Theorem psr1clfi 14617

Description: The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
psrring.s	|- S = ( I mPwSer R )
psrringfi.i	|- ( ph -> I e. Fin )
psrring.r	|- ( ph -> R e. Ring )
psr1cl.d	|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
psr1cl.z	|- .0. = ( 0g ` R )
psr1cl.o	|- .1. = ( 1r ` R )
psr1cl.u	|- U = ( x e. D |-> if ( x = ( I X. { 0 } ) , .1. , .0. ) )
psr1cl.b	|- B = ( Base ` S )

Assertion

Ref	Expression
psr1clfi	|- ( ph -> U e. B )

Distinct variable groups:   x, f, .0.   f, I, x   x, B   R, f, x   x, D   ph, x   x, S   x, .1.

Allowed substitution hints:   ph( f)   B( f)   D( f)   S( f)   U( x, f)   .1. ( f)

Proof of Theorem psr1clfi

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrring.r	. . . . . . 7 |- ( ph -> R e. Ring )
2		eqid 2209	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
3		psr1cl.o	. . . . . . . 8 |- .1. = ( 1r ` R )
4	2, 3	ringidcl 13949	. . . . . . 7 |- ( R e. Ring -> .1. e. ( Base ` R ) )
5	1, 4	syl 14	. . . . . 6 |- ( ph -> .1. e. ( Base ` R ) )
6	5	adantr 276	. . . . 5 |- ( ( ph /\ x e. D ) -> .1. e. ( Base ` R ) )
7		psr1cl.z	. . . . . . . 8 |- .0. = ( 0g ` R )
8	2, 7	ring0cl 13950	. . . . . . 7 |- ( R e. Ring -> .0. e. ( Base ` R ) )
9	1, 8	syl 14	. . . . . 6 |- ( ph -> .0. e. ( Base ` R ) )
10	9	adantr 276	. . . . 5 |- ( ( ph /\ x e. D ) -> .0. e. ( Base ` R ) )
11		psrringfi.i	. . . . . . . . 9 |- ( ph -> I e. Fin )
12		0z 9425	. . . . . . . . . . 11 |- 0 e. ZZ
13		cnveq 4873	. . . . . . . . . . . . . . . . . . 19 |- ( f = x -> `' f = `' x )
14	13	imaeq1d 5043	. . . . . . . . . . . . . . . . . 18 |- ( f = x -> ( `' f " NN ) = ( `' x " NN ) )
15	14	eleq1d 2278	. . . . . . . . . . . . . . . . 17 |- ( f = x -> ( ( `' f " NN ) e. Fin <-> ( `' x " NN ) e. Fin ) )
16		psr1cl.d	. . . . . . . . . . . . . . . . 17 |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
17	15, 16	elrab2 2942	. . . . . . . . . . . . . . . 16 |- ( x e. D <-> ( x e. ( NN0 ^m I ) /\ ( `' x " NN ) e. Fin ) )
18	17	simplbi 274	. . . . . . . . . . . . . . 15 |- ( x e. D -> x e. ( NN0 ^m I ) )
19	18	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. D ) -> x e. ( NN0 ^m I ) )
20		nn0ex 9343	. . . . . . . . . . . . . . . 16 |- NN0 e. _V
21	20	a1i 9	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. D ) -> NN0 e. _V )
22	11	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. D ) -> I e. Fin )
23	21, 22	elmapd 6779	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. D ) -> ( x e. ( NN0 ^m I ) <-> x : I --> NN0 ) )
24	19, 23	mpbid 147	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. D ) -> x : I --> NN0 )
25	24	ffvelcdmda 5743	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. D ) /\ z e. I ) -> ( x ` z ) e. NN0 )
26	25	nn0zd 9535	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. D ) /\ z e. I ) -> ( x ` z ) e. ZZ )
27		zdceq 9490	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ ( x ` z ) e. ZZ ) -> DECID 0 = ( x ` z ) )
28	12, 26, 27	sylancr 414	. . . . . . . . . 10 |- ( ( ( ph /\ x e. D ) /\ z e. I ) -> DECID 0 = ( x ` z ) )
29	28	ralrimiva 2583	. . . . . . . . 9 |- ( ( ph /\ x e. D ) -> A. z e. I DECID 0 = ( x ` z ) )
30		dcfi 7116	. . . . . . . . 9 |- ( ( I e. Fin /\ A. z e. I DECID 0 = ( x ` z ) ) -> DECID A. z e. I 0 = ( x ` z ) )
31	11, 29, 30	syl2an2r 597	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> DECID A. z e. I 0 = ( x ` z ) )
32		0nn0 9352	. . . . . . . . . . 11 |- 0 e. NN0
33	32	rgenw 2565	. . . . . . . . . 10 |- A. z e. I 0 e. NN0
34		mpteqb 5698	. . . . . . . . . 10 |- ( A. z e. I 0 e. NN0 -> ( ( z e. I |-> 0 ) = ( z e. I |-> ( x ` z ) ) <-> A. z e. I 0 = ( x ` z ) ) )
35	33, 34	ax-mp 5	. . . . . . . . 9 |- ( ( z e. I |-> 0 ) = ( z e. I |-> ( x ` z ) ) <-> A. z e. I 0 = ( x ` z ) )
36	35	dcbii 844	. . . . . . . 8 |- (DECID ( z e. I |-> 0 ) = ( z e. I |-> ( x ` z ) ) <-> DECID A. z e. I 0 = ( x ` z ) )
37	31, 36	sylibr 134	. . . . . . 7 |- ( ( ph /\ x e. D ) -> DECID ( z e. I |-> 0 ) = ( z e. I |-> ( x ` z ) ) )
38		eqcom 2211	. . . . . . . 8 |- ( ( z e. I |-> 0 ) = ( z e. I |-> ( x ` z ) ) <-> ( z e. I |-> ( x ` z ) ) = ( z e. I |-> 0 ) )
39	38	dcbii 844	. . . . . . 7 |- (DECID ( z e. I |-> 0 ) = ( z e. I |-> ( x ` z ) ) <-> DECID ( z e. I |-> ( x ` z ) ) = ( z e. I |-> 0 ) )
40	37, 39	sylib 122	. . . . . 6 |- ( ( ph /\ x e. D ) -> DECID ( z e. I |-> ( x ` z ) ) = ( z e. I |-> 0 ) )
41	24	feqmptd 5660	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> x = ( z e. I |-> ( x ` z ) ) )
42		fconstmpt 4743	. . . . . . . . 9 |- ( I X. { 0 } ) = ( z e. I |-> 0 )
43	42	a1i 9	. . . . . . . 8 |- ( ( ph /\ x e. D ) -> ( I X. { 0 } ) = ( z e. I |-> 0 ) )
44	41, 43	eqeq12d 2224	. . . . . . 7 |- ( ( ph /\ x e. D ) -> ( x = ( I X. { 0 } ) <-> ( z e. I |-> ( x ` z ) ) = ( z e. I |-> 0 ) ) )
45	44	dcbid 842	. . . . . 6 |- ( ( ph /\ x e. D ) -> (DECID x = ( I X. { 0 } ) <-> DECID ( z e. I |-> ( x ` z ) ) = ( z e. I |-> 0 ) ) )
46	40, 45	mpbird 167	. . . . 5 |- ( ( ph /\ x e. D ) -> DECID x = ( I X. { 0 } ) )
47	6, 10, 46	ifcldcd 3620	. . . 4 |- ( ( ph /\ x e. D ) -> if ( x = ( I X. { 0 } ) , .1. , .0. ) e. ( Base ` R ) )
48		psr1cl.u	. . . 4 |- U = ( x e. D |-> if ( x = ( I X. { 0 } ) , .1. , .0. ) )
49	47, 48	fmptd 5762	. . 3 |- ( ph -> U : D --> ( Base ` R ) )
50		basfn 13057	. . . . 5 |- Base Fn _V
51	1	elexd 2793	. . . . 5 |- ( ph -> R e. _V )
52		funfvex 5620	. . . . . 6 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
53	52	funfni 5399	. . . . 5 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
54	50, 51, 53	sylancr 414	. . . 4 |- ( ph -> ( Base ` R ) e. _V )
55		fnmap 6772	. . . . . 6 |- ^m Fn ( _V X. _V )
56	11	elexd 2793	. . . . . 6 |- ( ph -> I e. _V )
57		fnovex 6007	. . . . . 6 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ I e. _V ) -> ( NN0 ^m I ) e. _V )
58	55, 20, 56, 57	mp3an12i 1356	. . . . 5 |- ( ph -> ( NN0 ^m I ) e. _V )
59	16, 58	rabexd 4208	. . . 4 |- ( ph -> D e. _V )
60	54, 59	elmapd 6779	. . 3 |- ( ph -> ( U e. ( ( Base ` R ) ^m D ) <-> U : D --> ( Base ` R ) ) )
61	49, 60	mpbird 167	. 2 |- ( ph -> U e. ( ( Base ` R ) ^m D ) )
62		psrring.s	. . 3 |- S = ( I mPwSer R )
63		psr1cl.b	. . 3 |- B = ( Base ` S )
64	62, 2, 16, 63, 11, 1	psrbasg 14603	. 2 |- ( ph -> B = ( ( Base ` R ) ^m D ) )
65	61, 64	eleqtrrd 2289	1 |- ( ph -> U e. B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 838   = wceq 1375   e. wcel 2180   A.wral 2488   {crab 2492   _Vcvv 2779   ifcif 3582   {csn 3646   |-> cmpt 4124   X. cxp 4694   `'ccnv 4695   "cima 4699   Fn wfn 5289   -->wf 5290   ` cfv 5294  (class class class)co 5974   ^m cmap 6765   Fincfn 6857   0cc0 7967   NNcn 9078   NN0cn0 9337   ZZcz 9414   Basecbs 12998   0gc0g 13255   1rcur 13888   Ringcrg 13925   mPwSer cmps 14590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-tp 3654  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-of 6188  df-1st 6256  df-2nd 6257  df-er 6650  df-map 6767  df-ixp 6816  df-en 6858  df-fin 6860  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-z 9415  df-uz 9691  df-fz 10173  df-struct 13000  df-ndx 13001  df-slot 13002  df-base 13004  df-sets 13005  df-plusg 13089  df-mulr 13090  df-sca 13092  df-vsca 13093  df-tset 13095  df-rest 13240  df-topn 13241  df-0g 13257  df-topgen 13259  df-pt 13260  df-mgm 13355  df-sgrp 13401  df-mnd 13416  df-grp 13502  df-mgp 13850  df-ur 13889  df-ring 13927  df-psr 14592

This theorem is referenced by: (None)