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Theorem matassa 18713
Description: Existence of the matrix algebra, see also the statement "Then Matn(R) is an algebra over R" in [Lang] p. 505. (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 2467 . . 3  |-  ( Base `  R )  =  (
Base `  R )
31, 2matbas2 18690 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( ( Base `  R
)  ^m  ( N  X.  N ) )  =  ( Base `  A
) )
41matsca2 18689 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  =  (Scalar `  A
) )
5 eqidd 2468 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( Base `  R )  =  ( Base `  R
) )
6 eqidd 2468 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( .s `  A
)  =  ( .s
`  A ) )
7 eqid 2467 . . 3  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
81, 7matmulr 18707 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
9 crngrng 16996 . . 3  |-  ( R  e.  CRing  ->  R  e.  Ring )
101matlmod 18698 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
119, 10sylan2 474 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  LMod )
121matrng 18712 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
139, 12sylan2 474 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  Ring )
14 simpr 461 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  e.  CRing )
159ad2antlr 726 . . . 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 753 . . . 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 2467 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
18 simpr1 1002 . . . 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 1003 . . . 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 1004 . . . 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 18674 . . 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 } )  oF ( .r `  R ) y ) ( R maMul  <. N ,  N ,  N >. ) z )  =  ( ( ( N  X.  N )  X.  {
x } )  oF ( .r `  R ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
223adantr 465 . . . . . 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 2557 . . . . 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 2467 . . . . . 6  |-  ( Base `  A )  =  (
Base `  A )
25 eqid 2467 . . . . . 6  |-  ( .s
`  A )  =  ( .s `  A
)
26 eqid 2467 . . . . . 6  |-  ( N  X.  N )  =  ( N  X.  N
)
271, 24, 2, 25, 17, 26matvsca2 18697 . . . . 5  |-  ( ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  A
) )  ->  (
x ( .s `  A ) y )  =  ( ( ( N  X.  N )  X.  { x }
)  oF ( .r `  R ) y ) )
2818, 23, 27syl2anc 661 . . . 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 } )  oF ( .r
`  R ) y ) )
2928oveq1d 6297 . . 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 } )  oF ( .r
`  R ) y ) ( R maMul  <. N ,  N ,  N >. ) z ) )
302, 15, 7, 16, 16, 16, 19, 20mamucl 18670 . . . . 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 2557 . . . 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 18697 . . . 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 } )  oF ( .r `  R
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
3318, 31, 32syl2anc 661 . . 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 } )  oF ( .r `  R
) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
3421, 29, 333eqtr4d 2518 . 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 754 . . . 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 18675 . . 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 } )  oF ( .r
`  R ) z ) )  =  ( ( ( N  X.  N )  X.  {
x } )  oF ( .r `  R ) ( y ( R maMul  <. N ,  N ,  N >. ) z ) ) )
3720, 22eleqtrd 2557 . . . . 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 18697 . . . . 5  |-  ( ( x  e.  ( Base `  R )  /\  z  e.  ( Base `  A
) )  ->  (
x ( .s `  A ) z )  =  ( ( ( N  X.  N )  X.  { x }
)  oF ( .r `  R ) z ) )
3918, 37, 38syl2anc 661 . . . 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 } )  oF ( .r
`  R ) z ) )
4039oveq2d 6298 . . 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 } )  oF ( .r
`  R ) z ) ) )
4136, 40, 333eqtr4d 2518 . 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 17743 1  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {csn 4027   <.cotp 4035    X. cxp 4997   ` cfv 5586  (class class class)co 6282    oFcof 6520    ^m cmap 7417   Fincfn 7513   Basecbs 14486   .rcmulr 14552   .scvsca 14555   Ringcrg 16986   CRingccrg 16987   LModclmod 17295  AssAlgcasa 17729   maMul cmmul 18652   Mat cmat 18676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-hom 14575  df-cco 14576  df-0g 14693  df-gsum 14694  df-prds 14699  df-pws 14701  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-ghm 16060  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-cring 16989  df-subrg 17210  df-lmod 17297  df-lss 17362  df-sra 17601  df-rgmod 17602  df-assa 17732  df-dsmm 18530  df-frlm 18545  df-mamu 18653  df-mat 18677
This theorem is referenced by:  matinv  18946  cpmadugsumlemB  19142  cpmadugsumlemC  19143  cayhamlem2  19152
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