Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  matassa Unicode version

Theorem matassa 27144
Description: Existence of the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
matassa.a  |-  A  =  ( N Mat  R )
Assertion
Ref Expression
matassa  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )

Proof of Theorem matassa
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 matassa.a . . 3  |-  A  =  ( N Mat  R )
2 eqid 2381 . . 3  |-  ( Base `  R )  =  (
Base `  R )
31, 2matbas2 27138 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( ( Base `  R
)  ^m  ( N  X.  N ) )  =  ( Base `  A
) )
41matsca2 27137 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  =  (Scalar `  A
) )
5 eqidd 2382 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( Base `  R )  =  ( Base `  R
) )
6 eqidd 2382 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( .s `  A
)  =  ( .s
`  A ) )
7 eqid 2381 . . 3  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
81, 7matmulr 27130 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
9 crngrng 15595 . . 3  |-  ( R  e.  CRing  ->  R  e.  Ring )
101matlmod 27142 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
119, 10sylan2 461 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  LMod )
121matrng 27143 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
139, 12sylan2 461 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  Ring )
14 simpr 448 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  e.  CRing )
159ad2antlr 708 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  R  e.  Ring )
16 simpll 731 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  N  e.  Fin )
17 eqid 2381 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
18 simpr1 963 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  x  e.  ( Base `  R )
)
19 simpr2 964 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  y  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
20 simpr3 965 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
212, 15, 7, 16, 16, 16, 17, 18, 19, 20mamuvs1 27126 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( (
( ( N  X.  N )  X.  {
x } )  o F ( .r `  R ) y ) ( R maMul  <. N ,  N ,  N >. ) z )  =  ( ( ( N  X.  N )  X.  {
x } )  o F ( .r `  R ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
223adantr 452 . . . . . 6  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( ( Base `  R )  ^m  ( N  X.  N
) )  =  (
Base `  A )
)
2319, 22eleqtrd 2457 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  y  e.  ( Base `  A )
)
24 eqid 2381 . . . . . 6  |-  ( Base `  A )  =  (
Base `  A )
25 eqid 2381 . . . . . 6  |-  ( .s
`  A )  =  ( .s `  A
)
26 eqid 2381 . . . . . 6  |-  ( N  X.  N )  =  ( N  X.  N
)
271, 24, 2, 25, 17, 26matvsca2 27141 . . . . 5  |-  ( ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  A
) )  ->  (
x ( .s `  A ) y )  =  ( ( ( N  X.  N )  X.  { x }
)  o F ( .r `  R ) y ) )
2818, 23, 27syl2anc 643 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( x
( .s `  A
) y )  =  ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) y ) )
2928oveq1d 6029 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( (
x ( .s `  A ) y ) ( R maMul  <. N ,  N ,  N >. ) z )  =  ( ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) y ) ( R maMul  <. N ,  N ,  N >. ) z ) )
302, 15, 7, 16, 16, 16, 19, 20mamucl 27119 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) z )  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
3130, 22eleqtrd 2457 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) z )  e.  (
Base `  A )
)
321, 24, 2, 25, 17, 26matvsca2 27141 . . . 4  |-  ( ( x  e.  ( Base `  R )  /\  (
y ( R maMul  <. N ,  N ,  N >. ) z )  e.  (
Base `  A )
)  ->  ( x
( .s `  A
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) )  =  ( ( ( N  X.  N
)  X.  { x } )  o F ( .r `  R
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
3318, 31, 32syl2anc 643 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( x
( .s `  A
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) )  =  ( ( ( N  X.  N
)  X.  { x } )  o F ( .r `  R
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
3421, 29, 333eqtr4d 2423 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( (
x ( .s `  A ) y ) ( R maMul  <. N ,  N ,  N >. ) z )  =  ( x ( .s `  A ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
35 simplr 732 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  R  e.  CRing
)
3635, 2, 17, 7, 16, 16, 16, 19, 18, 20mamuvs2 27127 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) z ) )  =  ( ( ( N  X.  N )  X.  {
x } )  o F ( .r `  R ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
3720, 22eleqtrd 2457 . . . . 5  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  z  e.  ( Base `  A )
)
381, 24, 2, 25, 17, 26matvsca2 27141 . . . . 5  |-  ( ( x  e.  ( Base `  R )  /\  z  e.  ( Base `  A
) )  ->  (
x ( .s `  A ) z )  =  ( ( ( N  X.  N )  X.  { x }
)  o F ( .r `  R ) z ) )
3918, 37, 38syl2anc 643 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( x
( .s `  A
) z )  =  ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) z ) )
4039oveq2d 6030 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) ( x ( .s
`  A ) z ) )  =  ( y ( R maMul  <. N ,  N ,  N >. ) ( ( ( N  X.  N )  X. 
{ x } )  o F ( .r
`  R ) z ) ) )
4136, 40, 333eqtr4d 2423 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( ( Base `  R )  ^m  ( N  X.  N
) )  /\  z  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )  ->  ( y
( R maMul  <. N ,  N ,  N >. ) ( x ( .s
`  A ) z ) )  =  ( x ( .s `  A ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
423, 4, 5, 6, 8, 11, 13, 14, 34, 41isassad 16303 1  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {csn 3751   <.cotp 3755    X. cxp 4810   ` cfv 5388  (class class class)co 6014    o Fcof 6236    ^m cmap 6948   Fincfn 7039   Basecbs 13390   .rcmulr 13451   .scvsca 13454   Ringcrg 15581   CRingccrg 15582   LModclmod 15871  AssAlgcasa 16290   maMul cmmul 27102   Mat cmat 27103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-inf2 7523  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-ot 3761  df-uni 3952  df-int 3987  df-iun 4031  df-iin 4032  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-se 4477  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-isom 5397  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-of 6238  df-1st 6282  df-2nd 6283  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-oadd 6658  df-er 6835  df-map 6950  df-ixp 6994  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-sup 7375  df-oi 7406  df-card 7753  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-nn 9927  df-2 9984  df-3 9985  df-4 9986  df-5 9987  df-6 9988  df-7 9989  df-8 9990  df-9 9991  df-10 9992  df-n0 10148  df-z 10209  df-dec 10309  df-uz 10415  df-fz 10970  df-fzo 11060  df-seq 11245  df-hash 11540  df-struct 13392  df-ndx 13393  df-slot 13394  df-base 13395  df-sets 13396  df-ress 13397  df-plusg 13463  df-mulr 13464  df-sca 13466  df-vsca 13467  df-tset 13469  df-ple 13470  df-ds 13472  df-hom 13474  df-cco 13475  df-prds 13592  df-pws 13594  df-0g 13648  df-gsum 13649  df-mre 13732  df-mrc 13733  df-acs 13735  df-mnd 14611  df-mhm 14659  df-submnd 14660  df-grp 14733  df-minusg 14734  df-sbg 14735  df-mulg 14736  df-subg 14862  df-ghm 14925  df-cntz 15037  df-cmn 15335  df-abl 15336  df-mgp 15570  df-rng 15584  df-cring 15585  df-ur 15586  df-subrg 15787  df-lmod 15873  df-lss 15930  df-sra 16165  df-rgmod 16166  df-assa 16293  df-dsmm 26861  df-frlm 26877  df-mamu 27104  df-mat 27105
  Copyright terms: Public domain W3C validator