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Theorem prdsgsumOLD 16879
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 16878 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 2467 . . . 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 2467 . . . . . 6  |-  ( x  e.  I  |->  R )  =  ( x  e.  I  |->  R )
75, 6fmptd 6056 . . . . 5  |-  ( ph  ->  ( x  e.  I  |->  R ) : I -->CMnd )
8 ffn 5737 . . . . 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 16738 . . . . 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 2881 . . . . . . 7  |-  ( (
ph  /\  y  e.  J )  ->  A. x  e.  I  U  e.  B )
175ralrimiva 2881 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  I  R  e. CMnd )
18 prdsgsumOLD.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
191, 2, 3, 4, 17, 18prdsbasmpt2 14753 . . . . . . . 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 2467 . . . . . 6  |-  ( y  e.  J  |->  ( x  e.  I  |->  U ) )  =  ( y  e.  J  |->  ( x  e.  I  |->  U ) )
2321, 22fmptd 6056 . . . . 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 16797 . . . 4  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  e.  ( Base `  Y
) )
261, 2, 3, 4, 9, 25prdsbasfn 14742 . . 3  |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  Fn  I )
27 nfcv 2629 . . . . 5  |-  F/_ x Y
28 nfcv 2629 . . . . 5  |-  F/_ x  gsumg
29 nfcv 2629 . . . . . 6  |-  F/_ x J
30 nfmpt1 4542 . . . . . 6  |-  F/_ x
( x  e.  I  |->  U )
3129, 30nfmpt 4541 . . . . 5  |-  F/_ x
( y  e.  J  |->  ( x  e.  I  |->  U ) )
3227, 28, 31nfov 6318 . . . 4  |-  F/_ x
( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )
3332dffn5f 5929 . . 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 2467 . . . . . . . 8  |-  ( x  e.  I  |->  U )  =  ( x  e.  I  |->  U )
3837fvmpt2 5964 . . . . . . 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 4539 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
y  e.  J  |->  ( ( x  e.  I  |->  U ) `  x
) )  =  ( y  e.  J  |->  U ) )
4140oveq2d 6311 . . . 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 16684 . . . . . 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 6056 . . . . . . . 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 15869 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
a  e.  ( Base `  Y )  |->  ( a `
 x ) )  e.  ( Y MndHom  (
( x  e.  I  |->  R ) `  x
) ) )
516fvmpt2 5964 . . . . . . . 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 6311 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( Y MndHom  ( ( x  e.  I  |->  R ) `  x ) )  =  ( Y MndHom  R ) )
5450, 53eleqtrd 2557 . . . . 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 5871 . . . . 5  |-  ( a  =  ( x  e.  I  |->  U )  -> 
( a `  x
)  =  ( ( x  e.  I  |->  U ) `  x ) )
58 fveq1 5871 . . . . 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 16833 . . . 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 2510 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  ( R  gsumg  ( y  e.  J  |->  U ) )  =  ( ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) ) `  x
) )
6160mpteq2dva 4539 . 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 2511 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 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    \ cdif 3478   {csn 4033    |-> cmpt 4511   `'ccnv 5004   "cima 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   Fincfn 7528   Basecbs 14506   0gc0g 14711    gsumg cgsu 14712   X_scprds 14717   Mndcmnd 15792   MndHom cmhm 15836  CMndccmn 16669
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-plusg 14584  df-mulr 14585  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-hom 14595  df-cco 14596  df-0g 14713  df-gsum 14714  df-prds 14719  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-mhm 15838  df-cntz 16226  df-cmn 16671
This theorem is referenced by:  pwsgsumOLD  16881
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