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Theorem matgsum 31047
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 6059 . . . 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 6228 . . . . 5  |-  ( N Mat 
R )  e.  _V
64, 5eqeltri 2538 . . . 4  |-  A  e. 
_V
76a1i 11 . . 3  |-  ( ph  ->  A  e.  _V )
8 ovex 6228 . . . 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 2454 . . . . . 6  |-  ( R freeLMod  ( N  X.  N
) )  =  ( R freeLMod  ( N  X.  N ) )
134, 12matbas 18442 . . . . 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 2462 . . 3  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  ( R freeLMod  ( N  X.  N ) ) ) )
164, 12matplusg 18443 . . . . 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 2462 . . 3  |-  ( ph  ->  ( +g  `  A
)  =  ( +g  `  ( R freeLMod  ( N  X.  N ) ) ) )
193, 7, 9, 15, 18gsumpropd 15626 . 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 6751 . . . . . 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 4490 . . . 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 6219 . . 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 2454 . . . 4  |-  ( Base `  ( R freeLMod  ( N  X.  N ) ) )  =  ( Base `  ( R freeLMod  ( N  X.  N
) ) )
25 eqid 2454 . . . 4  |-  ( 0g
`  ( R freeLMod  ( N  X.  N ) ) )  =  ( 0g
`  ( R freeLMod  ( N  X.  N ) ) )
26 xpfi 7697 . . . . 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 2552 . . . . 5  |-  ( (
ph  /\  y  e.  J )  ->  (
i  e.  N , 
j  e.  N  |->  U )  e.  ( Base `  A ) )
3120eqcomi 2467 . . . . . 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 2557 . . . 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 4486 . . . . . 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 2467 . . . . . 6  |-  ( 0g
`  A )  =  .0.
4137, 38, 403brtr4g 4435 . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  ( z  e.  ( N  X.  N ) 
|->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) finSupp  ( 0g `  A ) )
424, 12mat0 18446 . . . . . 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 4429 . . . 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 18324 . . 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 2495 . 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 5812 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
48 csbov2g 6239 . . . . . . . 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 3799 . . . . . 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 5812 . . . . . . 7  |-  ( 1st `  z )  e.  _V
52 csbov2g 6239 . . . . . . 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 4734 . . . . . . . . . 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 3799 . . . . . . . 8  |-  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ ( y  e.  J  |->  U )  =  [_ ( 1st `  z )  /  i ]_ (
y  e.  J  |->  [_ ( 2nd `  z )  /  j ]_ U
)
57 csbmpt2 4734 . . . . . . . . 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 2483 . . . . . . 7  |-  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ ( y  e.  J  |->  U )  =  ( y  e.  J  |->  [_ ( 1st `  z )  /  i ]_ [_ ( 2nd `  z )  / 
j ]_ U )
6059oveq2i 6214 . . . . . 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 2488 . . . . 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 4486 . . . 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 6751 . . . 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 2486 . . 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 2499 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 1370    e. wcel 1758   _Vcvv 3078   [_csb 3398   class class class wbr 4403    |-> cmpt 4461    X. cxp 4949   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   Fincfn 7423   finSupp cfsupp 7734   Basecbs 14295   +g cplusg 14360   0gc0g 14500    gsumg cgsu 14501   Ringcrg 16771   freeLMod cfrlm 18299   Mat cmat 18408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-ot 3997  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-sca 14376  df-vsca 14377  df-ip 14378  df-tset 14379  df-ple 14380  df-ds 14382  df-hom 14384  df-cco 14385  df-0g 14502  df-gsum 14503  df-prds 14508  df-pws 14510  df-mnd 15537  df-mhm 15586  df-grp 15667  df-minusg 15668  df-sbg 15669  df-subg 15800  df-cntz 15957  df-cmn 16403  df-abl 16404  df-mgp 16717  df-ur 16729  df-rng 16773  df-subrg 16989  df-lmod 17076  df-lss 17140  df-sra 17379  df-rgmod 17380  df-dsmm 18285  df-frlm 18300  df-mat 18410
This theorem is referenced by:  decpmatmul  31279  pmatcollpw2  31285
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