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Theorem matgsum 18806
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 6123 . . . 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 6305 . . . . 5  |-  ( N Mat 
R )  e.  _V
64, 5eqeltri 2525 . . . 4  |-  A  e. 
_V
76a1i 11 . . 3  |-  ( ph  ->  A  e.  _V )
8 ovex 6305 . . . 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 2441 . . . . . 6  |-  ( R freeLMod  ( N  X.  N
) )  =  ( R freeLMod  ( N  X.  N ) )
134, 12matbas 18782 . . . . 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 2449 . . 3  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  ( R freeLMod  ( N  X.  N ) ) ) )
164, 12matplusg 18783 . . . . 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 2449 . . 3  |-  ( ph  ->  ( +g  `  A
)  =  ( +g  `  ( R freeLMod  ( N  X.  N ) ) ) )
193, 7, 9, 15, 18gsumpropd 15768 . 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 6845 . . . . . 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 4520 . . . 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 6293 . . 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 2441 . . . 4  |-  ( Base `  ( R freeLMod  ( N  X.  N ) ) )  =  ( Base `  ( R freeLMod  ( N  X.  N
) ) )
25 eqid 2441 . . . 4  |-  ( 0g
`  ( R freeLMod  ( N  X.  N ) ) )  =  ( 0g
`  ( R freeLMod  ( N  X.  N ) ) )
26 xpfi 7789 . . . . 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 2539 . . . . 5  |-  ( (
ph  /\  y  e.  J )  ->  (
i  e.  N , 
j  e.  N  |->  U )  e.  ( Base `  A ) )
3120eqcomi 2454 . . . . . 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 2544 . . . 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 4516 . . . . . 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 2454 . . . . . 6  |-  ( 0g
`  A )  =  .0.
4137, 38, 403brtr4g 4465 . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  ( z  e.  ( N  X.  N ) 
|->  [_ ( 1st `  z
)  /  i ]_ [_ ( 2nd `  z
)  /  j ]_ U ) ) finSupp  ( 0g `  A ) )
424, 12mat0 18786 . . . . . 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 4459 . . . 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 18669 . . 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 2482 . 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 5862 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
48 csbov2g 6316 . . . . . . . 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 3818 . . . . . 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 5862 . . . . . . 7  |-  ( 1st `  z )  e.  _V
52 csbov2g 6316 . . . . . . 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 4768 . . . . . . . . . 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 3818 . . . . . . . 8  |-  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ ( y  e.  J  |->  U )  =  [_ ( 1st `  z )  /  i ]_ (
y  e.  J  |->  [_ ( 2nd `  z )  /  j ]_ U
)
57 csbmpt2 4768 . . . . . . . . 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 2470 . . . . . . 7  |-  [_ ( 1st `  z )  / 
i ]_ [_ ( 2nd `  z )  /  j ]_ ( y  e.  J  |->  U )  =  ( y  e.  J  |->  [_ ( 1st `  z )  /  i ]_ [_ ( 2nd `  z )  / 
j ]_ U )
6059oveq2i 6288 . . . . . 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 2475 . . . . 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 4516 . . . 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 6845 . . . 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 2473 . . 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 2486 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 1381    e. wcel 1802   _Vcvv 3093   [_csb 3417   class class class wbr 4433    |-> cmpt 4491    X. cxp 4983   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6779   2ndc2nd 6780   Fincfn 7514   finSupp cfsupp 7827   Basecbs 14504   +g cplusg 14569   0gc0g 14709    gsumg cgsu 14710   Ringcrg 17066   freeLMod cfrlm 18644   Mat cmat 18776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-ot 4019  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  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 10980  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-hash 12380  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-hom 14593  df-cco 14594  df-0g 14711  df-gsum 14712  df-prds 14717  df-pws 14719  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-mhm 15835  df-grp 15926  df-minusg 15927  df-sbg 15928  df-subg 16067  df-cntz 16224  df-cmn 16669  df-abl 16670  df-mgp 17010  df-ur 17022  df-ring 17068  df-subrg 17295  df-lmod 17382  df-lss 17447  df-sra 17686  df-rgmod 17687  df-dsmm 18630  df-frlm 18645  df-mat 18777
This theorem is referenced by:  decpmatmul  19140  pmatcollpw2  19146
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