Theorem psrlinv 14826

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 6888	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9498	. . . . 5 |- NN0 e. _V
4		psrgrp.i	. . . . . 6 |- ( ph -> I e. V )
5	4	elexd 2826	. . . . 5 |- ( ph -> I e. _V )
6		fnovex 6082	. . . . 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 4256	. . 3 |- ( ph -> D e. _V )
9		psrgrp.r	. . . 4 |- ( ph -> R e. Grp )
10		psrgrp.s	. . . . . 6 |- S = ( I mPwSer R )
11		eqid 2232	. . . . . 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 14817	. . . . 5 |- ( ph -> X : D --> ( Base ` R ) )
15	14	ffvelcdmda 5811	. . . 4 |- ( ( ph /\ x e. D ) -> ( X ` x ) e. ( Base ` R ) )
16		psrnegcl.i	. . . . 5 |- N = ( invg ` R )
17	11, 16	grpinvcl 13750	. . . 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 5729	. . . 4 |- ( ph -> X = ( x e. D |-> ( X ` x ) ) )
20	11, 16, 9	grpinvf1o 13772	. . . . . 6 |- ( ph -> N : ( Base ` R ) -1-1-onto-> ( Base ` R ) )
21		f1of 5613	. . . . . 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 5729	. . . 4 |- ( ph -> N = ( y e. ( Base ` R ) |-> ( N ` y ) ) )
24		fveq2 5669	. . . 4 |- ( y = ( X ` x ) -> ( N ` y ) = ( N ` ( X ` x ) ) )
25	15, 19, 23, 24	fmptco 5842	. . 3 |- ( ph -> ( N o. X ) = ( x e. D |-> ( N ` ( X ` x ) ) ) )
26	8, 18, 15, 25, 19	offval2 6281	. 2 |- ( ph -> ( ( N o. X ) oF ( +g ` R ) X ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
27		eqid 2232	. . 3 |- ( +g ` R ) = ( +g ` R )
28		psrlinv.p	. . 3 |- .+ = ( +g ` S )
29	10, 4, 9, 1, 16, 12, 13	psrnegcl 14825	. . 3 |- ( ph -> ( N o. X ) e. B )
30	10, 12, 27, 28, 29, 13	psradd 14821	. 2 |- ( ph -> ( ( N o. X ) .+ X ) = ( ( N o. X ) oF ( +g ` R ) X ) )
31		fconstmpt 4796	. . 3 |- ( D X. { .0. } ) = ( x e. D |-> .0. )
32		psrlinv.o	. . . . . 6 |- .0. = ( 0g ` R )
33	11, 27, 32, 16	grplinv 13752	. . . . 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 4199	. . 3 |- ( ph -> ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) = ( x e. D |-> .0. ) )
36	31, 35	eqtr4id 2284	. 2 |- ( ph -> ( D X. { .0. } ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
37	26, 30, 36	3eqtr4d 2275	1 |- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   {crab 2524   _Vcvv 2812   {csn 3688   |-> cmpt 4170   X. cxp 4746   `'ccnv 4747   "cima 4751   o. ccom 4752   Fn wfn 5346   -->wf 5347   -1-1-onto->wf1o 5350   ` cfv 5351  (class class class)co 6049   oFcof 6263   ^m cmap 6881   Fincfn 6974   NNcn 9233   NN0cn0 9492   Basecbs 13201   +g cplusg 13279   0gc0g 13458   Grpcgrp 13702   invgcminusg 13703   mPwSer cmps 14796

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-of 6265  df-1st 6333  df-2nd 6334  df-map 6883  df-ixp 6933  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-uz 9850  df-fz 10339  df-struct 13203  df-ndx 13204  df-slot 13205  df-base 13207  df-plusg 13292  df-mulr 13293  df-sca 13295  df-vsca 13296  df-tset 13298  df-rest 13443  df-topn 13444  df-0g 13460  df-topgen 13462  df-pt 13463  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-grp 13705  df-minusg 13706  df-psr 14798

This theorem is referenced by:  psrneg  14829