 df-sbg 18688 df-subg 18858 df-cntz 19029 df-lsm 19347 df-cmn 19493 df-abl 19494 df-mgp 19826 df-ur 19843 df-ring 19890 df-lmod 20247 df-lss 20316 df-lsp 20356 |
This theorem is referenced by: lsppreli 20474 lsmelpr 20475 lsppr0 20476 lspprabs 20479 lspabs2 20504 lspindpi 20516 lsmsat 37356 dvh4dimlem 39792 dvh3dim3N 39798 lclkrlem2c 39858 lcfrlem20 39911 lcfrlem23 39914 mapdindp 40020 mapdindp2 40070 mapdindp4 40072 mapdh6dN 40088 lspindp5 40119 hdmap1l6d 40162 hdmaprnlem3eN 40207 |
Copyright terms: Public domain | W3C validator |