Theorem gfsumz 16918

Description: Value of a finite group sum over the zero element. (Contributed by Jim Kingdon, 24-May-2026.)

Hypothesis

Ref	Expression
gfsumz.z	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
gfsumz	|- ( ( G e. CMnd /\ A e. Fin ) -> ( G gfsumgf ( k e. A |-> .0. ) ) = .0. )

Distinct variable groups:   .0. , k   A, k   k, G

Proof of Theorem gfsumz

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

Step	Hyp	Ref	Expression
1		mpteq1 4196	. . . 4 |- ( w = (/) -> ( k e. w |-> .0. ) = ( k e. (/) |-> .0. ) )
2	1	oveq2d 6068	. . 3 |- ( w = (/) -> ( G gfsumgf ( k e. w |-> .0. ) ) = ( G gfsumgf ( k e. (/) |-> .0. ) ) )
3	2	eqeq1d 2243	. 2 |- ( w = (/) -> ( ( G gfsumgf ( k e. w |-> .0. ) ) = .0. <-> ( G gfsumgf ( k e. (/) |-> .0. ) ) = .0. ) )
4		mpteq1 4196	. . . 4 |- ( w = y -> ( k e. w |-> .0. ) = ( k e. y |-> .0. ) )
5	4	oveq2d 6068	. . 3 |- ( w = y -> ( G gfsumgf ( k e. w |-> .0. ) ) = ( G gfsumgf ( k e. y |-> .0. ) ) )
6	5	eqeq1d 2243	. 2 |- ( w = y -> ( ( G gfsumgf ( k e. w |-> .0. ) ) = .0. <-> ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) )
7		mpteq1 4196	. . . 4 |- ( w = ( y u. { z } ) -> ( k e. w |-> .0. ) = ( k e. ( y u. { z } ) |-> .0. ) )
8	7	oveq2d 6068	. . 3 |- ( w = ( y u. { z } ) -> ( G gfsumgf ( k e. w |-> .0. ) ) = ( G gfsumgf ( k e. ( y u. { z } ) |-> .0. ) ) )
9	8	eqeq1d 2243	. 2 |- ( w = ( y u. { z } ) -> ( ( G gfsumgf ( k e. w |-> .0. ) ) = .0. <-> ( G gfsumgf ( k e. ( y u. { z } ) |-> .0. ) ) = .0. ) )
10		mpteq1 4196	. . . 4 |- ( w = A -> ( k e. w |-> .0. ) = ( k e. A |-> .0. ) )
11	10	oveq2d 6068	. . 3 |- ( w = A -> ( G gfsumgf ( k e. w |-> .0. ) ) = ( G gfsumgf ( k e. A |-> .0. ) ) )
12	11	eqeq1d 2243	. 2 |- ( w = A -> ( ( G gfsumgf ( k e. w |-> .0. ) ) = .0. <-> ( G gfsumgf ( k e. A |-> .0. ) ) = .0. ) )
13		gfsum0 16913	. . . 4 |- ( G e. CMnd -> ( G gfsumgf (/) ) = ( 0g ` G ) )
14		mpt0 5488	. . . . 5 |- ( k e. (/) |-> .0. ) = (/)
15	14	oveq2i 6063	. . . 4 |- ( G gfsumgf ( k e. (/) |-> .0. ) ) = ( G gfsumgf (/) )
16		gfsumz.z	. . . 4 |- .0. = ( 0g ` G )
17	13, 15, 16	3eqtr4g 2292	. . 3 |- ( G e. CMnd -> ( G gfsumgf ( k e. (/) |-> .0. ) ) = .0. )
18	17	adantr 276	. 2 |- ( ( G e. CMnd /\ A e. Fin ) -> ( G gfsumgf ( k e. (/) |-> .0. ) ) = .0. )
19		eqid 2234	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
20		eqid 2234	. . . . . 6 |- ( +g ` G ) = ( +g ` G )
21		simplll 535	. . . . . 6 |- ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> G e. CMnd )
22		cmnmnd 14039	. . . . . . . . . 10 |- ( G e. CMnd -> G e. Mnd )
23	19, 16	mndidcl 13664	. . . . . . . . . 10 |- ( G e. Mnd -> .0. e. ( Base ` G ) )
24	22, 23	syl 14	. . . . . . . . 9 |- ( G e. CMnd -> .0. e. ( Base ` G ) )
25	24	adantr 276	. . . . . . . 8 |- ( ( G e. CMnd /\ k e. ( y u. { z } ) ) -> .0. e. ( Base ` G ) )
26	25	fmpttd 5834	. . . . . . 7 |- ( G e. CMnd -> ( k e. ( y u. { z } ) |-> .0. ) : ( y u. { z } ) --> ( Base ` G ) )
27	21, 26	syl 14	. . . . . 6 |- ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( k e. ( y u. { z } ) |-> .0. ) : ( y u. { z } ) --> ( Base ` G ) )
28		simplr 529	. . . . . 6 |- ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> y e. Fin )
29		simprr 533	. . . . . 6 |- ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> z e. ( A \ y ) )
30	29	eldifbd 3225	. . . . . 6 |- ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> -. z e. y )
31	19, 20, 21, 27, 28, 29, 30	gfsump1 16917	. . . . 5 |- ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( G gfsumgf ( k e. ( y u. { z } ) |-> .0. ) ) = ( ( G gfsumgf ( ( k e. ( y u. { z } ) |-> .0. ) |` y ) ) ( +g ` G ) ( ( k e. ( y u. { z } ) |-> .0. ) ` z ) ) )
32	31	adantr 276	. . . 4 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> ( G gfsumgf ( k e. ( y u. { z } ) |-> .0. ) ) = ( ( G gfsumgf ( ( k e. ( y u. { z } ) |-> .0. ) |` y ) ) ( +g ` G ) ( ( k e. ( y u. { z } ) |-> .0. ) ` z ) ) )
33		ssun1 3384	. . . . . . . . 9 |- y C_ ( y u. { z } )
34	33	a1i 9	. . . . . . . 8 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> y C_ ( y u. { z } ) )
35	34	resmptd 5091	. . . . . . 7 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> ( ( k e. ( y u. { z } ) |-> .0. ) |` y ) = ( k e. y |-> .0. ) )
36	35	oveq2d 6068	. . . . . 6 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> ( G gfsumgf ( ( k e. ( y u. { z } ) |-> .0. ) |` y ) ) = ( G gfsumgf ( k e. y |-> .0. ) ) )
37		simpr 110	. . . . . 6 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> ( G gfsumgf ( k e. y |-> .0. ) ) = .0. )
38	36, 37	eqtrd 2267	. . . . 5 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> ( G gfsumgf ( ( k e. ( y u. { z } ) |-> .0. ) |` y ) ) = .0. )
39		eqid 2234	. . . . . . 7 |- ( k e. ( y u. { z } ) |-> .0. ) = ( k e. ( y u. { z } ) |-> .0. )
40		eqidd 2235	. . . . . . 7 |- ( k = z -> .0. = .0. )
41		vsnid 3723	. . . . . . . 8 |- z e. { z }
42		elun2 3389	. . . . . . . 8 |- ( z e. { z } -> z e. ( y u. { z } ) )
43	41, 42	mp1i 10	. . . . . . 7 |- ( G e. CMnd -> z e. ( y u. { z } ) )
44	39, 40, 43, 24	fvmptd3 5773	. . . . . 6 |- ( G e. CMnd -> ( ( k e. ( y u. { z } ) |-> .0. ) ` z ) = .0. )
45	44	ad4antr 494	. . . . 5 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> ( ( k e. ( y u. { z } ) |-> .0. ) ` z ) = .0. )
46	38, 45	oveq12d 6070	. . . 4 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> ( ( G gfsumgf ( ( k e. ( y u. { z } ) |-> .0. ) |` y ) ) ( +g ` G ) ( ( k e. ( y u. { z } ) |-> .0. ) ` z ) ) = ( .0. ( +g ` G ) .0. ) )
47	22	ad4antr 494	. . . . 5 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> G e. Mnd )
48	19, 20, 16	mndlid 13669	. . . . 5 |- ( ( G e. Mnd /\ .0. e. ( Base ` G ) ) -> ( .0. ( +g ` G ) .0. ) = .0. )
49	47, 23, 48	syl2anc2 412	. . . 4 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> ( .0. ( +g ` G ) .0. ) = .0. )
50	32, 46, 49	3eqtrd 2271	. . 3 |- ( ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) /\ ( G gfsumgf ( k e. y |-> .0. ) ) = .0. ) -> ( G gfsumgf ( k e. ( y u. { z } ) |-> .0. ) ) = .0. )
51	50	ex 115	. 2 |- ( ( ( ( G e. CMnd /\ A e. Fin ) /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( ( G gfsumgf ( k e. y |-> .0. ) ) = .0. -> ( G gfsumgf ( k e. ( y u. { z } ) |-> .0. ) ) = .0. ) )
52		simpr 110	. 2 |- ( ( G e. CMnd /\ A e. Fin ) -> A e. Fin )
53	3, 6, 9, 12, 18, 51, 52	findcard2sd 7151	1 |- ( ( G e. CMnd /\ A e. Fin ) -> ( G gfsumgf ( k e. A |-> .0. ) ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   \ cdif 3210   u. cun 3211   C_ wss 3213   (/)c0 3510   {csn 3691   |-> cmpt 4173   |` cres 4753   -->wf 5350   ` cfv 5354  (class class class)co 6052   Fincfn 6977   Basecbs 13233   +g cplusg 13311   0gc0g 13490   Mndcmnd 13650  CMndccmn 14022   gfsum_gf cgfsu 16909

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-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  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-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  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-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  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-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-inn 9243  df-2 9301  df-n0 9502  df-z 9583  df-uz 9860  df-fz 10349  df-fzo 10484  df-seqfrec 10817  df-ihash 11147  df-ndx 13236  df-slot 13237  df-base 13239  df-plusg 13324  df-0g 13492  df-igsum 13493  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-minusg 13738  df-mulg 13858  df-cmn 14024  df-gfsum 16910

This theorem is referenced by: (None)