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Theorem psrplusgpropd 18041
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 994 . . . . . . . 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 2460 . . . . . . . . . . 11  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
3 eqid 2460 . . . . . . . . . . 11  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2460 . . . . . . . . . . 11  |-  { c  e.  ( NN0  ^m  I )  |  ( `' c " NN )  e.  Fin }  =  { c  e.  ( NN0  ^m  I )  |  ( `' c
" NN )  e. 
Fin }
5 eqid 2460 . . . . . . . . . . 11  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
6 simp2 992 . . . . . . . . . . 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 17796 . . . . . . . . . 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 6012 . . . . . . . . 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 2551 . . . . . . . 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 993 . . . . . . . . . . 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 17796 . . . . . . . . . 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 6012 . . . . . . . . 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 2551 . . . . . . . 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 14934 . . . . . . . 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 1221 . . . . . . 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 4526 . . . . . 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 5722 . . . . . . . 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 5722 . . . . . . . 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 6300 . . . . . . . . 9  |-  ( NN0 
^m  I )  e. 
_V
2524rabex 4591 . . . . . . . 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 3700 . . . . . . 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 2461 . . . . . . 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 2461 . . . . . . 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 6522 . . . . . 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 6522 . . . . . 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 2511 . . . . 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 6336 . . . 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 2503 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  S ) )
3635psrbaspropd 18040 . . . . 5  |-  ( ph  ->  ( Base `  (
I mPwSer  R ) )  =  ( Base `  (
I mPwSer  S ) ) )
37 mpt2eq12 6332 . . . . 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 661 . . . 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 2501 . . 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 6770 . . 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 6770 . . 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 2526 . 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 2460 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
44 eqid 2460 . . 3  |-  ( +g  `  ( I mPwSer  R ) )  =  ( +g  `  ( I mPwSer  R ) )
452, 5, 43, 44psrplusg 17798 . 2  |-  ( +g  `  ( I mPwSer  R ) )  =  (  oF ( +g  `  R
)  |`  ( ( Base `  ( I mPwSer  R ) )  X.  ( Base `  ( I mPwSer  R ) ) ) )
46 eqid 2460 . . 3  |-  ( I mPwSer  S )  =  ( I mPwSer  S )
47 eqid 2460 . . 3  |-  ( Base `  ( I mPwSer  S ) )  =  ( Base `  ( I mPwSer  S ) )
48 eqid 2460 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
49 eqid 2460 . . 3  |-  ( +g  `  ( I mPwSer  S ) )  =  ( +g  `  ( I mPwSer  S ) )
5046, 47, 48, 49psrplusg 17798 . 2  |-  ( +g  `  ( I mPwSer  S ) )  =  (  oF ( +g  `  S
)  |`  ( ( Base `  ( I mPwSer  S ) )  X.  ( Base `  ( I mPwSer  S ) ) ) )
5142, 45, 503eqtr4g 2526 1  |-  ( ph  ->  ( +g  `  (
I mPwSer  R ) )  =  ( +g  `  (
I mPwSer  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {crab 2811   _Vcvv 3106    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991    |` cres 4994   "cima 4995    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277    oFcof 6513    ^m cmap 7410   Fincfn 7506   NNcn 10525   NN0cn0 10784   Basecbs 14479   +g cplusg 14544   mPwSer cmps 17764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-tset 14563  df-psr 17769
This theorem is referenced by:  ply1plusgpropd  18049
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