Theorem gsumval1 17277

Description: Value of the group sum operation when every element being summed is an identity of G. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
gsumval1.b	|- B = ( Base ` G )
gsumval1.z	|- .0. = ( 0g ` G )
gsumval1.p	|- .+ = ( +g ` G )
gsumval1.o	|- O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }
gsumval1.g	|- ( ph -> G e. V )
gsumval1.a	|- ( ph -> A e. W )
gsumval1.f	|- ( ph -> F : A --> O )

Assertion

Ref	Expression
gsumval1	|- ( ph -> ( G gsumg F ) = .0. )

Distinct variable groups:   x, y, B   x, .+ , y

Allowed substitution hints:   ph( x, y)   A( x, y)   F( x, y)   G( x, y)   O( x, y)   V( x, y)   W( x, y)   .0. ( x, y)

Proof of Theorem gsumval1

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

Step	Hyp	Ref	Expression
1		gsumval1.b	. . 3 |- B = ( Base ` G )
2		gsumval1.z	. . 3 |- .0. = ( 0g ` G )
3		gsumval1.p	. . 3 |- .+ = ( +g ` G )
4		gsumval1.o	. . 3 |- O = { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) }
5		eqidd 2623	. . 3 |- ( ph -> ( `' F " ( _V \ O ) ) = ( `' F " ( _V \ O ) ) )
6		gsumval1.g	. . 3 |- ( ph -> G e. V )
7		gsumval1.a	. . 3 |- ( ph -> A e. W )
8		gsumval1.f	. . . 4 |- ( ph -> F : A --> O )
9		ssrab2 3687	. . . . 5 |- { x e. B | A. y e. B ( ( x .+ y ) = y /\ ( y .+ x ) = y ) } C_ B
10	4, 9	eqsstri 3635	. . . 4 |- O C_ B
11		fss 6056	. . . 4 |- ( ( F : A --> O /\ O C_ B ) -> F : A --> B )
12	8, 10, 11	sylancl 694	. . 3 |- ( ph -> F : A --> B )
13	1, 2, 3, 4, 5, 6, 7, 12	gsumval 17271	. 2 |- ( ph -> ( G gsumg F ) = if ( ran F C_ O , .0. , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ O ) ) ) ) -1-1-onto-> ( `' F " ( _V \ O ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ O ) ) ) ) ) ) ) ) )
14		frn 6053	. . 3 |- ( F : A --> O -> ran F C_ O )
15		iftrue 4092	. . 3 |- ( ran F C_ O -> if ( ran F C_ O , .0. , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ O ) ) ) ) -1-1-onto-> ( `' F " ( _V \ O ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ O ) ) ) ) ) ) ) ) = .0. )
16	8, 14, 15	3syl 18	. 2 |- ( ph -> if ( ran F C_ O , .0. , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ O ) ) ) ) -1-1-onto-> ( `' F " ( _V \ O ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ O ) ) ) ) ) ) ) ) = .0. )
17	13, 16	eqtrd 2656	1 |- ( ph -> ( G gsumg F ) = .0. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ifcif 4086   `'ccnv 5113   ran crn 5115   "cima 5117   o. ccom 5118   iotacio 5849   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1c1 9937   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101

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-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-ral 2917  df-rex 2918  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-gsum 16103

This theorem is referenced by:  gsum0  17278  gsumval2  17280  gsumz  17374