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

Theorem matassa 27349
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 2404 . . 3  |-  ( Base `  R )  =  (
Base `  R )
31, 2matbas2 27343 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( ( Base `  R
)  ^m  ( N  X.  N ) )  =  ( Base `  A
) )
41matsca2 27342 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  =  (Scalar `  A
) )
5 eqidd 2405 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( Base `  R )  =  ( Base `  R
) )
6 eqidd 2405 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( .s `  A
)  =  ( .s
`  A ) )
7 eqid 2404 . . 3  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
81, 7matmulr 27335 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
9 crngrng 15629 . . 3  |-  ( R  e.  CRing  ->  R  e.  Ring )
101matlmod 27347 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
119, 10sylan2 461 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  LMod )
121matrng 27348 . . 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 2404 . . . 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 27331 . . 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 2480 . . . . 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 2404 . . . . . 6  |-  ( Base `  A )  =  (
Base `  A )
25 eqid 2404 . . . . . 6  |-  ( .s
`  A )  =  ( .s `  A
)
26 eqid 2404 . . . . . 6  |-  ( N  X.  N )  =  ( N  X.  N
)
271, 24, 2, 25, 17, 26matvsca2 27346 . . . . 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 6055 . . 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 27324 . . . . 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 2480 . . . 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 27346 . . . 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 2446 . 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 27332 . . 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 2480 . . . . 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 27346 . . . . 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 6056 . . 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 2446 . 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 16337 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 1721   {csn 3774   <.cotp 3778    X. cxp 4835   ` cfv 5413  (class class class)co 6040    o Fcof 6262    ^m cmap 6977   Fincfn 7068   Basecbs 13424   .rcmulr 13485   .scvsca 13488   Ringcrg 15615   CRingccrg 15616   LModclmod 15905  AssAlgcasa 16324   maMul cmmul 27307   Mat cmat 27308
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-lmod 15907  df-lss 15964  df-sra 16199  df-rgmod 16200  df-assa 16327  df-dsmm 27066  df-frlm 27082  df-mamu 27309  df-mat 27310
  Copyright terms: Public domain W3C validator