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Theorem prdsgsumOLD 16462
Description: Finite commutative sums in a product structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.) Obsolete version of prdsgsum 16461 as of 9-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
prdsgsumOLD.y  |-  Y  =  ( S X_s ( x  e.  I  |->  R ) )
prdsgsumOLD.b  |-  B  =  ( Base `  R
)
prdsgsumOLD.z  |-  .0.  =  ( 0g `  Y )
prdsgsumOLD.i  |-  ( ph  ->  I  e.  V )
prdsgsumOLD.j  |-  ( ph  ->  J  e.  W )
prdsgsumOLD.s  |-  ( ph  ->  S  e.  X )
prdsgsumOLD.r  |-  ( (
ph  /\  x  e.  I )  ->  R  e. CMnd )
prdsgsumOLD.f  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  J ) )  ->  U  e.  B )
prdsgsumOLD.w  |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V 
\  {  .0.  }
) )  e.  Fin )
Assertion
Ref Expression
prdsgsumOLD  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
Distinct variable groups:    x, y, I    x, J, y    x, Y, y    ph, x, y
Allowed substitution hints:    B( x, y)    R( x, y)    S( x, y)    U( x, y)    V( x, y)    W( x, y)    X( x, y)    .0. ( x, y)

Proof of Theorem prdsgsumOLD
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 prdsgsumOLD.y . . . 4  |-  Y  =  ( S X_s ( x  e.  I  |->  R ) )
2 eqid 2441 . . . 4  |-  ( Base `  Y )  =  (
Base `  Y )
3 prdsgsumOLD.s . . . 4  |-  ( ph  ->  S  e.  X )
4 prdsgsumOLD.i . . . 4  |-  ( ph  ->  I  e.  V )
5 prdsgsumOLD.r . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  R  e. CMnd )
6 eqid 2441 . . . . . 6  |-  ( x  e.  I  |->  R )  =  ( x  e.  I  |->  R )
75, 6fmptd 5864 . . . . 5  |-  ( ph  ->  ( x  e.  I  |->  R ) : I -->CMnd )
8 ffn 5556 . . . . 5  |-  ( ( x  e.  I  |->  R ) : I -->CMnd  ->  ( x  e.  I  |->  R )  Fn  I )
97, 8syl 16 . . . 4  |-  ( ph  ->  ( x  e.  I  |->  R )  Fn  I
)
10 prdsgsumOLD.z . . . . 5  |-  .0.  =  ( 0g `  Y )
111, 4, 3, 7prdscmnd 16336 . . . . 5  |-  ( ph  ->  Y  e. CMnd )
12 prdsgsumOLD.j . . . . 5  |-  ( ph  ->  J  e.  W )
13 prdsgsumOLD.f . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  J ) )  ->  U  e.  B )
1413anassrs 643 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  U  e.  B )
1514an32s 797 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  J )  /\  x  e.  I )  ->  U  e.  B )
1615ralrimiva 2797 . . . . . . 7  |-  ( (
ph  /\  y  e.  J )  ->  A. x  e.  I  U  e.  B )
175ralrimiva 2797 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  I  R  e. CMnd )
18 prdsgsumOLD.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
191, 2, 3, 4, 17, 18prdsbasmpt2 14416 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  I  |->  U )  e.  ( Base `  Y
)  <->  A. x  e.  I  U  e.  B )
)
2019adantr 462 . . . . . . 7  |-  ( (
ph  /\  y  e.  J )  ->  (
( x  e.  I  |->  U )  e.  (
Base `  Y )  <->  A. x  e.  I  U  e.  B ) )
2116, 20mpbird 232 . . . . . 6  |-  ( (
ph  /\  y  e.  J )  ->  (
x  e.  I  |->  U )  e.  ( Base `  Y ) )
22 eqid 2441 . . . . . 6  |-  ( y  e.  J  |->  ( x  e.  I  |->  U ) )  =  ( y  e.  J  |->  ( x  e.  I  |->  U ) )
2321, 22fmptd 5864 . . . . 5  |-  ( ph  ->  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) : J --> ( Base `  Y
) )
24 prdsgsumOLD.w . . . . 5  |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V 
\  {  .0.  }
) )  e.  Fin )
252, 10, 11, 12, 23, 24gsumclOLD 16393 . . . 4  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  e.  ( Base `  Y
) )
261, 2, 3, 4, 9, 25prdsbasfn 14405 . . 3  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  Fn  I )
27 nfcv 2577 . . . . 5  |-  F/_ x Y
28 nfcv 2577 . . . . 5  |-  F/_ x  gsumg
29 nfcv 2577 . . . . . 6  |-  F/_ x J
30 nfmpt1 4378 . . . . . 6  |-  F/_ x
( x  e.  I  |->  U )
3129, 30nfmpt 4377 . . . . 5  |-  F/_ x
( y  e.  J  |->  ( x  e.  I  |->  U ) )
3227, 28, 31nfov 6113 . . . 4  |-  F/_ x
( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )
3332dffn5f 5743 . . 3  |-  ( ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  Fn  I  <->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) `  x
) ) )
3426, 33sylib 196 . 2  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( ( Y 
gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) `
 x ) ) )
35 simpr 458 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
3635adantr 462 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  x  e.  I )
37 eqid 2441 . . . . . . . 8  |-  ( x  e.  I  |->  U )  =  ( x  e.  I  |->  U )
3837fvmpt2 5778 . . . . . . 7  |-  ( ( x  e.  I  /\  U  e.  B )  ->  ( ( x  e.  I  |->  U ) `  x )  =  U )
3936, 14, 38syl2anc 656 . . . . . 6  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  (
( x  e.  I  |->  U ) `  x
)  =  U )
4039mpteq2dva 4375 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
y  e.  J  |->  ( ( x  e.  I  |->  U ) `  x
) )  =  ( y  e.  J  |->  U ) )
4140oveq2d 6106 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( R  gsumg  ( y  e.  J  |->  ( ( x  e.  I  |->  U ) `  x ) ) )  =  ( R  gsumg  ( y  e.  J  |->  U ) ) )
4211adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  Y  e. CMnd )
43 cmnmnd 16285 . . . . . 6  |-  ( R  e. CMnd  ->  R  e.  Mnd )
445, 43syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  R  e.  Mnd )
4512adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  J  e.  W )
464adantr 462 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  V )
473adantr 462 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  X )
4844, 6fmptd 5864 . . . . . . . 8  |-  ( ph  ->  ( x  e.  I  |->  R ) : I --> Mnd )
4948adantr 462 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
x  e.  I  |->  R ) : I --> Mnd )
501, 2, 46, 47, 49, 35prdspjmhm 15490 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
a  e.  ( Base `  Y )  |->  ( a `
 x ) )  e.  ( Y MndHom  (
( x  e.  I  |->  R ) `  x
) ) )
516fvmpt2 5778 . . . . . . . 8  |-  ( ( x  e.  I  /\  R  e. CMnd )  ->  ( ( x  e.  I  |->  R ) `  x
)  =  R )
5235, 5, 51syl2anc 656 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  R ) `  x
)  =  R )
5352oveq2d 6106 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( Y MndHom  ( ( x  e.  I  |->  R ) `  x ) )  =  ( Y MndHom  R ) )
5450, 53eleqtrd 2517 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
a  e.  ( Base `  Y )  |->  ( a `
 x ) )  e.  ( Y MndHom  R
) )
5521adantlr 709 . . . . 5  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  (
x  e.  I  |->  U )  e.  ( Base `  Y ) )
5624adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
57 fveq1 5687 . . . . 5  |-  ( a  =  ( x  e.  I  |->  U )  -> 
( a `  x
)  =  ( ( x  e.  I  |->  U ) `  x ) )
58 fveq1 5687 . . . . 5  |-  ( a  =  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  ->  (
a `  x )  =  ( ( Y 
gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) `
 x ) )
592, 10, 42, 44, 45, 54, 55, 56, 57, 58gsummhm2OLD 16427 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( R  gsumg  ( y  e.  J  |->  ( ( x  e.  I  |->  U ) `  x ) ) )  =  ( ( Y 
gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) `
 x ) )
6041, 59eqtr3d 2475 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) `  x
) )
6160mpteq2dva 4375 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) )  =  ( x  e.  I  |->  ( ( Y 
gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) `
 x ) ) )
6234, 61eqtr4d 2476 1  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    \ cdif 3322   {csn 3874    e. cmpt 4347   `'ccnv 4835   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   Fincfn 7306   Basecbs 14170   0gc0g 14374    gsumg cgsu 14375   X_scprds 14380   Mndcmnd 15405   MndHom cmhm 15458  CMndccmn 16270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-hom 14258  df-cco 14259  df-0g 14376  df-gsum 14377  df-prds 14382  df-mnd 15411  df-mhm 15460  df-cntz 15828  df-cmn 16272
This theorem is referenced by:  pwsgsumOLD  16464
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