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Theorem psrplusgpropd 16592
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 960 . . . . . . . 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 2412 . . . . . . . . . . 11  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
3 eqid 2412 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2412 . . . . . . . . . . 11  |-  { c  e.  ( NN0  ^m  I )  |  ( `' c " NN )  e.  Fin }  =  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin }
5 eqid 2412 . . . . . . . . . . 11  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
6 simp2 958 . . . . . . . . . . 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 16407 . . . . . . . . . 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 5837 . . . . . . . . 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 2489 . . . . . . . 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 959 . . . . . . . . . . 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 16407 . . . . . . . . . 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 5837 . . . . . . . . 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 2489 . . . . . . . 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 13878 . . . . . . . 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 1182 . . . . . . 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 4263 . . . . . 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 5558 . . . . . . . 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 5558 . . . . . . . 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 6073 . . . . . . . . 9  |-  ( NN0 
^m  I )  e. 
_V
2524rabex 4322 . . . . . . . 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 3518 . . . . . . 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 2413 . . . . . . 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 2413 . . . . . . 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 6279 . . . . . 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 6279 . . . . . 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 2454 . . . . 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 6105 . . . 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 2446 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  S ) )
3635psrbaspropd 16591 . . . . 5  |-  ( ph  ->  ( Base `  (
I mPwSer  R ) )  =  ( Base `  (
I mPwSer  S ) ) )
37 mpt2eq12 6101 . . . . 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 643 . . . 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 2444 . . 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 6310 . . 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 6310 . . 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 2469 . 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 2412 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
44 eqid 2412 . . 3  |-  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  ( I mPwSer  R ) )
452, 5, 43, 44psrplusg 16408 . 2  |-  ( +g  `  ( I mPwSer  R ) )  =  (  o F ( +g  `  R
)  |`  ( ( Base `  ( I mPwSer  R ) )  X.  ( Base `  ( I mPwSer  R ) ) ) )
46 eqid 2412 . . 3  |-  ( I mPwSer  S )  =  ( I mPwSer  S )
47 eqid 2412 . . 3  |-  ( Base `  ( I mPwSer  S ) )  =  ( Base `  ( I mPwSer  S ) )
48 eqid 2412 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
49 eqid 2412 . . 3  |-  ( +g  `  ( I mPwSer  S ) )  =  ( +g  `  ( I mPwSer  S ) )
5046, 47, 48, 49psrplusg 16408 . 2  |-  ( +g  `  ( I mPwSer  S ) )  =  (  o F ( +g  `  S
)  |`  ( ( Base `  ( I mPwSer  S ) )  X.  ( Base `  ( I mPwSer  S ) ) ) )
5142, 45, 503eqtr4g 2469 1  |-  ( ph  ->  ( +g  `  (
I mPwSer  R ) )  =  ( +g  `  (
I mPwSer  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {crab 2678   _Vcvv 2924    e. cmpt 4234    X. cxp 4843   `'ccnv 4844    |` cres 4847   "cima 4848    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050    o Fcof 6270    ^m cmap 6985   Fincfn 7076   NNcn 9964   NN0cn0 10185   Basecbs 13432   +g cplusg 13492   mPwSer cmps 16369
This theorem is referenced by:  ply1plusgpropd  16601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-tset 13511  df-psr 16380
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