Theorem psrplusgpropd 19606

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.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . . . 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 2622	. . . . . . . . . . 11 |- ( I mPwSer R ) = ( I mPwSer R )
3		eqid 2622	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
4		eqid 2622	. . . . . . . . . . 11 |- { c e. ( NN0 ^m I ) | ( `' c " NN ) e. Fin } = { c e. ( NN0 ^m I ) | ( `' c " NN ) e. Fin }
5		eqid 2622	. . . . . . . . . . 11 |- ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer R ) )
6		simp2 1062	. . . . . . . . . . 11 |- ( ( ph /\ a e. ( Base ` ( I mPwSer R ) ) /\ b e. ( Base ` ( I mPwSer R ) ) ) -> a e. ( Base ` ( I mPwSer R ) ) )
7	2, 3, 4, 5, 6	psrelbas 19379	. . . . . . . . . 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	7	ffvelrnda 6359	. . . . . . . . 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 ) )
10	1, 9	syl 17	. . . . . . . . 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 ) )
11	8, 10	eleqtrrd 2704	. . . . . . . 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 1063	. . . . . . . . . . 11 |- ( ( ph /\ a e. ( Base ` ( I mPwSer R ) ) /\ b e. ( Base ` ( I mPwSer R ) ) ) -> b e. ( Base ` ( I mPwSer R ) ) )
13	2, 3, 4, 5, 12	psrelbas 19379	. . . . . . . . . 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 ) )
14	13	ffvelrnda 6359	. . . . . . . . 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 ) )
15	14, 10	eleqtrrd 2704	. . . . . . . 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 ) )
17	16	oveqrspc2v 6673	. . . . . . . 8 |- ( ( ph /\ ( ( a ` d ) e. B /\ ( b ` d ) e. B ) ) -> ( ( a ` d ) ( +g ` R ) ( b ` d ) ) = ( ( a ` d ) ( +g ` S ) ( b ` d ) ) )
18	1, 11, 15, 17	syl12anc 1324	. . . . . . 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 ) ) )
19	18	mpteq2dva 4744	. . . . . 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 6045	. . . . . . . 8 |- ( a : { c e. ( NN0 ^m I ) | ( `' c " NN ) e. Fin } --> ( Base ` R ) -> a Fn { c e. ( NN0 ^m I ) | ( `' c " NN ) e. Fin } )
21	7, 20	syl 17	. . . . . . 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 6045	. . . . . . . 8 |- ( b : { c e. ( NN0 ^m I ) | ( `' c " NN ) e. Fin } --> ( Base ` R ) -> b Fn { c e. ( NN0 ^m I ) | ( `' c " NN ) e. Fin } )
23	13, 22	syl 17	. . . . . . 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 6678	. . . . . . . . 9 |- ( NN0 ^m I ) e. _V
25	24	rabex 4813	. . . . . . . 8 |- { c e. ( NN0 ^m I ) | ( `' c " NN ) e. Fin } e. _V
26	25	a1i 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 3822	. . . . . . 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 2623	. . . . . . 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 2623	. . . . . . 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 ) )
30	21, 23, 26, 26, 27, 28, 29	offval 6904	. . . . . 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 ) ) ) )
31	21, 23, 26, 26, 27, 28, 29	offval 6904	. . . . . 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 ) ) ) )
32	19, 30, 31	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ a e. ( Base ` ( I mPwSer R ) ) /\ b e. ( Base ` ( I mPwSer R ) ) ) -> ( a oF ( +g ` R ) b ) = ( a oF ( +g ` S ) b ) )
33	32	mpt2eq3dva 6719	. . . 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 ) )
35	9, 34	eqtr3d 2658	. . . . . 6 |- ( ph -> ( Base ` R ) = ( Base ` S ) )
36	35	psrbaspropd 19605	. . . . 5 |- ( ph -> ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer S ) ) )
37		mpt2eq12 6715	. . . . 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 ) ) )
38	36, 36, 37	syl2anc 693	. . . 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 ) ) )
39	33, 38	eqtrd 2656	. . 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 7164	. . 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 7164	. . 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 ) )
42	39, 40, 41	3eqtr4g 2681	. 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 2622	. . 3 |- ( +g ` R ) = ( +g ` R )
44		eqid 2622	. . 3 |- ( +g ` ( I mPwSer R ) ) = ( +g ` ( I mPwSer R ) )
45	2, 5, 43, 44	psrplusg 19381	. 2 |- ( +g ` ( I mPwSer R ) ) = ( oF ( +g ` R ) |` ( ( Base ` ( I mPwSer R ) ) X. ( Base ` ( I mPwSer R ) ) ) )
46		eqid 2622	. . 3 |- ( I mPwSer S ) = ( I mPwSer S )
47		eqid 2622	. . 3 |- ( Base ` ( I mPwSer S ) ) = ( Base ` ( I mPwSer S ) )
48		eqid 2622	. . 3 |- ( +g ` S ) = ( +g ` S )
49		eqid 2622	. . 3 |- ( +g ` ( I mPwSer S ) ) = ( +g ` ( I mPwSer S ) )
50	46, 47, 48, 49	psrplusg 19381	. 2 |- ( +g ` ( I mPwSer S ) ) = ( oF ( +g ` S ) |` ( ( Base ` ( I mPwSer S ) ) X. ( Base ` ( I mPwSer S ) ) ) )
51	42, 45, 50	3eqtr4g 2681	1 |- ( ph -> ( +g ` ( I mPwSer R ) ) = ( +g ` ( I mPwSer S ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   ^m cmap 7857   Fincfn 7955   NNcn 11020   NN0cn0 11292   Basecbs 15857   +g cplusg 15941   mPwSer cmps 19351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-psr 19356

This theorem is referenced by:  ply1plusgpropd  19614