Theorem resspsradd 19416

Description: A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)

Hypotheses

Ref	Expression
resspsr.s	|- S = ( I mPwSer R )
resspsr.h	|- H = ( R ↾s T )
resspsr.u	|- U = ( I mPwSer H )
resspsr.b	|- B = ( Base ` U )
resspsr.p	|- P = ( S ↾s B )
resspsr.2	|- ( ph -> T e. (SubRing ` R ) )

Assertion

Ref	Expression
resspsradd	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` U ) Y ) = ( X ( +g ` P ) Y ) )

Proof of Theorem resspsradd

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resspsr.u	. . 3 |- U = ( I mPwSer H )
2		resspsr.b	. . 3 |- B = ( Base ` U )
3		eqid 2622	. . 3 |- ( +g ` H ) = ( +g ` H )
4		eqid 2622	. . 3 |- ( +g ` U ) = ( +g ` U )
5		simprl 794	. . 3 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> X e. B )
6		simprr 796	. . 3 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
7	1, 2, 3, 4, 5, 6	psradd 19382	. 2 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` U ) Y ) = ( X oF ( +g ` H ) Y ) )
8		resspsr.s	. . . 4 |- S = ( I mPwSer R )
9		eqid 2622	. . . 4 |- ( Base ` S ) = ( Base ` S )
10		eqid 2622	. . . 4 |- ( +g ` R ) = ( +g ` R )
11		eqid 2622	. . . 4 |- ( +g ` S ) = ( +g ` S )
12		fvex 6201	. . . . . . . 8 |- ( Base ` R ) e. _V
13		resspsr.2	. . . . . . . . . 10 |- ( ph -> T e. (SubRing ` R ) )
14		resspsr.h	. . . . . . . . . . 11 |- H = ( R ↾s T )
15	14	subrgbas 18789	. . . . . . . . . 10 |- ( T e. (SubRing ` R ) -> T = ( Base ` H ) )
16	13, 15	syl 17	. . . . . . . . 9 |- ( ph -> T = ( Base ` H ) )
17		eqid 2622	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
18	17	subrgss 18781	. . . . . . . . . 10 |- ( T e. (SubRing ` R ) -> T C_ ( Base ` R ) )
19	13, 18	syl 17	. . . . . . . . 9 |- ( ph -> T C_ ( Base ` R ) )
20	16, 19	eqsstr3d 3640	. . . . . . . 8 |- ( ph -> ( Base ` H ) C_ ( Base ` R ) )
21		mapss 7900	. . . . . . . 8 |- ( ( ( Base ` R ) e. _V /\ ( Base ` H ) C_ ( Base ` R ) ) -> ( ( Base ` H ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) C_ ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) )
22	12, 20, 21	sylancr 695	. . . . . . 7 |- ( ph -> ( ( Base ` H ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) C_ ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) )
23	22	adantr 481	. . . . . 6 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( ( Base ` H ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) C_ ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) )
24		eqid 2622	. . . . . . 7 |- ( Base ` H ) = ( Base ` H )
25		eqid 2622	. . . . . . 7 |- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
26		reldmpsr 19361	. . . . . . . . . 10 |- Rel dom mPwSer
27	26, 1, 2	elbasov 15921	. . . . . . . . 9 |- ( X e. B -> ( I e. _V /\ H e. _V ) )
28	27	ad2antrl 764	. . . . . . . 8 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( I e. _V /\ H e. _V ) )
29	28	simpld 475	. . . . . . 7 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> I e. _V )
30	1, 24, 25, 2, 29	psrbas 19378	. . . . . 6 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> B = ( ( Base ` H ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) )
31	8, 17, 25, 9, 29	psrbas 19378	. . . . . 6 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( Base ` S ) = ( ( Base ` R ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) )
32	23, 30, 31	3sstr4d 3648	. . . . 5 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> B C_ ( Base ` S ) )
33	32, 5	sseldd 3604	. . . 4 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> X e. ( Base ` S ) )
34	32, 6	sseldd 3604	. . . 4 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> Y e. ( Base ` S ) )
35	8, 9, 10, 11, 33, 34	psradd 19382	. . 3 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` S ) Y ) = ( X oF ( +g ` R ) Y ) )
36	14, 10	ressplusg 15993	. . . . . . 7 |- ( T e. (SubRing ` R ) -> ( +g ` R ) = ( +g ` H ) )
37	13, 36	syl 17	. . . . . 6 |- ( ph -> ( +g ` R ) = ( +g ` H ) )
38	37	adantr 481	. . . . 5 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( +g ` R ) = ( +g ` H ) )
39		ofeq 6899	. . . . 5 |- ( ( +g ` R ) = ( +g ` H ) -> oF ( +g ` R ) = oF ( +g ` H ) )
40	38, 39	syl 17	. . . 4 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> oF ( +g ` R ) = oF ( +g ` H ) )
41	40	oveqd 6667	. . 3 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X oF ( +g ` R ) Y ) = ( X oF ( +g ` H ) Y ) )
42	35, 41	eqtrd 2656	. 2 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` S ) Y ) = ( X oF ( +g ` H ) Y ) )
43		fvex 6201	. . . . 5 |- ( Base ` U ) e. _V
44	2, 43	eqeltri 2697	. . . 4 |- B e. _V
45		resspsr.p	. . . . 5 |- P = ( S ↾s B )
46	45, 11	ressplusg 15993	. . . 4 |- ( B e. _V -> ( +g ` S ) = ( +g ` P ) )
47	44, 46	mp1i 13	. . 3 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( +g ` S ) = ( +g ` P ) )
48	47	oveqd 6667	. 2 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` S ) Y ) = ( X ( +g ` P ) Y ) )
49	7, 42, 48	3eqtr2d 2662	1 |- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( +g ` U ) Y ) = ( X ( +g ` P ) Y ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   Fincfn 7955   NNcn 11020   NN0cn0 11292   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941  SubRingcsubrg 18776   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-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-subg 17591  df-ring 18549  df-subrg 18778  df-psr 19356

This theorem is referenced by:  subrgpsr  19419  ressmpladd  19457