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Mirrors > Home > MPE Home > Th. List > matmulr | Structured version Visualization version GIF version |
Description: Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Ref | Expression |
---|---|
matmulr.a | β’ π΄ = (π Mat π ) |
matmulr.t | β’ Β· = (π maMul β¨π, π, πβ©) |
Ref | Expression |
---|---|
matmulr | β’ ((π β Fin β§ π β π) β Β· = (.rβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7444 | . . . 4 β’ (π freeLMod (π Γ π)) β V | |
2 | matmulr.t | . . . . 5 β’ Β· = (π maMul β¨π, π, πβ©) | |
3 | 2 | ovexi 7445 | . . . 4 β’ Β· β V |
4 | 1, 3 | pm3.2i 469 | . . 3 β’ ((π freeLMod (π Γ π)) β V β§ Β· β V) |
5 | mulridx 17243 | . . . 4 β’ .r = Slot (.rβndx) | |
6 | 5 | setsid 17145 | . . 3 β’ (((π freeLMod (π Γ π)) β V β§ Β· β V) β Β· = (.rβ((π freeLMod (π Γ π)) sSet β¨(.rβndx), Β· β©))) |
7 | 4, 6 | mp1i 13 | . 2 β’ ((π β Fin β§ π β π) β Β· = (.rβ((π freeLMod (π Γ π)) sSet β¨(.rβndx), Β· β©))) |
8 | matmulr.a | . . . 4 β’ π΄ = (π Mat π ) | |
9 | eqid 2730 | . . . 4 β’ (π freeLMod (π Γ π)) = (π freeLMod (π Γ π)) | |
10 | 8, 9, 2 | matval 22131 | . . 3 β’ ((π β Fin β§ π β π) β π΄ = ((π freeLMod (π Γ π)) sSet β¨(.rβndx), Β· β©)) |
11 | 10 | fveq2d 6894 | . 2 β’ ((π β Fin β§ π β π) β (.rβπ΄) = (.rβ((π freeLMod (π Γ π)) sSet β¨(.rβndx), Β· β©))) |
12 | 7, 11 | eqtr4d 2773 | 1 β’ ((π β Fin β§ π β π) β Β· = (.rβπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 Vcvv 3472 β¨cop 4633 β¨cotp 4635 Γ cxp 5673 βcfv 6542 (class class class)co 7411 Fincfn 8941 sSet csts 17100 ndxcnx 17130 .rcmulr 17202 freeLMod cfrlm 21520 maMul cmmul 22105 Mat cmat 22127 |
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-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-1cn 11170 ax-addcl 11172 |
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-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-ot 4636 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-nn 12217 df-2 12279 df-3 12280 df-sets 17101 df-slot 17119 df-ndx 17131 df-mulr 17215 df-mat 22128 |
This theorem is referenced by: matring 22165 matassa 22166 matmulcell 22167 mpomatmul 22168 mat1 22169 mattposm 22181 mat1dimmul 22198 dmatmul 22219 mdetmul 22345 madurid 22366 slesolinv 22402 slesolinvbi 22403 cramerimplem3 22407 mat2pmatmul 22453 decpmatmullem 22493 decpmatmul 22494 matunitlindflem2 36788 |
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