Theorem psrlinv 14490

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 6749	. . . . 5 |- ^m Fn ( _V X. _V )
3		nn0ex 9308	. . . . 5 |- NN0 e. _V
4		psrgrp.i	. . . . . 6 |- ( ph -> I e. V )
5	4	elexd 2786	. . . . 5 |- ( ph -> I e. _V )
6		fnovex 5984	. . . . 5 |- ( ( ^m Fn ( _V X. _V ) /\ NN0 e. _V /\ I e. _V ) -> ( NN0 ^m I ) e. _V )
7	2, 3, 5, 6	mp3an12i 1354	. . . 4 |- ( ph -> ( NN0 ^m I ) e. _V )
8	1, 7	rabexd 4193	. . 3 |- ( ph -> D e. _V )
9		psrgrp.r	. . . 4 |- ( ph -> R e. Grp )
10		psrgrp.s	. . . . . 6 |- S = ( I mPwSer R )
11		eqid 2206	. . . . . 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 14481	. . . . 5 |- ( ph -> X : D --> ( Base ` R ) )
15	14	ffvelcdmda 5722	. . . 4 |- ( ( ph /\ x e. D ) -> ( X ` x ) e. ( Base ` R ) )
16		psrnegcl.i	. . . . 5 |- N = ( invg ` R )
17	11, 16	grpinvcl 13424	. . . 4 |- ( ( R e. Grp /\ ( X ` x ) e. ( Base ` R ) ) -> ( N ` ( X ` x ) ) e. ( Base ` R ) )
18	9, 15, 17	syl2an2r 595	. . 3 |- ( ( ph /\ x e. D ) -> ( N ` ( X ` x ) ) e. ( Base ` R ) )
19	14	feqmptd 5639	. . . 4 |- ( ph -> X = ( x e. D |-> ( X ` x ) ) )
20	11, 16, 9	grpinvf1o 13446	. . . . . 6 |- ( ph -> N : ( Base ` R ) -1-1-onto-> ( Base ` R ) )
21		f1of 5529	. . . . . 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 5639	. . . 4 |- ( ph -> N = ( y e. ( Base ` R ) |-> ( N ` y ) ) )
24		fveq2 5583	. . . 4 |- ( y = ( X ` x ) -> ( N ` y ) = ( N ` ( X ` x ) ) )
25	15, 19, 23, 24	fmptco 5753	. . 3 |- ( ph -> ( N o. X ) = ( x e. D |-> ( N ` ( X ` x ) ) ) )
26	8, 18, 15, 25, 19	offval2 6181	. 2 |- ( ph -> ( ( N o. X ) oF ( +g ` R ) X ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
27		eqid 2206	. . 3 |- ( +g ` R ) = ( +g ` R )
28		psrlinv.p	. . 3 |- .+ = ( +g ` S )
29	10, 4, 9, 1, 16, 12, 13	psrnegcl 14489	. . 3 |- ( ph -> ( N o. X ) e. B )
30	10, 12, 27, 28, 29, 13	psradd 14485	. 2 |- ( ph -> ( ( N o. X ) .+ X ) = ( ( N o. X ) oF ( +g ` R ) X ) )
31		fconstmpt 4726	. . 3 |- ( D X. { .0. } ) = ( x e. D |-> .0. )
32		psrlinv.o	. . . . . 6 |- .0. = ( 0g ` R )
33	11, 27, 32, 16	grplinv 13426	. . . . 5 |- ( ( R e. Grp /\ ( X ` x ) e. ( Base ` R ) ) -> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) = .0. )
34	9, 15, 33	syl2an2r 595	. . . 4 |- ( ( ph /\ x e. D ) -> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) = .0. )
35	34	mpteq2dva 4138	. . 3 |- ( ph -> ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) = ( x e. D |-> .0. ) )
36	31, 35	eqtr4id 2258	. 2 |- ( ph -> ( D X. { .0. } ) = ( x e. D |-> ( ( N ` ( X ` x ) ) ( +g ` R ) ( X ` x ) ) ) )
37	26, 30, 36	3eqtr4d 2249	1 |- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   {crab 2489   _Vcvv 2773   {csn 3634   |-> cmpt 4109   X. cxp 4677   `'ccnv 4678   "cima 4682   o. ccom 4683   Fn wfn 5271   -->wf 5272   -1-1-onto->wf1o 5275   ` cfv 5276  (class class class)co 5951   oFcof 6163   ^m cmap 6742   Fincfn 6834   NNcn 9043   NN0cn0 9302   Basecbs 12876   +g cplusg 12953   0gc0g 13132   Grpcgrp 13376   invgcminusg 13377   mPwSer cmps 14467

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-addass 8034  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-tp 3642  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-of 6165  df-1st 6233  df-2nd 6234  df-map 6744  df-ixp 6793  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-9 9109  df-n0 9303  df-z 9380  df-uz 9656  df-fz 10138  df-struct 12878  df-ndx 12879  df-slot 12880  df-base 12882  df-plusg 12966  df-mulr 12967  df-sca 12969  df-vsca 12970  df-tset 12972  df-rest 13117  df-topn 13118  df-0g 13134  df-topgen 13136  df-pt 13137  df-mgm 13232  df-sgrp 13278  df-mnd 13293  df-grp 13379  df-minusg 13380  df-psr 14469

This theorem is referenced by:  psrneg  14493