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Theorem psrplusgpropd 16313
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 958 . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
3 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2283 . . . . . . . . . . 11  |-  { c  e.  ( NN0  ^m  I )  |  ( `' c " NN )  e.  Fin }  =  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin }
5 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
6 simp2 956 . . . . . . . . . . 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 16125 . . . . . . . . . 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
) )
8 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( a : { c  e.  ( NN0  ^m  I )  |  ( `' c " NN )  e.  Fin } --> ( Base `  R )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  (
a `  d )  e.  ( Base `  R
) )
97, 8sylan 457 . . . . . . . . 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
) )
10 psrplusgpropd.b1 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( Base `  R ) )
111, 10syl 15 . . . . . . . . 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
) )
129, 11eleqtrrd 2360 . . . . . . . 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 )
13 simp3 957 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  b  e.  (
Base `  ( I mPwSer  R ) ) )
142, 3, 4, 5, 13psrelbas 16125 . . . . . . . . . 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
) )
15 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( b : { c  e.  ( NN0  ^m  I )  |  ( `' c " NN )  e.  Fin } --> ( Base `  R )  /\  d  e.  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )  ->  (
b `  d )  e.  ( Base `  R
) )
1614, 15sylan 457 . . . . . . . . 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
) )
1716, 11eleqtrrd 2360 . . . . . . . 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 )
18 psrplusgpropd.p . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  S ) y ) )
1918proplem 13592 . . . . . . . 8  |-  ( (
ph  /\  ( (
a `  d )  e.  B  /\  (
b `  d )  e.  B ) )  -> 
( ( a `  d ) ( +g  `  R ) ( b `
 d ) )  =  ( ( a `
 d ) ( +g  `  S ) ( b `  d
) ) )
201, 12, 17, 19syl12anc 1180 . . . . . . 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 ) ) )
2120mpteq2dva 4106 . . . . . 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 ) ) ) )
22 ffn 5389 . . . . . . . 8  |-  ( a : { c  e.  ( NN0  ^m  I
)  |  ( `' c " NN )  e.  Fin } --> ( Base `  R )  ->  a  Fn  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )
237, 22syl 15 . . . . . . 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 } )
24 ffn 5389 . . . . . . . 8  |-  ( b : { c  e.  ( NN0  ^m  I
)  |  ( `' c " NN )  e.  Fin } --> ( Base `  R )  ->  b  Fn  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin } )
2514, 24syl 15 . . . . . . 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 } )
26 ovex 5883 . . . . . . . . 9  |-  ( NN0 
^m  I )  e. 
_V
2726rabex 4165 . . . . . . . 8  |-  { c  e.  ( NN0  ^m  I )  |  ( `' c " NN )  e.  Fin }  e.  _V
2827a1i 10 . . . . . . 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 )
29 inidm 3378 . . . . . . 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 }
30 eqidd 2284 . . . . . . 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 ) )
31 eqidd 2284 . . . . . . 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 ) )
3223, 25, 28, 28, 29, 30, 31offval 6085 . . . . . 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 ) ) ) )
3323, 25, 28, 28, 29, 30, 31offval 6085 . . . . . 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 ) ) ) )
3421, 32, 333eqtr4d 2325 . . . . 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 ) )
3534mpt2eq3dva 5912 . . . 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 ) ) )
36 psrplusgpropd.b2 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  S ) )
3710, 36eqtr3d 2317 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  S ) )
3837psrbaspropd 16312 . . . . 5  |-  ( ph  ->  ( Base `  (
I mPwSer  R ) )  =  ( Base `  (
I mPwSer  S ) ) )
39 mpt2eq12 5908 . . . . 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 ) ) )
4038, 38, 39syl2anc 642 . . . 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 ) ) )
4135, 40eqtrd 2315 . . 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 ) ) )
42 ofmres 6116 . . 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 ) )
43 ofmres 6116 . . 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 ) )
4441, 42, 433eqtr4g 2340 . 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 ) ) ) ) )
45 eqid 2283 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
46 eqid 2283 . . 3  |-  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  ( I mPwSer  R ) )
472, 5, 45, 46psrplusg 16126 . 2  |-  ( +g  `  ( I mPwSer  R ) )  =  (  o F ( +g  `  R
)  |`  ( ( Base `  ( I mPwSer  R ) )  X.  ( Base `  ( I mPwSer  R ) ) ) )
48 eqid 2283 . . 3  |-  ( I mPwSer  S )  =  ( I mPwSer  S )
49 eqid 2283 . . 3  |-  ( Base `  ( I mPwSer  S ) )  =  ( Base `  ( I mPwSer  S ) )
50 eqid 2283 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
51 eqid 2283 . . 3  |-  ( +g  `  ( I mPwSer  S ) )  =  ( +g  `  ( I mPwSer  S ) )
5248, 49, 50, 51psrplusg 16126 . 2  |-  ( +g  `  ( I mPwSer  S ) )  =  (  o F ( +g  `  S
)  |`  ( ( Base `  ( I mPwSer  S ) )  X.  ( Base `  ( I mPwSer  S ) ) ) )
5344, 47, 523eqtr4g 2340 1  |-  ( ph  ->  ( +g  `  (
I mPwSer  R ) )  =  ( +g  `  (
I mPwSer  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    e. cmpt 4077    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    o Fcof 6076    ^m cmap 6772   Fincfn 6863   NNcn 9746   NN0cn0 9965   Basecbs 13148   +g cplusg 13208   mPwSer cmps 16087
This theorem is referenced by:  ply1plusgpropd  16322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-psr 16098
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