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Theorem prdsgsumOLD 16477
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 16476 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 2443 . . . 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 2443 . . . . . 6  |-  ( x  e.  I  |->  R )  =  ( x  e.  I  |->  R )
75, 6fmptd 5872 . . . . 5  |-  ( ph  ->  ( x  e.  I  |->  R ) : I -->CMnd )
8 ffn 5564 . . . . 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 16348 . . . . 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 648 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  U  e.  B )
1514an32s 802 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  J )  /\  x  e.  I )  ->  U  e.  B )
1615ralrimiva 2804 . . . . . . 7  |-  ( (
ph  /\  y  e.  J )  ->  A. x  e.  I  U  e.  B )
175ralrimiva 2804 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  I  R  e. CMnd )
18 prdsgsumOLD.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
191, 2, 3, 4, 17, 18prdsbasmpt2 14425 . . . . . . . 8  |-  ( ph  ->  ( ( x  e.  I  |->  U )  e.  ( Base `  Y
)  <->  A. x  e.  I  U  e.  B )
)
2019adantr 465 . . . . . . 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 2443 . . . . . 6  |-  ( y  e.  J  |->  ( x  e.  I  |->  U ) )  =  ( y  e.  J  |->  ( x  e.  I  |->  U ) )
2321, 22fmptd 5872 . . . . 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 16405 . . . 4  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  e.  ( Base `  Y
) )
261, 2, 3, 4, 9, 25prdsbasfn 14414 . . 3  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  Fn  I )
27 nfcv 2584 . . . . 5  |-  F/_ x Y
28 nfcv 2584 . . . . 5  |-  F/_ x  gsumg
29 nfcv 2584 . . . . . 6  |-  F/_ x J
30 nfmpt1 4386 . . . . . 6  |-  F/_ x
( x  e.  I  |->  U )
3129, 30nfmpt 4385 . . . . 5  |-  F/_ x
( y  e.  J  |->  ( x  e.  I  |->  U ) )
3227, 28, 31nfov 6119 . . . 4  |-  F/_ x
( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )
3332dffn5f 5751 . . 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 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
3635adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  x  e.  I )
37 eqid 2443 . . . . . . . 8  |-  ( x  e.  I  |->  U )  =  ( x  e.  I  |->  U )
3837fvmpt2 5786 . . . . . . 7  |-  ( ( x  e.  I  /\  U  e.  B )  ->  ( ( x  e.  I  |->  U ) `  x )  =  U )
3936, 14, 38syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  (
( x  e.  I  |->  U ) `  x
)  =  U )
4039mpteq2dva 4383 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
y  e.  J  |->  ( ( x  e.  I  |->  U ) `  x
) )  =  ( y  e.  J  |->  U ) )
4140oveq2d 6112 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( R  gsumg  ( y  e.  J  |->  ( ( x  e.  I  |->  U ) `  x ) ) )  =  ( R  gsumg  ( y  e.  J  |->  U ) ) )
4211adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  Y  e. CMnd )
43 cmnmnd 16297 . . . . . 6  |-  ( R  e. CMnd  ->  R  e.  Mnd )
445, 43syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  R  e.  Mnd )
4512adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  J  e.  W )
464adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  V )
473adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  X )
4844, 6fmptd 5872 . . . . . . . 8  |-  ( ph  ->  ( x  e.  I  |->  R ) : I --> Mnd )
4948adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
x  e.  I  |->  R ) : I --> Mnd )
501, 2, 46, 47, 49, 35prdspjmhm 15500 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
a  e.  ( Base `  Y )  |->  ( a `
 x ) )  e.  ( Y MndHom  (
( x  e.  I  |->  R ) `  x
) ) )
516fvmpt2 5786 . . . . . . . 8  |-  ( ( x  e.  I  /\  R  e. CMnd )  ->  ( ( x  e.  I  |->  R ) `  x
)  =  R )
5235, 5, 51syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  (
( x  e.  I  |->  R ) `  x
)  =  R )
5352oveq2d 6112 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( Y MndHom  ( ( x  e.  I  |->  R ) `  x ) )  =  ( Y MndHom  R ) )
5450, 53eleqtrd 2519 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
a  e.  ( Base `  Y )  |->  ( a `
 x ) )  e.  ( Y MndHom  R
) )
5521adantlr 714 . . . . 5  |-  ( ( ( ph  /\  x  e.  I )  /\  y  e.  J )  ->  (
x  e.  I  |->  U )  e.  ( Base `  Y ) )
5624adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
57 fveq1 5695 . . . . 5  |-  ( a  =  ( x  e.  I  |->  U )  -> 
( a `  x
)  =  ( ( x  e.  I  |->  U ) `  x ) )
58 fveq1 5695 . . . . 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 16440 . . . 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 2477 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) `  x
) )
6160mpteq2dva 4383 . 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 2478 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 1369    e. wcel 1756   A.wral 2720   _Vcvv 2977    \ cdif 3330   {csn 3882    e. cmpt 4355   `'ccnv 4844   "cima 4848    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   Fincfn 7315   Basecbs 14179   0gc0g 14383    gsumg cgsu 14384   X_scprds 14389   Mndcmnd 15414   MndHom cmhm 15467  CMndccmn 16282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-hom 14267  df-cco 14268  df-0g 14385  df-gsum 14386  df-prds 14391  df-mnd 15420  df-mhm 15469  df-cntz 15840  df-cmn 16284
This theorem is referenced by:  pwsgsumOLD  16479
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