Theorem gsumval3a 18304

Description: Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by AV, 29-May-2019.)

Hypotheses

Ref	Expression
gsumval3.b	|- B = ( Base ` G )
gsumval3.0	|- .0. = ( 0g ` G )
gsumval3.p	|- .+ = ( +g ` G )
gsumval3.z	|- Z = (Cntz ` G )
gsumval3.g	|- ( ph -> G e. Mnd )
gsumval3.a	|- ( ph -> A e. V )
gsumval3.f	|- ( ph -> F : A --> B )
gsumval3.c	|- ( ph -> ran F C_ ( Z ` ran F ) )
gsumval3a.t	|- ( ph -> W e. Fin )
gsumval3a.n	|- ( ph -> W =/= (/) )
gsumval3a.w	|- W = ( F supp .0. )
gsumval3a.i	|- ( ph -> -. A e. ran ... )

Assertion

Ref	Expression
gsumval3a	|- ( ph -> ( G gsumg F ) = ( iota x E. f ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W /\ x = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) ) ) )

Distinct variable groups:   x, f, .+   A, f, x   ph, f, x   x, .0.   f, G, x   x, V   B, f, x   f, F, x   f, W, x

Allowed substitution hints:   V( f)   .0. ( f)   Z( x, f)

Proof of Theorem gsumval3a

Dummy variables m n y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval3.b	. . 3 |- B = ( Base ` G )
2		gsumval3.0	. . 3 |- .0. = ( 0g ` G )
3		gsumval3.p	. . 3 |- .+ = ( +g ` G )
4		eqid 2622	. . 3 |- { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } = { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) }
5		gsumval3a.w	. . . . 5 |- W = ( F supp .0. )
6	5	a1i 11	. . . 4 |- ( ph -> W = ( F supp .0. ) )
7		gsumval3.f	. . . . . . 7 |- ( ph -> F : A --> B )
8		gsumval3.a	. . . . . . 7 |- ( ph -> A e. V )
9	7, 8	jca 554	. . . . . 6 |- ( ph -> ( F : A --> B /\ A e. V ) )
10		fex 6490	. . . . . 6 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
11	9, 10	syl 17	. . . . 5 |- ( ph -> F e. _V )
12		fvex 6201	. . . . . 6 |- ( 0g ` G ) e. _V
13	2, 12	eqeltri 2697	. . . . 5 |- .0. e. _V
14		suppimacnv 7306	. . . . 5 |- ( ( F e. _V /\ .0. e. _V ) -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
15	11, 13, 14	sylancl 694	. . . 4 |- ( ph -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
16		gsumval3.g	. . . . . . . 8 |- ( ph -> G e. Mnd )
17	1, 2, 3, 4	gsumvallem2 17372	. . . . . . . 8 |- ( G e. Mnd -> { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } = { .0. } )
18	16, 17	syl 17	. . . . . . 7 |- ( ph -> { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } = { .0. } )
19	18	eqcomd 2628	. . . . . 6 |- ( ph -> { .0. } = { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } )
20	19	difeq2d 3728	. . . . 5 |- ( ph -> ( _V \ { .0. } ) = ( _V \ { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } ) )
21	20	imaeq2d 5466	. . . 4 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( `' F " ( _V \ { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } ) ) )
22	6, 15, 21	3eqtrd 2660	. . 3 |- ( ph -> W = ( `' F " ( _V \ { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } ) ) )
23	1, 2, 3, 4, 22, 16, 8, 7	gsumval 17271	. 2 |- ( ph -> ( G gsumg F ) = if ( ran F C_ { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } , .0. , if ( A e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W /\ x = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) ) ) ) ) )
24		gsumval3a.n	. . . 4 |- ( ph -> W =/= (/) )
25	18	sseq2d 3633	. . . . . 6 |- ( ph -> ( ran F C_ { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } <-> ran F C_ { .0. } ) )
26	5	a1i 11	. . . . . . . 8 |- ( ( ph /\ ran F C_ { .0. } ) -> W = ( F supp .0. ) )
27	9	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ran F C_ { .0. } ) -> ( F : A --> B /\ A e. V ) )
28	27, 10	syl 17	. . . . . . . . 9 |- ( ( ph /\ ran F C_ { .0. } ) -> F e. _V )
29	28, 13, 14	sylancl 694	. . . . . . . 8 |- ( ( ph /\ ran F C_ { .0. } ) -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
30		ffn 6045	. . . . . . . . . . . 12 |- ( F : A --> B -> F Fn A )
31	7, 30	syl 17	. . . . . . . . . . 11 |- ( ph -> F Fn A )
32	31	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ran F C_ { .0. } ) -> F Fn A )
33		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ ran F C_ { .0. } ) -> ran F C_ { .0. } )
34		df-f 5892	. . . . . . . . . 10 |- ( F : A --> { .0. } <-> ( F Fn A /\ ran F C_ { .0. } ) )
35	32, 33, 34	sylanbrc 698	. . . . . . . . 9 |- ( ( ph /\ ran F C_ { .0. } ) -> F : A --> { .0. } )
36		disjdif 4040	. . . . . . . . 9 |- ( { .0. } i^i ( _V \ { .0. } ) ) = (/)
37		fimacnvdisj 6083	. . . . . . . . 9 |- ( ( F : A --> { .0. } /\ ( { .0. } i^i ( _V \ { .0. } ) ) = (/) ) -> ( `' F " ( _V \ { .0. } ) ) = (/) )
38	35, 36, 37	sylancl 694	. . . . . . . 8 |- ( ( ph /\ ran F C_ { .0. } ) -> ( `' F " ( _V \ { .0. } ) ) = (/) )
39	26, 29, 38	3eqtrd 2660	. . . . . . 7 |- ( ( ph /\ ran F C_ { .0. } ) -> W = (/) )
40	39	ex 450	. . . . . 6 |- ( ph -> ( ran F C_ { .0. } -> W = (/) ) )
41	25, 40	sylbid 230	. . . . 5 |- ( ph -> ( ran F C_ { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } -> W = (/) ) )
42	41	necon3ad 2807	. . . 4 |- ( ph -> ( W =/= (/) -> -. ran F C_ { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } ) )
43	24, 42	mpd 15	. . 3 |- ( ph -> -. ran F C_ { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } )
44	43	iffalsed 4097	. 2 |- ( ph -> if ( ran F C_ { z e. B | A. y e. B ( ( z .+ y ) = y /\ ( y .+ z ) = y ) } , .0. , if ( A e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W /\ x = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) ) ) ) ) = if ( A e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W /\ x = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) ) ) ) )
45		gsumval3a.i	. . 3 |- ( ph -> -. A e. ran ... )
46	45	iffalsed 4097	. 2 |- ( ph -> if ( A e. ran ... , ( iota x E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) , ( iota x E. f ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W /\ x = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) ) ) ) = ( iota x E. f ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W /\ x = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) ) ) )
47	23, 44, 46	3eqtrd 2660	1 |- ( ph -> ( G gsumg F ) = ( iota x E. f ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W /\ x = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   `'ccnv 5113   ran crn 5115   "cima 5117   o. ccom 5118   iotacio 5849   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   1c1 9937   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  Cntzccntz 17748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295

This theorem is referenced by:  gsumval3lem2  18307