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Theorem psrplusgpropd 17588
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 984 . . . . . . . 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 2433 . . . . . . . . . . 11  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
3 eqid 2433 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2433 . . . . . . . . . . 11  |-  { c  e.  ( NN0  ^m  I )  |  ( `' c " NN )  e.  Fin }  =  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin }
5 eqid 2433 . . . . . . . . . . 11  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
6 simp2 982 . . . . . . . . . . 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 17384 . . . . . . . . . 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 5831 . . . . . . . . 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 2510 . . . . . . . 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 983 . . . . . . . . . . 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 17384 . . . . . . . . . 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 5831 . . . . . . . . 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 2510 . . . . . . . 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 14611 . . . . . . . 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 1209 . . . . . . 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 4366 . . . . . 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 5547 . . . . . . . 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 5547 . . . . . . . 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 6105 . . . . . . . . 9  |-  ( NN0 
^m  I )  e. 
_V
2524rabex 4431 . . . . . . . 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 3547 . . . . . . 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 2434 . . . . . . 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 2434 . . . . . . 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 6316 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  ( a  oF ( +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 6316 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  ( a  oF ( +g  `  S
) b )  =  ( d  e.  {
c  e.  ( NN0 
^m  I )  |  ( `' c " NN )  e.  Fin } 
|->  ( ( a `  d ) ( +g  `  S ) ( b `
 d ) ) ) )
3219, 30, 313eqtr4d 2475 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  ( I mPwSer  R ) )  /\  b  e.  ( Base `  (
I mPwSer  R ) ) )  ->  ( a  oF ( +g  `  R
) b )  =  ( a  oF ( +g  `  S
) b ) )
3332mpt2eq3dva 6139 . . . 4  |-  ( ph  ->  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  oF ( +g  `  R ) b ) )  =  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  oF ( +g  `  S ) b ) ) )
34 psrplusgpropd.b2 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  S ) )
359, 34eqtr3d 2467 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  S ) )
3635psrbaspropd 17587 . . . . 5  |-  ( ph  ->  ( Base `  (
I mPwSer  R ) )  =  ( Base `  (
I mPwSer  S ) ) )
37 mpt2eq12 6135 . . . . 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  oF ( +g  `  S ) b ) )  =  ( a  e.  (
Base `  ( I mPwSer  S ) ) ,  b  e.  ( Base `  (
I mPwSer  S ) )  |->  ( a  oF ( +g  `  S ) b ) ) )
3836, 36, 37syl2anc 654 . . . 4  |-  ( ph  ->  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  oF ( +g  `  S ) b ) )  =  ( a  e.  (
Base `  ( I mPwSer  S ) ) ,  b  e.  ( Base `  (
I mPwSer  S ) )  |->  ( a  oF ( +g  `  S ) b ) ) )
3933, 38eqtrd 2465 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  oF ( +g  `  R ) b ) )  =  ( a  e.  (
Base `  ( I mPwSer  S ) ) ,  b  e.  ( Base `  (
I mPwSer  S ) )  |->  ( a  oF ( +g  `  S ) b ) ) )
40 ofmres 6562 . . 3  |-  (  oF ( +g  `  R
)  |`  ( ( Base `  ( I mPwSer  R ) )  X.  ( Base `  ( I mPwSer  R ) ) ) )  =  ( a  e.  (
Base `  ( I mPwSer  R ) ) ,  b  e.  ( Base `  (
I mPwSer  R ) )  |->  ( a  oF ( +g  `  R ) b ) )
41 ofmres 6562 . . 3  |-  (  oF ( +g  `  S
)  |`  ( ( Base `  ( I mPwSer  S ) )  X.  ( Base `  ( I mPwSer  S ) ) ) )  =  ( a  e.  (
Base `  ( I mPwSer  S ) ) ,  b  e.  ( Base `  (
I mPwSer  S ) )  |->  ( a  oF ( +g  `  S ) b ) )
4239, 40, 413eqtr4g 2490 . 2  |-  ( ph  ->  (  oF ( +g  `  R )  |`  ( ( Base `  (
I mPwSer  R ) )  X.  ( Base `  (
I mPwSer  R ) ) ) )  =  (  oF ( +g  `  S
)  |`  ( ( Base `  ( I mPwSer  S ) )  X.  ( Base `  ( I mPwSer  S ) ) ) ) )
43 eqid 2433 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
44 eqid 2433 . . 3  |-  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  ( I mPwSer  R ) )
452, 5, 43, 44psrplusg 17386 . 2  |-  ( +g  `  ( I mPwSer  R ) )  =  (  oF ( +g  `  R
)  |`  ( ( Base `  ( I mPwSer  R ) )  X.  ( Base `  ( I mPwSer  R ) ) ) )
46 eqid 2433 . . 3  |-  ( I mPwSer  S )  =  ( I mPwSer  S )
47 eqid 2433 . . 3  |-  ( Base `  ( I mPwSer  S ) )  =  ( Base `  ( I mPwSer  S ) )
48 eqid 2433 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
49 eqid 2433 . . 3  |-  ( +g  `  ( I mPwSer  S ) )  =  ( +g  `  ( I mPwSer  S ) )
5046, 47, 48, 49psrplusg 17386 . 2  |-  ( +g  `  ( I mPwSer  S ) )  =  (  oF ( +g  `  S
)  |`  ( ( Base `  ( I mPwSer  S ) )  X.  ( Base `  ( I mPwSer  S ) ) ) )
5142, 45, 503eqtr4g 2490 1  |-  ( ph  ->  ( +g  `  (
I mPwSer  R ) )  =  ( +g  `  (
I mPwSer  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   {crab 2709   _Vcvv 2962    e. cmpt 4338    X. cxp 4825   `'ccnv 4826    |` cres 4829   "cima 4830    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082    oFcof 6307    ^m cmap 7202   Fincfn 7298   NNcn 10310   NN0cn0 10567   Basecbs 14157   +g cplusg 14221   mPwSer cmps 17340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-tset 14240  df-psr 17351
This theorem is referenced by:  ply1plusgpropd  17597
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