Theorem psrlinv 14613

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

Hypotheses

Ref	Expression
psrgrp.s	|- S = ( I mPwSer R )
psrgrp.i	|- ( ph -> I e. V )
psrgrp.r	|- ( ph -> R e. Grp )
psrnegcl.d	|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
psrnegcl.i	|- N = ( invg ` R )
psrnegcl.b	|- B = ( Base ` S )
psrnegcl.z	|- ( ph -> X e. B )
psrlinv.o	|- .0. = ( 0g ` R )
psrlinv.p	|- .+ = ( +g ` S )

Assertion

Ref	Expression
psrlinv	|- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) )

Distinct variable group:   f, I

Allowed substitution hints:   ph( f)   B( f)   D( f)   .+ ( f)   R( f)   S( f)   N( f)   V( f)   X( f)   .0. ( f)

Proof of Theorem psrlinv

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrnegcl.d	. . . 4 |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }
2		fnmap 6772	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9343	. . . . 5 |- NN0 e. _V
4		psrgrp.i	. . . . . 6 |- ( ph -> I e. V )
5	4	elexd 2793	. . . . 5 |- ( ph -> I e. _V )
6		fnovex 6007	. . . . 5 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ I e. _V ) -> ( NN0 ^m I ) e. _V )
7	2, 3, 5, 6	mp3an12i 1356	. . . 4 |- ( ph -> ( NN0 ^m I ) e. _V )
8	1, 7	rabexd 4208	. . 3 |- ( ph -> D e. _V )
9		psrgrp.r	. . . 4 |- ( ph -> R e. Grp )
10		psrgrp.s	. . . . . 6 |- S = ( I mPwSer R )
11		eqid 2209	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
12		psrnegcl.b	. . . . . 6 |- B = ( Base ` S )
13		psrnegcl.z	. . . . . 6 |- ( ph -> X e. B )
14	10, 11, 1, 12, 13	psrelbas 14604	. . . . 5 |- ( ph -> X : D --> ( Base ` R ) )
15	14	ffvelcdmda 5743	. . . 4 |- ( ( ph /\ x e. D ) -> ( X ` x ) e. ( Base ` R ) )
16		psrnegcl.i	. . . . 5 |- N = ( invg ` R )
17	11, 16	grpinvcl 13547	. . . 4 |- ( ( R e. Grp /\ ( X ` x ) e. ( Base ` R ) ) -> ( N ` ( X ` x ) ) e. ( Base ` R ) )
18	9, 15, 17	syl2an2r 597	. . 3 |- ( ( ph /\ x e. D ) -> ( N ` ( X ` x ) ) e. ( Base ` R ) )
19	14	feqmptd 5660	. . . 4 |- ( ph -> X = ( x e. D |-> ( X ` x ) ) )
20	11, 16, 9	grpinvf1o 13569	. . . . . 6 |- ( ph -> N : ( Base ` R ) -1-1-onto-> ( Base ` R ) )
21		f1of 5548	. . . . . 6 |- ( N : ( Base ` R ) -1-1-onto-> ( Base ` R ) -> N : ( Base ` R ) --> ( Base ` R ) )
22	20, 21	syl 14	. . . . 5 |- ( ph -> N : ( Base ` R ) --> ( Base ` R ) )
23	22	feqmptd 5660	. . . 4 |- ( ph -> N = ( y e. ( Base ` R ) |-> ( N ` y ) ) )
24		fveq2 5603	. . . 4 |- ( y = ( X ` x ) -> ( N ` y ) = ( N ` ( X ` x ) ) )
25	15, 19, 23, 24	fmptco 5774	. . 3 |- ( ph -> ( N o. X ) = ( x e. D |-> ( N ` ( X ` x ) ) ) )
26	8, 18, 15, 25, 19	offval2 6204	. 2 |- ( ph -> ( ( N o. X ) oF ( +g ` R ) X ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
27		eqid 2209	. . 3 |- ( +g ` R ) = ( +g ` R )
28		psrlinv.p	. . 3 |- .+ = ( +g ` S )
29	10, 4, 9, 1, 16, 12, 13	psrnegcl 14612	. . 3 |- ( ph -> ( N o. X ) e. B )
30	10, 12, 27, 28, 29, 13	psradd 14608	. 2 |- ( ph -> ( ( N o. X ) .+ X ) = ( ( N o. X ) oF ( +g ` R ) X ) )
31		fconstmpt 4743	. . 3 |- ( D X. { .0. } ) = ( x e. D |-> .0. )
32		psrlinv.o	. . . . . 6 |- .0. = ( 0g ` R )
33	11, 27, 32, 16	grplinv 13549	. . . . 5 |- ( ( R e. Grp /\ ( X ` x ) e. ( Base ` R ) ) -> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) = .0. )
34	9, 15, 33	syl2an2r 597	. . . 4 |- ( ( ph /\ x e. D ) -> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) = .0. )
35	34	mpteq2dva 4153	. . 3 |- ( ph -> ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) = ( x e. D |-> .0. ) )
36	31, 35	eqtr4id 2261	. 2 |- ( ph -> ( D X. { .0. } ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
37	26, 30, 36	3eqtr4d 2252	1 |- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) )

Syntax hints:   -> wi 4   = wceq 1375   e. wcel 2180   {crab 2492   _Vcvv 2779   {csn 3646   |-> cmpt 4124   X. cxp 4694   `'ccnv 4695   "cima 4699   o. ccom 4700   Fn wfn 5289   -->wf 5290   -1-1-onto->wf1o 5293   ` cfv 5294  (class class class)co 5974   oFcof 6186   ^m cmap 6765   Fincfn 6857   NNcn 9078   NN0cn0 9337   Basecbs 12998   +g cplusg 13076   0gc0g 13255   Grpcgrp 13499   invgcminusg 13500   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-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  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-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-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-id 4361  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-map 6767  df-ixp 6816  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-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-minusg 13503  df-psr 14592

This theorem is referenced by:  psrneg  14616