MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  scmatscmid Structured version   Visualization version   GIF version

Theorem scmatscmid 21871
Description: A scalar matrix can be expressed as a multiplication of a scalar with the identity matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 18-Dec-2019.)
Hypotheses
Ref Expression
scmatval.k ๐พ = (Baseโ€˜๐‘…)
scmatval.a ๐ด = (๐‘ Mat ๐‘…)
scmatval.b ๐ต = (Baseโ€˜๐ด)
scmatval.1 1 = (1rโ€˜๐ด)
scmatval.t ยท = ( ยท๐‘  โ€˜๐ด)
scmatval.s ๐‘† = (๐‘ ScMat ๐‘…)
Assertion
Ref Expression
scmatscmid ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰ โˆง ๐‘€ โˆˆ ๐‘†) โ†’ โˆƒ๐‘ โˆˆ ๐พ ๐‘€ = (๐‘ ยท 1 ))
Distinct variable groups:   ๐พ,๐‘   ๐‘,๐‘   ๐‘…,๐‘   ๐‘€,๐‘
Allowed substitution hints:   ๐ด(๐‘)   ๐ต(๐‘)   ๐‘†(๐‘)   ยท (๐‘)   1 (๐‘)   ๐‘‰(๐‘)

Proof of Theorem scmatscmid
StepHypRef Expression
1 scmatval.k . . . 4 ๐พ = (Baseโ€˜๐‘…)
2 scmatval.a . . . 4 ๐ด = (๐‘ Mat ๐‘…)
3 scmatval.b . . . 4 ๐ต = (Baseโ€˜๐ด)
4 scmatval.1 . . . 4 1 = (1rโ€˜๐ด)
5 scmatval.t . . . 4 ยท = ( ยท๐‘  โ€˜๐ด)
6 scmatval.s . . . 4 ๐‘† = (๐‘ ScMat ๐‘…)
71, 2, 3, 4, 5, 6scmatel 21870 . . 3 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ (๐‘€ โˆˆ ๐‘† โ†” (๐‘€ โˆˆ ๐ต โˆง โˆƒ๐‘ โˆˆ ๐พ ๐‘€ = (๐‘ ยท 1 ))))
87simplbda 501 . 2 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โˆง ๐‘€ โˆˆ ๐‘†) โ†’ โˆƒ๐‘ โˆˆ ๐พ ๐‘€ = (๐‘ ยท 1 ))
983impa 1111 1 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰ โˆง ๐‘€ โˆˆ ๐‘†) โ†’ โˆƒ๐‘ โˆˆ ๐พ ๐‘€ = (๐‘ ยท 1 ))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆง wa 397   โˆง w3a 1088   = wceq 1542   โˆˆ wcel 2107  โˆƒwrex 3074  โ€˜cfv 6501  (class class class)co 7362  Fincfn 8890  Basecbs 17090   ยท๐‘  cvsca 17144  1rcur 19920   Mat cmat 21770   ScMat cscmat 21854
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-scmat 21856
This theorem is referenced by:  scmate  21875  scmatscm  21878  scmataddcl  21881  scmatsubcl  21882  scmatfo  21895
  Copyright terms: Public domain W3C validator