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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 4767 | . . 3 β’ (π β {π} β π) |
4 | lsmpr.y | . . . 4 β’ (π β π β π) | |
5 | 4 | snssd 4767 | . . 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 1371 | . 2 β’ (π β (πβ({π} βͺ {π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
10 | df-pr 4587 | . . . 4 β’ {π, π} = ({π} βͺ {π}) | |
11 | 10 | fveq2i 6840 | . . 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 2737 | . . . . 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 584 | . . 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 584 | . . 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 1371 | . 2 β’ (π β ((πβ{π}) β (πβ{π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
21 | 9, 12, 20 | 3eqtr4d 2787 | 1 β’ (π β (πβ{π, π}) = ((πβ{π}) β (πβ{π}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 βͺ cun 3906 β wss 3908 {csn 4584 {cpr 4586 βcfv 6491 (class class class)co 7349 Basecbs 17017 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 1797 ax-4 1811 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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-ress 17047 df-plusg 17080 df-0g 17257 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-submnd 18536 df-grp 18685 df-minusg 18686 df-sbg 18687 df-subg 18857 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 37365 dvh4dimlem 39801 dvh3dim3N 39807 lclkrlem2c 39867 lcfrlem20 39920 lcfrlem23 39923 mapdindp 40029 mapdindp2 40079 mapdindp4 40081 mapdh6dN 40097 lspindp5 40128 hdmap1l6d 40171 hdmaprnlem3eN 40216 |
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