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Mirrors > Home > MPE Home > Th. List > matvsca | Structured version Visualization version GIF version |
Description: The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.) |
Ref | Expression |
---|---|
matbas.a | β’ π΄ = (π Mat π ) |
matbas.g | β’ πΊ = (π freeLMod (π Γ π)) |
Ref | Expression |
---|---|
matvsca | β’ ((π β Fin β§ π β π) β ( Β·π βπΊ) = ( Β·π βπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vscaid 17308 | . . 3 β’ Β·π = Slot ( Β·π βndx) | |
2 | vscandxnmulrndx 17311 | . . 3 β’ ( Β·π βndx) β (.rβndx) | |
3 | 1, 2 | setsnid 17185 | . 2 β’ ( Β·π βπΊ) = ( Β·π β(πΊ sSet β¨(.rβndx), (π maMul β¨π, π, πβ©)β©)) |
4 | matbas.a | . . . 4 β’ π΄ = (π Mat π ) | |
5 | matbas.g | . . . 4 β’ πΊ = (π freeLMod (π Γ π)) | |
6 | eqid 2728 | . . . 4 β’ (π maMul β¨π, π, πβ©) = (π maMul β¨π, π, πβ©) | |
7 | 4, 5, 6 | matval 22331 | . . 3 β’ ((π β Fin β§ π β π) β π΄ = (πΊ sSet β¨(.rβndx), (π maMul β¨π, π, πβ©)β©)) |
8 | 7 | fveq2d 6906 | . 2 β’ ((π β Fin β§ π β π) β ( Β·π βπ΄) = ( Β·π β(πΊ sSet β¨(.rβndx), (π maMul β¨π, π, πβ©)β©))) |
9 | 3, 8 | eqtr4id 2787 | 1 β’ ((π β Fin β§ π β π) β ( Β·π βπΊ) = ( Β·π βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β¨cop 4638 β¨cotp 4640 Γ cxp 5680 βcfv 6553 (class class class)co 7426 Fincfn 8970 sSet csts 17139 ndxcnx 17169 .rcmulr 17241 Β·π cvsca 17244 freeLMod cfrlm 21687 maMul cmmul 22305 Mat cmat 22327 |
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-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-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-2nd 8000 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-4 12315 df-5 12316 df-6 12317 df-sets 17140 df-slot 17158 df-ndx 17170 df-mulr 17254 df-vsca 17257 df-mat 22328 |
This theorem is referenced by: matvsca2 22350 matlmod 22351 matdim 33346 |
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