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Theorem psrplusgpropd 16660
Description: Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
Hypotheses
Ref Expression
psrplusgpropd.b1  |-  ( ph  ->  B  =  ( Base `  R ) )
psrplusgpropd.b2  |-  ( ph  ->  B  =  ( Base `  S ) )
psrplusgpropd.p  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  S ) y ) )
Assertion
Ref Expression
psrplusgpropd  |-  ( ph  ->  ( +g  `  (
I mPwSer  R ) )  =  ( +g  `  (
I mPwSer  S ) ) )
Distinct variable groups:    ph, y, x   
x, B, y    y, R, x    y, S, x
Allowed substitution hints:    I( x, y)

Proof of Theorem psrplusgpropd
Dummy variables  a 
b  d  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 961 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  (
I mPwSer  R ) )  /\  b  e.  ( Base `  ( I mPwSer  R ) ) )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  ph )
2 eqid 2442 . . . . . . . . . . 11  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
3 eqid 2442 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2442 . . . . . . . . . . 11  |-  { c  e.  ( NN0  ^m  I )  |  ( `' c " NN )  e.  Fin }  =  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin }
5 eqid 2442 . . . . . . . . . . 11  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
6 simp2 959 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  a  e.  (
Base `  ( I mPwSer  R ) ) )
72, 3, 4, 5, 6psrelbas 16475 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  a : {
c  e.  ( NN0 
^m  I )  |  ( `' c " NN )  e.  Fin } --> ( Base `  R
) )
87ffvelrnda 5899 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ( Base `  (
I mPwSer  R ) )  /\  b  e.  ( Base `  ( I mPwSer  R ) ) )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  (
a `  d )  e.  ( Base `  R
) )
9 psrplusgpropd.b1 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( Base `  R ) )
101, 9syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ( Base `  (
I mPwSer  R ) )  /\  b  e.  ( Base `  ( I mPwSer  R ) ) )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  B  =  ( Base `  R
) )
118, 10eleqtrrd 2519 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  (
I mPwSer  R ) )  /\  b  e.  ( Base `  ( I mPwSer  R ) ) )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  (
a `  d )  e.  B )
12 simp3 960 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  b  e.  (
Base `  ( I mPwSer  R ) ) )
132, 3, 4, 5, 12psrelbas 16475 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  b : {
c  e.  ( NN0 
^m  I )  |  ( `' c " NN )  e.  Fin } --> ( Base `  R
) )
1413ffvelrnda 5899 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  ( Base `  (
I mPwSer  R ) )  /\  b  e.  ( Base `  ( I mPwSer  R ) ) )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  (
b `  d )  e.  ( Base `  R
) )
1514, 10eleqtrrd 2519 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  (
I mPwSer  R ) )  /\  b  e.  ( Base `  ( I mPwSer  R ) ) )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  (
b `  d )  e.  B )
16 psrplusgpropd.p . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  S ) y ) )
1716proplem 13946 . . . . . . . 8  |-  ( (
ph  /\  ( (
a `  d )  e.  B  /\  (
b `  d )  e.  B ) )  -> 
( ( a `  d ) ( +g  `  R ) ( b `
 d ) )  =  ( ( a `
 d ) ( +g  `  S ) ( b `  d
) ) )
181, 11, 15, 17syl12anc 1183 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( Base `  (
I mPwSer  R ) )  /\  b  e.  ( Base `  ( I mPwSer  R ) ) )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  (
( a `  d
) ( +g  `  R
) ( b `  d ) )  =  ( ( a `  d ) ( +g  `  S ) ( b `
 d ) ) )
1918mpteq2dva 4320 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  ( d  e. 
{ c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin }  |->  ( ( a `  d ) ( +g  `  R
) ( b `  d ) ) )  =  ( d  e. 
{ c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin }  |->  ( ( a `  d ) ( +g  `  S
) ( b `  d ) ) ) )
20 ffn 5620 . . . . . . . 8  |-  ( a : { c  e.  ( NN0  ^m  I
)  |  ( `' c " NN )  e.  Fin } --> ( Base `  R )  ->  a  Fn  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )
217, 20syl 16 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  a  Fn  {
c  e.  ( NN0 
^m  I )  |  ( `' c " NN )  e.  Fin } )
22 ffn 5620 . . . . . . . 8  |-  ( b : { c  e.  ( NN0  ^m  I
)  |  ( `' c " NN )  e.  Fin } --> ( Base `  R )  ->  b  Fn  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )
2313, 22syl 16 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  b  Fn  {
c  e.  ( NN0 
^m  I )  |  ( `' c " NN )  e.  Fin } )
24 ovex 6135 . . . . . . . . 9  |-  ( NN0 
^m  I )  e. 
_V
2524rabex 4383 . . . . . . . 8  |-  { c  e.  ( NN0  ^m  I )  |  ( `' c " NN )  e.  Fin }  e.  _V
2625a1i 11 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  { c  e.  ( NN0  ^m  I
)  |  ( `' c " NN )  e.  Fin }  e.  _V )
27 inidm 3535 . . . . . . 7  |-  ( { c  e.  ( NN0 
^m  I )  |  ( `' c " NN )  e.  Fin }  i^i  { c  e.  ( NN0  ^m  I
)  |  ( `' c " NN )  e.  Fin } )  =  { c  e.  ( NN0  ^m  I
)  |  ( `' c " NN )  e.  Fin }
28 eqidd 2443 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( Base `  (
I mPwSer  R ) )  /\  b  e.  ( Base `  ( I mPwSer  R ) ) )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  (
a `  d )  =  ( a `  d ) )
29 eqidd 2443 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( Base `  (
I mPwSer  R ) )  /\  b  e.  ( Base `  ( I mPwSer  R ) ) )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  (
b `  d )  =  ( b `  d ) )
3021, 23, 26, 26, 27, 28, 29offval 6341 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  ( a  o F ( +g  `  R
) b )  =  ( d  e.  {
c  e.  ( NN0 
^m  I )  |  ( `' c " NN )  e.  Fin } 
|->  ( ( a `  d ) ( +g  `  R ) ( b `
 d ) ) ) )
3121, 23, 26, 26, 27, 28, 29offval 6341 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  ( a  o F ( +g  `  S
) b )  =  ( d  e.  {
c  e.  ( NN0 
^m  I )  |  ( `' c " NN )  e.  Fin } 
|->  ( ( a `  d ) ( +g  `  S ) ( b `
 d ) ) ) )
3219, 30, 313eqtr4d 2484 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  ( a  o F ( +g  `  R
) b )  =  ( a  o F ( +g  `  S
) b ) )
3332mpt2eq3dva 6167 . . . 4  |-  ( ph  ->  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  o F ( +g  `  R ) b ) )  =  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  o F ( +g  `  S ) b ) ) )
34 psrplusgpropd.b2 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  S ) )
359, 34eqtr3d 2476 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  S ) )
3635psrbaspropd 16659 . . . . 5  |-  ( ph  ->  ( Base `  (
I mPwSer  R ) )  =  ( Base `  (
I mPwSer  S ) ) )
37 mpt2eq12 6163 . . . . 5  |-  ( ( ( Base `  (
I mPwSer  R ) )  =  ( Base `  (
I mPwSer  S ) )  /\  ( Base `  ( I mPwSer  R ) )  =  (
Base `  ( I mPwSer  S ) ) )  -> 
( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  o F ( +g  `  S ) b ) )  =  ( a  e.  (
Base `  ( I mPwSer  S ) ) ,  b  e.  ( Base `  (
I mPwSer  S ) )  |->  ( a  o F ( +g  `  S ) b ) ) )
3836, 36, 37syl2anc 644 . . . 4  |-  ( ph  ->  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  o F ( +g  `  S ) b ) )  =  ( a  e.  (
Base `  ( I mPwSer  S ) ) ,  b  e.  ( Base `  (
I mPwSer  S ) )  |->  ( a  o F ( +g  `  S ) b ) ) )
3933, 38eqtrd 2474 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  o F ( +g  `  R ) b ) )  =  ( a  e.  (
Base `  ( I mPwSer  S ) ) ,  b  e.  ( Base `  (
I mPwSer  S ) )  |->  ( a  o F ( +g  `  S ) b ) ) )
40 ofmres 6372 . . 3  |-  (  o F ( +g  `  R
)  |`  ( ( Base `  ( I mPwSer  R ) )  X.  ( Base `  ( I mPwSer  R ) ) ) )  =  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  o F ( +g  `  R ) b ) )
41 ofmres 6372 . . 3  |-  (  o F ( +g  `  S
)  |`  ( ( Base `  ( I mPwSer  S ) )  X.  ( Base `  ( I mPwSer  S ) ) ) )  =  ( a  e.  (
Base `  ( I mPwSer  S ) ) ,  b  e.  ( Base `  (
I mPwSer  S ) )  |->  ( a  o F ( +g  `  S ) b ) )
4239, 40, 413eqtr4g 2499 . 2  |-  ( ph  ->  (  o F ( +g  `  R )  |`  ( ( Base `  (
I mPwSer  R ) )  X.  ( Base `  (
I mPwSer  R ) ) ) )  =  (  o F ( +g  `  S
)  |`  ( ( Base `  ( I mPwSer  S ) )  X.  ( Base `  ( I mPwSer  S ) ) ) ) )
43 eqid 2442 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
44 eqid 2442 . . 3  |-  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  ( I mPwSer  R ) )
452, 5, 43, 44psrplusg 16476 . 2  |-  ( +g  `  ( I mPwSer  R ) )  =  (  o F ( +g  `  R
)  |`  ( ( Base `  ( I mPwSer  R ) )  X.  ( Base `  ( I mPwSer  R ) ) ) )
46 eqid 2442 . . 3  |-  ( I mPwSer  S )  =  ( I mPwSer  S )
47 eqid 2442 . . 3  |-  ( Base `  ( I mPwSer  S ) )  =  ( Base `  ( I mPwSer  S ) )
48 eqid 2442 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
49 eqid 2442 . . 3  |-  ( +g  `  ( I mPwSer  S ) )  =  ( +g  `  ( I mPwSer  S ) )
5046, 47, 48, 49psrplusg 16476 . 2  |-  ( +g  `  ( I mPwSer  S ) )  =  (  o F ( +g  `  S
)  |`  ( ( Base `  ( I mPwSer  S ) )  X.  ( Base `  ( I mPwSer  S ) ) ) )
5142, 45, 503eqtr4g 2499 1  |-  ( ph  ->  ( +g  `  (
I mPwSer  R ) )  =  ( +g  `  (
I mPwSer  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   {crab 2715   _Vcvv 2962    e. cmpt 4291    X. cxp 4905   `'ccnv 4906    |` cres 4909   "cima 4910    Fn wfn 5478   -->wf 5479   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112    o Fcof 6332    ^m cmap 7047   Fincfn 7138   NNcn 10031   NN0cn0 10252   Basecbs 13500   +g cplusg 13560   mPwSer cmps 16437
This theorem is referenced by:  ply1plusgpropd  16669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-tset 13579  df-psr 16448
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