Theorem psrlinv 14888

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 6891	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9507	. . . . 5 |- NN0 e. _V
4		psrgrp.i	. . . . . 6 |- ( ph -> I e. V )
5	4	elexd 2829	. . . . 5 |- ( ph -> I e. _V )
6		fnovex 6085	. . . . 5 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ I e. _V ) -> ( NN0 ^m I ) e. _V )
7	2, 3, 5, 6	mp3an12i 1378	. . . 4 |- ( ph -> ( NN0 ^m I ) e. _V )
8	1, 7	rabexd 4259	. . 3 |- ( ph -> D e. _V )
9		psrgrp.r	. . . 4 |- ( ph -> R e. Grp )
10		psrgrp.s	. . . . . 6 |- S = ( I mPwSer R )
11		eqid 2234	. . . . . 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 14879	. . . . 5 |- ( ph -> X : D --> ( Base ` R ) )
15	14	ffvelcdmda 5814	. . . 4 |- ( ( ph /\ x e. D ) -> ( X ` x ) e. ( Base ` R ) )
16		psrnegcl.i	. . . . 5 |- N = ( invg ` R )
17	11, 16	grpinvcl 13782	. . . 4 |- ( ( R e. Grp /\ ( X ` x ) e. ( Base ` R ) ) -> ( N ` ( X ` x ) ) e. ( Base ` R ) )
18	9, 15, 17	syl2an2r 599	. . 3 |- ( ( ph /\ x e. D ) -> ( N ` ( X ` x ) ) e. ( Base ` R ) )
19	14	feqmptd 5732	. . . 4 |- ( ph -> X = ( x e. D |-> ( X ` x ) ) )
20	11, 16, 9	grpinvf1o 13804	. . . . . 6 |- ( ph -> N : ( Base ` R ) -1-1-onto-> ( Base ` R ) )
21		f1of 5616	. . . . . 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 5732	. . . 4 |- ( ph -> N = ( y e. ( Base ` R ) |-> ( N ` y ) ) )
24		fveq2 5672	. . . 4 |- ( y = ( X ` x ) -> ( N ` y ) = ( N ` ( X ` x ) ) )
25	15, 19, 23, 24	fmptco 5845	. . 3 |- ( ph -> ( N o. X ) = ( x e. D |-> ( N ` ( X ` x ) ) ) )
26	8, 18, 15, 25, 19	offval2 6284	. 2 |- ( ph -> ( ( N o. X ) oF ( +g ` R ) X ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
27		eqid 2234	. . 3 |- ( +g ` R ) = ( +g ` R )
28		psrlinv.p	. . 3 |- .+ = ( +g ` S )
29	10, 4, 9, 1, 16, 12, 13	psrnegcl 14887	. . 3 |- ( ph -> ( N o. X ) e. B )
30	10, 12, 27, 28, 29, 13	psradd 14883	. 2 |- ( ph -> ( ( N o. X ) .+ X ) = ( ( N o. X ) oF ( +g ` R ) X ) )
31		fconstmpt 4799	. . 3 |- ( D X. { .0. } ) = ( x e. D |-> .0. )
32		psrlinv.o	. . . . . 6 |- .0. = ( 0g ` R )
33	11, 27, 32, 16	grplinv 13784	. . . . 5 |- ( ( R e. Grp /\ ( X ` x ) e. ( Base ` R ) ) -> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) = .0. )
34	9, 15, 33	syl2an2r 599	. . . 4 |- ( ( ph /\ x e. D ) -> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) = .0. )
35	34	mpteq2dva 4202	. . 3 |- ( ph -> ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) = ( x e. D |-> .0. ) )
36	31, 35	eqtr4id 2286	. 2 |- ( ph -> ( D X. { .0. } ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
37	26, 30, 36	3eqtr4d 2277	1 |- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   {csn 3691   |-> cmpt 4173   X. cxp 4749   `'ccnv 4750   "cima 4754   o. ccom 4755   Fn wfn 5349   -->wf 5350   -1-1-onto->wf1o 5353   ` cfv 5354  (class class class)co 6052   oFcof 6266   ^m cmap 6884   Fincfn 6977   NNcn 9242   NN0cn0 9501   Basecbs 13233   +g cplusg 13311   0gc0g 13490   Grpcgrp 13734   invgcminusg 13735   mPwSer cmps 14858

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-of 6268  df-1st 6336  df-2nd 6337  df-map 6886  df-ixp 6936  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-z 9583  df-uz 9860  df-fz 10349  df-struct 13235  df-ndx 13236  df-slot 13237  df-base 13239  df-plusg 13324  df-mulr 13325  df-sca 13327  df-vsca 13328  df-tset 13330  df-rest 13475  df-topn 13476  df-0g 13492  df-topgen 13494  df-pt 13495  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-grp 13737  df-minusg 13738  df-psr 14860

This theorem is referenced by:  psrneg  14891