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Theorem matgsum 18746
Description: Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
Hypotheses
Ref Expression
matgsum.a  |-  A  =  ( N Mat  R )
matgsum.b  |-  B  =  ( Base `  A
)
matgsum.z  |-  .0.  =  ( 0g `  A )
matgsum.i  |-  ( ph  ->  N  e.  Fin )
matgsum.j  |-  ( ph  ->  J  e.  W )
matgsum.r  |-  ( ph  ->  R  e.  Ring )
matgsum.f  |-  ( (
ph  /\  y  e.  J )  ->  (
i  e.  N , 
j  e.  N  |->  U )  e.  B )
matgsum.w  |-  ( ph  ->  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) finSupp  .0.  )
Assertion
Ref Expression
matgsum  |-  ( ph  ->  ( A  gsumg  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
Distinct variable groups:    i, J, j, y    i, N, j, y    R, i, j, y    ph, y
Allowed substitution hints:    ph( i, j)    A( y, i, j)    B( y, i, j)    U( y, i, j)    W( y, i, j)    .0. ( y,
i, j)

Proof of Theorem matgsum
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 matgsum.j . . . 4  |-  ( ph  ->  J  e.  W )
2 mptexg 6131 . . . 4  |-  ( J  e.  W  ->  (
y  e.  J  |->  ( i  e.  N , 
j  e.  N  |->  U ) )  e.  _V )
31, 2syl 16 . . 3  |-  ( ph  ->  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) )  e. 
_V )
4 matgsum.a . . . . 5  |-  A  =  ( N Mat  R )
5 ovex 6310 . . . . 5  |-  ( N Mat 
R )  e.  _V
64, 5eqeltri 2551 . . . 4  |-  A  e. 
_V
76a1i 11 . . 3  |-  ( ph  ->  A  e.  _V )
8 ovex 6310 . . . 4  |-  ( R freeLMod  ( N  X.  N
) )  e.  _V
98a1i 11 . . 3  |-  ( ph  ->  ( R freeLMod  ( N  X.  N ) )  e. 
_V )
10 matgsum.i . . . . 5  |-  ( ph  ->  N  e.  Fin )
11 matgsum.r . . . . 5  |-  ( ph  ->  R  e.  Ring )
12 eqid 2467 . . . . . 6  |-  ( R freeLMod  ( N  X.  N
) )  =  ( R freeLMod  ( N  X.  N ) )
134, 12matbas 18722 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( Base `  ( R freeLMod  ( N  X.  N ) ) )  =  (
Base `  A )
)
1410, 11, 13syl2anc 661 . . . 4  |-  ( ph  ->  ( Base `  ( R freeLMod  ( N  X.  N
) ) )  =  ( Base `  A
) )
1514eqcomd 2475 . . 3  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  ( R freeLMod  ( N  X.  N ) ) ) )
164, 12matplusg 18723 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( +g  `  ( R freeLMod  ( N  X.  N
) ) )  =  ( +g  `  A
) )
1710, 11, 16syl2anc 661 . . . 4  |-  ( ph  ->  ( +g  `  ( R freeLMod  ( N  X.  N
) ) )  =  ( +g  `  A
) )
1817eqcomd 2475 . . 3  |-  ( ph  ->  ( +g  `  A
)  =  ( +g  `  ( R freeLMod  ( N  X.  N ) ) ) )
193, 7, 9, 15, 18gsumpropd 15829 . 2  |-  ( ph  ->  ( A  gsumg  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) )  =  ( ( R freeLMod  ( N  X.  N
) )  gsumg  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) ) )
20 mpt2mpts 6849 . . . . . 6  |-  ( i  e.  N ,  j  e.  N  |->  U )  =  ( z  e.  ( N  X.  N
)  |->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U )
2120a1i 11 . . . . 5  |-  ( ph  ->  ( i  e.  N ,  j  e.  N  |->  U )  =  ( z  e.  ( N  X.  N )  |->  [_ ( 1st `  z )  /  i ]_ [_ ( 2nd `  z )  / 
j ]_ U ) )
2221mpteq2dv 4534 . . . 4  |-  ( ph  ->  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) )  =  ( y  e.  J  |->  ( z  e.  ( N  X.  N ) 
|->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) )
2322oveq2d 6301 . . 3  |-  ( ph  ->  ( ( R freeLMod  ( N  X.  N ) ) 
gsumg  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) )  =  ( ( R freeLMod  ( N  X.  N
) )  gsumg  ( y  e.  J  |->  ( z  e.  ( N  X.  N ) 
|->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) ) )
24 eqid 2467 . . . 4  |-  ( Base `  ( R freeLMod  ( N  X.  N ) ) )  =  ( Base `  ( R freeLMod  ( N  X.  N
) ) )
25 eqid 2467 . . . 4  |-  ( 0g
`  ( R freeLMod  ( N  X.  N ) ) )  =  ( 0g
`  ( R freeLMod  ( N  X.  N ) ) )
26 xpfi 7792 . . . . 5  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( N  X.  N
)  e.  Fin )
2710, 10, 26syl2anc 661 . . . 4  |-  ( ph  ->  ( N  X.  N
)  e.  Fin )
28 matgsum.f . . . . . 6  |-  ( (
ph  /\  y  e.  J )  ->  (
i  e.  N , 
j  e.  N  |->  U )  e.  B )
29 matgsum.b . . . . . 6  |-  B  =  ( Base `  A
)
3028, 29syl6eleq 2565 . . . . 5  |-  ( (
ph  /\  y  e.  J )  ->  (
i  e.  N , 
j  e.  N  |->  U )  e.  ( Base `  A ) )
3120eqcomi 2480 . . . . . 6  |-  ( z  e.  ( N  X.  N )  |->  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ U )  =  ( i  e.  N , 
j  e.  N  |->  U )
3231a1i 11 . . . . 5  |-  ( (
ph  /\  y  e.  J )  ->  (
z  e.  ( N  X.  N )  |->  [_ ( 1st `  z )  /  i ]_ [_ ( 2nd `  z )  / 
j ]_ U )  =  ( i  e.  N ,  j  e.  N  |->  U ) )
3310, 11jca 532 . . . . . . 7  |-  ( ph  ->  ( N  e.  Fin  /\  R  e.  Ring )
)
3433adantr 465 . . . . . 6  |-  ( (
ph  /\  y  e.  J )  ->  ( N  e.  Fin  /\  R  e.  Ring ) )
3534, 13syl 16 . . . . 5  |-  ( (
ph  /\  y  e.  J )  ->  ( Base `  ( R freeLMod  ( N  X.  N ) ) )  =  ( Base `  A ) )
3630, 32, 353eltr4d 2570 . . . 4  |-  ( (
ph  /\  y  e.  J )  ->  (
z  e.  ( N  X.  N )  |->  [_ ( 1st `  z )  /  i ]_ [_ ( 2nd `  z )  / 
j ]_ U )  e.  ( Base `  ( R freeLMod  ( N  X.  N
) ) ) )
37 matgsum.w . . . . . 6  |-  ( ph  ->  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) finSupp  .0.  )
3831mpteq2i 4530 . . . . . 6  |-  ( y  e.  J  |->  ( z  e.  ( N  X.  N )  |->  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ U ) )  =  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) )
39 matgsum.z . . . . . . 7  |-  .0.  =  ( 0g `  A )
4039eqcomi 2480 . . . . . 6  |-  ( 0g
`  A )  =  .0.
4137, 38, 403brtr4g 4479 . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  ( z  e.  ( N  X.  N ) 
|->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) finSupp  ( 0g `  A ) )
424, 12mat0 18726 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> 
( 0g `  ( R freeLMod  ( N  X.  N
) ) )  =  ( 0g `  A
) )
4310, 11, 42syl2anc 661 . . . . 5  |-  ( ph  ->  ( 0g `  ( R freeLMod  ( N  X.  N
) ) )  =  ( 0g `  A
) )
4441, 43breqtrrd 4473 . . . 4  |-  ( ph  ->  ( y  e.  J  |->  ( z  e.  ( N  X.  N ) 
|->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) finSupp  ( 0g `  ( R freeLMod  ( N  X.  N ) ) ) )
4512, 24, 25, 27, 1, 11, 36, 44frlmgsum 18609 . . 3  |-  ( ph  ->  ( ( R freeLMod  ( N  X.  N ) ) 
gsumg  ( y  e.  J  |->  ( z  e.  ( N  X.  N ) 
|->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) )  =  ( z  e.  ( N  X.  N
)  |->  ( R  gsumg  ( y  e.  J  |->  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ U ) ) ) )
4623, 45eqtrd 2508 . 2  |-  ( ph  ->  ( ( R freeLMod  ( N  X.  N ) ) 
gsumg  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) )  =  ( z  e.  ( N  X.  N
)  |->  ( R  gsumg  ( y  e.  J  |->  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ U ) ) ) )
47 fvex 5876 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
48 csbov2g 6321 . . . . . . . 8  |-  ( ( 2nd `  z )  e.  _V  ->  [_ ( 2nd `  z )  / 
j ]_ ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( R 
gsumg  [_ ( 2nd `  z
)  /  j ]_ ( y  e.  J  |->  U ) ) )
4947, 48ax-mp 5 . . . . . . 7  |-  [_ ( 2nd `  z )  / 
j ]_ ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( R 
gsumg  [_ ( 2nd `  z
)  /  j ]_ ( y  e.  J  |->  U ) )
5049csbeq2i 3836 . . . . . 6  |-  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ ( R  gsumg  ( y  e.  J  |->  U ) )  = 
[_ ( 1st `  z
)  /  i ]_ ( R  gsumg  [_ ( 2nd `  z
)  /  j ]_ ( y  e.  J  |->  U ) )
51 fvex 5876 . . . . . . 7  |-  ( 1st `  z )  e.  _V
52 csbov2g 6321 . . . . . . 7  |-  ( ( 1st `  z )  e.  _V  ->  [_ ( 1st `  z )  / 
i ]_ ( R  gsumg  [_ ( 2nd `  z )  / 
j ]_ ( y  e.  J  |->  U ) )  =  ( R  gsumg  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ ( y  e.  J  |->  U ) ) )
5351, 52ax-mp 5 . . . . . 6  |-  [_ ( 1st `  z )  / 
i ]_ ( R  gsumg  [_ ( 2nd `  z )  / 
j ]_ ( y  e.  J  |->  U ) )  =  ( R  gsumg  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ ( y  e.  J  |->  U ) )
54 csbmpt2 4782 . . . . . . . . . 10  |-  ( ( 2nd `  z )  e.  _V  ->  [_ ( 2nd `  z )  / 
j ]_ ( y  e.  J  |->  U )  =  ( y  e.  J  |-> 
[_ ( 2nd `  z
)  /  j ]_ U ) )
5547, 54ax-mp 5 . . . . . . . . 9  |-  [_ ( 2nd `  z )  / 
j ]_ ( y  e.  J  |->  U )  =  ( y  e.  J  |-> 
[_ ( 2nd `  z
)  /  j ]_ U )
5655csbeq2i 3836 . . . . . . . 8  |-  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ ( y  e.  J  |->  U )  =  [_ ( 1st `  z )  /  i ]_ (
y  e.  J  |->  [_ ( 2nd `  z )  /  j ]_ U
)
57 csbmpt2 4782 . . . . . . . . 9  |-  ( ( 1st `  z )  e.  _V  ->  [_ ( 1st `  z )  / 
i ]_ ( y  e.  J  |->  [_ ( 2nd `  z
)  /  j ]_ U )  =  ( y  e.  J  |->  [_ ( 1st `  z )  /  i ]_ [_ ( 2nd `  z )  / 
j ]_ U ) )
5851, 57ax-mp 5 . . . . . . . 8  |-  [_ ( 1st `  z )  / 
i ]_ ( y  e.  J  |->  [_ ( 2nd `  z
)  /  j ]_ U )  =  ( y  e.  J  |->  [_ ( 1st `  z )  /  i ]_ [_ ( 2nd `  z )  / 
j ]_ U )
5956, 58eqtri 2496 . . . . . . 7  |-  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ ( y  e.  J  |->  U )  =  ( y  e.  J  |->  [_ ( 1st `  z )  /  i ]_ [_ ( 2nd `  z )  / 
j ]_ U )
6059oveq2i 6296 . . . . . 6  |-  ( R 
gsumg  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ ( y  e.  J  |->  U ) )  =  ( R  gsumg  ( y  e.  J  |-> 
[_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) )
6150, 53, 603eqtrri 2501 . . . . 5  |-  ( R 
gsumg  ( y  e.  J  |-> 
[_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) )  = 
[_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ ( R  gsumg  ( y  e.  J  |->  U ) )
6261mpteq2i 4530 . . . 4  |-  ( z  e.  ( N  X.  N )  |->  ( R 
gsumg  ( y  e.  J  |-> 
[_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) )  =  ( z  e.  ( N  X.  N
)  |->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ ( R  gsumg  ( y  e.  J  |->  U ) ) )
63 mpt2mpts 6849 . . . 4  |-  ( i  e.  N ,  j  e.  N  |->  ( R 
gsumg  ( y  e.  J  |->  U ) ) )  =  ( z  e.  ( N  X.  N
)  |->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ ( R  gsumg  ( y  e.  J  |->  U ) ) )
6462, 63eqtr4i 2499 . . 3  |-  ( z  e.  ( N  X.  N )  |->  ( R 
gsumg  ( y  e.  J  |-> 
[_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) )
6564a1i 11 . 2  |-  ( ph  ->  ( z  e.  ( N  X.  N ) 
|->  ( R  gsumg  ( y  e.  J  |-> 
[_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
6619, 46, 653eqtrd 2512 1  |-  ( ph  ->  ( A  gsumg  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   [_csb 3435   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   1stc1st 6783   2ndc2nd 6784   Fincfn 7517   finSupp cfsupp 7830   Basecbs 14493   +g cplusg 14558   0gc0g 14698    gsumg cgsu 14699   Ringcrg 17012   freeLMod cfrlm 18584   Mat cmat 18716
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-fzo 11794  df-seq 12077  df-hash 12375  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-hom 14582  df-cco 14583  df-0g 14700  df-gsum 14701  df-prds 14706  df-pws 14708  df-mnd 15735  df-mhm 15789  df-grp 15871  df-minusg 15872  df-sbg 15873  df-subg 16012  df-cntz 16169  df-cmn 16615  df-abl 16616  df-mgp 16956  df-ur 16968  df-rng 17014  df-subrg 17239  df-lmod 17326  df-lss 17391  df-sra 17630  df-rgmod 17631  df-dsmm 18570  df-frlm 18585  df-mat 18717
This theorem is referenced by:  decpmatmul  19080  pmatcollpw2  19086
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