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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 7394 | . . . 4 β’ (π freeLMod (π Γ π)) β V | |
2 | matmulr.t | . . . . 5 β’ Β· = (π maMul β¨π, π, πβ©) | |
3 | 2 | ovexi 7395 | . . . 4 β’ Β· β V |
4 | 1, 3 | pm3.2i 472 | . . 3 β’ ((π freeLMod (π Γ π)) β V β§ Β· β V) |
5 | mulrid 17183 | . . . 4 β’ .r = Slot (.rβndx) | |
6 | 5 | setsid 17088 | . . 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 2733 | . . . 4 β’ (π freeLMod (π Γ π)) = (π freeLMod (π Γ π)) | |
10 | 8, 9, 2 | matval 21781 | . . 3 β’ ((π β Fin β§ π β π) β π΄ = ((π freeLMod (π Γ π)) sSet β¨(.rβndx), Β· β©)) |
11 | 10 | fveq2d 6850 | . 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 2776 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3447 β¨cop 4596 β¨cotp 4598 Γ cxp 5635 βcfv 6500 (class class class)co 7361 Fincfn 8889 sSet csts 17043 ndxcnx 17073 .rcmulr 17142 freeLMod cfrlm 21175 maMul cmmul 21755 Mat cmat 21777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-1cn 11117 ax-addcl 11119 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-ot 4599 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-nn 12162 df-2 12224 df-3 12225 df-sets 17044 df-slot 17062 df-ndx 17074 df-mulr 17155 df-mat 21778 |
This theorem is referenced by: matring 21815 matassa 21816 matmulcell 21817 mpomatmul 21818 mat1 21819 mattposm 21831 mat1dimmul 21848 dmatmul 21869 mdetmul 21995 madurid 22016 slesolinv 22052 slesolinvbi 22053 cramerimplem3 22057 mat2pmatmul 22103 decpmatmullem 22143 decpmatmul 22144 matunitlindflem2 36125 |
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