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Theorem matassa 19238
Description: Existence of the matrix algebra, see also the statement in [Lang] p. 505:"Then Matn(R) is an algebra over R" . (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 2402 . . 3  |-  ( Base `  R )  =  (
Base `  R )
31, 2matbas2 19215 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( ( Base `  R
)  ^m  ( N  X.  N ) )  =  ( Base `  A
) )
41matsca2 19214 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  =  (Scalar `  A
) )
5 eqidd 2403 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( Base `  R )  =  ( Base `  R
) )
6 eqidd 2403 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( .s `  A
)  =  ( .s
`  A ) )
7 eqid 2402 . . 3  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
81, 7matmulr 19232 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
9 crngring 17529 . . 3  |-  ( R  e.  CRing  ->  R  e.  Ring )
101matlmod 19223 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
119, 10sylan2 472 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  LMod )
121matring 19237 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
139, 12sylan2 472 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e.  Ring )
14 simpr 459 . 2  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  R  e.  CRing )
159ad2antlr 725 . . . 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 752 . . . 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 2402 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
18 simpr1 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 ) ) ) )  ->  x  e.  ( Base `  R )
)
19 simpr2 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 ) ) ) )  ->  y  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
20 simpr3 1005 . . . 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 19199 . . 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 463 . . . . . 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 2492 . . . . 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 2402 . . . . . 6  |-  ( Base `  A )  =  (
Base `  A )
25 eqid 2402 . . . . . 6  |-  ( .s
`  A )  =  ( .s `  A
)
26 eqid 2402 . . . . . 6  |-  ( N  X.  N )  =  ( N  X.  N
)
271, 24, 2, 25, 17, 26matvsca2 19222 . . . . 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 659 . . . 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 6293 . . 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 19195 . . . . 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 2492 . . . 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 19222 . . . 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 659 . . 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 2453 . 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 19200 . . 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 2492 . . . . 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 19222 . . . . 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 659 . . . 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 6294 . . 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 2453 . 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 18292 1  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {csn 3972   <.cotp 3980    X. cxp 4821   ` cfv 5569  (class class class)co 6278    oFcof 6519    ^m cmap 7457   Fincfn 7554   Basecbs 14841   .rcmulr 14910   .scvsca 14913   Ringcrg 17518   CRingccrg 17519   LModclmod 17832  AssAlgcasa 18278   maMul cmmul 19177   Mat cmat 19201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-hom 14933  df-cco 14934  df-0g 15056  df-gsum 15057  df-prds 15062  df-pws 15064  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-mhm 16290  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-mulg 16384  df-subg 16522  df-ghm 16589  df-cntz 16679  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-cring 17521  df-subrg 17747  df-lmod 17834  df-lss 17899  df-sra 18138  df-rgmod 18139  df-assa 18281  df-dsmm 19061  df-frlm 19076  df-mamu 19178  df-mat 19202
This theorem is referenced by:  matinv  19471  cpmadugsumlemB  19667  cpmadugsumlemC  19668  cayhamlem2  19677
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