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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 4813 | . . 3 β’ (π β {π} β π) |
| 4 | lsmpr.y | . . . 4 β’ (π β π β π) | |
| 5 | 4 | snssd 4813 | . . 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 20744 | . . 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 1369 | . 2 β’ (π β (πβ({π} βͺ {π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
| 10 | df-pr 4632 | . . . 4 β’ {π, π} = ({π} βͺ {π}) | |
| 11 | 10 | fveq2i 6895 | . . 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 2730 | . . . . 5 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 14 | 6, 13, 7 | lspsncl 20734 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
| 15 | 1, 2, 14 | syl2anc 582 | . . 3 β’ (π β (πβ{π}) β (LSubSpβπ)) |
| 16 | 6, 13, 7 | lspsncl 20734 | . . . 4 β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
| 17 | 1, 4, 16 | syl2anc 582 | . . 3 β’ (π β (πβ{π}) β (LSubSpβπ)) |
| 18 | lsmpr.p | . . . 4 β’ β = (LSSumβπ) | |
| 19 | 13, 7, 18 | lsmsp 20843 | . . 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 1369 | . 2 β’ (π β ((πβ{π}) β (πβ{π})) = (πβ((πβ{π}) βͺ (πβ{π})))) |
| 21 | 9, 12, 20 | 3eqtr4d 2780 | 1 β’ (π β (πβ{π, π}) = ((πβ{π}) β (πβ{π}))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1539 β wcel 2104 βͺ cun 3947 β wss 3949 {csn 4629 {cpr 4631 βcfv 6544 (class class class)co 7413 Basecbs 17150 LSSumclsm 19545 LModclmod 20616 LSubSpclss 20688 LSpanclspn 20728 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-0g 17393 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18708 df-grp 18860 df-minusg 18861 df-sbg 18862 df-subg 19041 df-cntz 19224 df-lsm 19547 df-cmn 19693 df-abl 19694 df-mgp 20031 df-ur 20078 df-ring 20131 df-lmod 20618 df-lss 20689 df-lsp 20729 |
| This theorem is referenced by: lsppreli 20847 lsmelpr 20848 lsppr0 20849 lspprabs 20852 lspabs2 20880 lspindpi 20892 lsmsat 38183 dvh4dimlem 40619 dvh3dim3N 40625 lclkrlem2c 40685 lcfrlem20 40738 lcfrlem23 40741 mapdindp 40847 mapdindp2 40897 mapdindp4 40899 mapdh6dN 40915 lspindp5 40946 hdmap1l6d 40989 hdmaprnlem3eN 41034 |
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