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Mirrors > Home > MPE Home > Th. List > mavmumamul1 | Structured version Visualization version GIF version |
Description: The multiplication of an NxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an NxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.) |
Ref | Expression |
---|---|
mavmumamul1.a | โข ๐ด = (๐ Mat ๐ ) |
mavmumamul1.m | โข ร = (๐ maMul โจ๐, ๐, {โ }โฉ) |
mavmumamul1.t | โข ยท = (๐ maVecMul โจ๐, ๐โฉ) |
mavmumamul1.b | โข ๐ต = (Baseโ๐ ) |
mavmumamul1.r | โข (๐ โ ๐ โ Ring) |
mavmumamul1.n | โข (๐ โ ๐ โ Fin) |
mavmumamul1.x | โข (๐ โ ๐ โ (Baseโ๐ด)) |
mavmumamul1.y | โข (๐ โ ๐ โ (๐ต โm ๐)) |
mavmumamul1.z | โข (๐ โ ๐ โ (๐ต โm (๐ ร {โ }))) |
Ref | Expression |
---|---|
mavmumamul1 | โข (๐ โ (โ๐ โ ๐ (๐โ๐) = (๐๐โ ) โ โ๐ โ ๐ ((๐ ยท ๐)โ๐) = (๐(๐ ร ๐)โ ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mavmumamul1.m | . 2 โข ร = (๐ maMul โจ๐, ๐, {โ }โฉ) | |
2 | mavmumamul1.t | . 2 โข ยท = (๐ maVecMul โจ๐, ๐โฉ) | |
3 | mavmumamul1.b | . 2 โข ๐ต = (Baseโ๐ ) | |
4 | mavmumamul1.r | . 2 โข (๐ โ ๐ โ Ring) | |
5 | mavmumamul1.n | . 2 โข (๐ โ ๐ โ Fin) | |
6 | mavmumamul1.x | . . 3 โข (๐ โ ๐ โ (Baseโ๐ด)) | |
7 | mavmumamul1.a | . . . . 5 โข ๐ด = (๐ Mat ๐ ) | |
8 | 7, 3 | matbas2 22343 | . . . 4 โข ((๐ โ Fin โง ๐ โ Ring) โ (๐ต โm (๐ ร ๐)) = (Baseโ๐ด)) |
9 | 5, 4, 8 | syl2anc 582 | . . 3 โข (๐ โ (๐ต โm (๐ ร ๐)) = (Baseโ๐ด)) |
10 | 6, 9 | eleqtrrd 2832 | . 2 โข (๐ โ ๐ โ (๐ต โm (๐ ร ๐))) |
11 | mavmumamul1.y | . 2 โข (๐ โ ๐ โ (๐ต โm ๐)) | |
12 | mavmumamul1.z | . 2 โข (๐ โ ๐ โ (๐ต โm (๐ ร {โ }))) | |
13 | 1, 2, 3, 4, 5, 5, 10, 11, 12 | mvmumamul1 22476 | 1 โข (๐ โ (โ๐ โ ๐ (๐โ๐) = (๐๐โ ) โ โ๐ โ ๐ ((๐ ยท ๐)โ๐) = (๐(๐ ร ๐)โ ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โwral 3058 โ c0 4326 {csn 4632 โจcop 4638 โจcotp 4640 ร cxp 5680 โcfv 6553 (class class class)co 7426 โm cmap 8851 Fincfn 8970 Basecbs 17187 Ringcrg 20180 maMul cmmul 22305 Mat cmat 22327 maVecMul cmvmul 22462 |
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-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-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 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-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 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-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-prds 17436 df-pws 17438 df-sra 21065 df-rgmod 21066 df-dsmm 21673 df-frlm 21688 df-mamu 22306 df-mat 22328 df-mvmul 22463 |
This theorem is referenced by: (None) |
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