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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 7442 | . . . 4 β’ (π freeLMod (π Γ π)) β V | |
2 | matmulr.t | . . . . 5 β’ Β· = (π maMul β¨π, π, πβ©) | |
3 | 2 | ovexi 7443 | . . . 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 | mulridx 17239 | . . . 4 β’ .r = Slot (.rβndx) | |
6 | 5 | setsid 17141 | . . 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 21911 | . . 3 β’ ((π β Fin β§ π β π) β π΄ = ((π freeLMod (π Γ π)) sSet β¨(.rβndx), Β· β©)) |
11 | 10 | fveq2d 6896 | . 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 3475 β¨cop 4635 β¨cotp 4637 Γ cxp 5675 βcfv 6544 (class class class)co 7409 Fincfn 8939 sSet csts 17096 ndxcnx 17126 .rcmulr 17198 freeLMod cfrlm 21301 maMul cmmul 21885 Mat cmat 21907 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-1cn 11168 ax-addcl 11170 |
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 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 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-ot 4638 df-uni 4910 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-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-nn 12213 df-2 12275 df-3 12276 df-sets 17097 df-slot 17115 df-ndx 17127 df-mulr 17211 df-mat 21908 |
This theorem is referenced by: matring 21945 matassa 21946 matmulcell 21947 mpomatmul 21948 mat1 21949 mattposm 21961 mat1dimmul 21978 dmatmul 21999 mdetmul 22125 madurid 22146 slesolinv 22182 slesolinvbi 22183 cramerimplem3 22187 mat2pmatmul 22233 decpmatmullem 22273 decpmatmul 22274 matunitlindflem2 36485 |
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