Theorem psrlinv 14669

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 6815	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9391	. . . . 5 |- NN0 e. _V
4		psrgrp.i	. . . . . 6 |- ( ph -> I e. V )
5	4	elexd 2813	. . . . 5 |- ( ph -> I e. _V )
6		fnovex 6043	. . . . 5 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ I e. _V ) -> ( NN0 ^m I ) e. _V )
7	2, 3, 5, 6	mp3an12i 1375	. . . 4 |- ( ph -> ( NN0 ^m I ) e. _V )
8	1, 7	rabexd 4230	. . 3 |- ( ph -> D e. _V )
9		psrgrp.r	. . . 4 |- ( ph -> R e. Grp )
10		psrgrp.s	. . . . . 6 |- S = ( I mPwSer R )
11		eqid 2229	. . . . . 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 14660	. . . . 5 |- ( ph -> X : D --> ( Base ` R ) )
15	14	ffvelcdmda 5775	. . . 4 |- ( ( ph /\ x e. D ) -> ( X ` x ) e. ( Base ` R ) )
16		psrnegcl.i	. . . . 5 |- N = ( invg ` R )
17	11, 16	grpinvcl 13602	. . . 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 5692	. . . 4 |- ( ph -> X = ( x e. D |-> ( X ` x ) ) )
20	11, 16, 9	grpinvf1o 13624	. . . . . 6 |- ( ph -> N : ( Base ` R ) -1-1-onto-> ( Base ` R ) )
21		f1of 5577	. . . . . 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 5692	. . . 4 |- ( ph -> N = ( y e. ( Base ` R ) |-> ( N ` y ) ) )
24		fveq2 5632	. . . 4 |- ( y = ( X ` x ) -> ( N ` y ) = ( N ` ( X ` x ) ) )
25	15, 19, 23, 24	fmptco 5806	. . 3 |- ( ph -> ( N o. X ) = ( x e. D |-> ( N ` ( X ` x ) ) ) )
26	8, 18, 15, 25, 19	offval2 6243	. 2 |- ( ph -> ( ( N o. X ) oF ( +g ` R ) X ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
27		eqid 2229	. . 3 |- ( +g ` R ) = ( +g ` R )
28		psrlinv.p	. . 3 |- .+ = ( +g ` S )
29	10, 4, 9, 1, 16, 12, 13	psrnegcl 14668	. . 3 |- ( ph -> ( N o. X ) e. B )
30	10, 12, 27, 28, 29, 13	psradd 14664	. 2 |- ( ph -> ( ( N o. X ) .+ X ) = ( ( N o. X ) oF ( +g ` R ) X ) )
31		fconstmpt 4768	. . 3 |- ( D X. { .0. } ) = ( x e. D |-> .0. )
32		psrlinv.o	. . . . . 6 |- .0. = ( 0g ` R )
33	11, 27, 32, 16	grplinv 13604	. . . . 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 4174	. . 3 |- ( ph -> ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) = ( x e. D |-> .0. ) )
36	31, 35	eqtr4id 2281	. 2 |- ( ph -> ( D X. { .0. } ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
37	26, 30, 36	3eqtr4d 2272	1 |- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   {csn 3666   |-> cmpt 4145   X. cxp 4718   `'ccnv 4719   "cima 4723   o. ccom 4724   Fn wfn 5316   -->wf 5317   -1-1-onto->wf1o 5320   ` cfv 5321  (class class class)co 6010   oFcof 6225   ^m cmap 6808   Fincfn 6900   NNcn 9126   NN0cn0 9385   Basecbs 13053   +g cplusg 13131   0gc0g 13310   Grpcgrp 13554   invgcminusg 13555   mPwSer cmps 14646

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-of 6227  df-1st 6295  df-2nd 6296  df-map 6810  df-ixp 6859  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-z 9463  df-uz 9739  df-fz 10222  df-struct 13055  df-ndx 13056  df-slot 13057  df-base 13059  df-plusg 13144  df-mulr 13145  df-sca 13147  df-vsca 13148  df-tset 13150  df-rest 13295  df-topn 13296  df-0g 13312  df-topgen 13314  df-pt 13315  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-grp 13557  df-minusg 13558  df-psr 14648

This theorem is referenced by:  psrneg  14672