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| Mirrors > Home > MPE Home > Th. List > lspindp3 | Structured version Visualization version GIF version | ||
| Description: Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.) |
| Ref | Expression |
|---|---|
| lspindp3.v | β’ π = (Baseβπ) |
| lspindp3.p | β’ + = (+gβπ) |
| lspindp3.o | β’ 0 = (0gβπ) |
| lspindp3.n | β’ π = (LSpanβπ) |
| lspindp3.w | β’ (π β π β LVec) |
| lspindp3.x | β’ (π β π β π) |
| lspindp3.y | β’ (π β π β (π β { 0 })) |
| lspindp3.e | β’ (π β (πβ{π}) β (πβ{π})) |
| Ref | Expression |
|---|---|
| lspindp3 | β’ (π β (πβ{π}) β (πβ{(π + π)})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lspindp3.e | . 2 β’ (π β (πβ{π}) β (πβ{π})) | |
| 2 | lspindp3.v | . . . . 5 β’ π = (Baseβπ) | |
| 3 | lspindp3.p | . . . . 5 β’ + = (+gβπ) | |
| 4 | lspindp3.o | . . . . 5 β’ 0 = (0gβπ) | |
| 5 | lspindp3.n | . . . . 5 β’ π = (LSpanβπ) | |
| 6 | lspindp3.w | . . . . . 6 β’ (π β π β LVec) | |
| 7 | 6 | adantr 479 | . . . . 5 β’ ((π β§ (πβ{π}) = (πβ{(π + π)})) β π β LVec) |
| 8 | lspindp3.x | . . . . . 6 β’ (π β π β π) | |
| 9 | 8 | adantr 479 | . . . . 5 β’ ((π β§ (πβ{π}) = (πβ{(π + π)})) β π β π) |
| 10 | lspindp3.y | . . . . . 6 β’ (π β π β (π β { 0 })) | |
| 11 | 10 | adantr 479 | . . . . 5 β’ ((π β§ (πβ{π}) = (πβ{(π + π)})) β π β (π β { 0 })) |
| 12 | simpr 483 | . . . . 5 β’ ((π β§ (πβ{π}) = (πβ{(π + π)})) β (πβ{π}) = (πβ{(π + π)})) | |
| 13 | 2, 3, 4, 5, 7, 9, 11, 12 | lspabs2 21015 | . . . 4 β’ ((π β§ (πβ{π}) = (πβ{(π + π)})) β (πβ{π}) = (πβ{π})) |
| 14 | 13 | ex 411 | . . 3 β’ (π β ((πβ{π}) = (πβ{(π + π)}) β (πβ{π}) = (πβ{π}))) |
| 15 | 14 | necon3d 2958 | . 2 β’ (π β ((πβ{π}) β (πβ{π}) β (πβ{π}) β (πβ{(π + π)}))) |
| 16 | 1, 15 | mpd 15 | 1 β’ (π β (πβ{π}) β (πβ{(π + π)})) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 β cdif 3946 {csn 4632 βcfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 0gc0g 17428 LSpanclspn 20862 LVecclvec 20994 |
| 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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 |
| This theorem is referenced by: mapdindp4 41228 hdmaprnlem3uN 41356 |
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