Theorem gsumsnfdcmn 14257

Description: Group sum of a singleton, deduction form, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Revised by AV, 11-Dec-2019.)

Hypotheses

Ref	Expression
gsumsnd.b	|- B = ( Base ` G )
gsumsndcmn.g	|- ( ph -> G e. CMnd )
gsumsnd.m	|- ( ph -> M e. V )
gsumsnd.c	|- ( ph -> C e. B )
gsumsnd.s	|- ( ( ph /\ k = M ) -> A = C )
gsumsnfd.p	|- F/ k ph
gsumsnfd.c	|- F/_ k C

Assertion

Ref	Expression
gsumsnfdcmn	|- ( ph -> ( G gsumg ( k e. { M } |-> A ) ) = C )

Distinct variable group:   k, M

Allowed substitution hints:   ph( k)   A( k)   B( k)   C( k)   G( k)   V( k)

Proof of Theorem gsumsnfdcmn

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumsnd.b	. . 3 |- B = ( Base ` G )
2		eqid 2238	. . 3 |- ( +g ` G ) = ( +g ` G )
3		gsumsndcmn.g	. . 3 |- ( ph -> G e. CMnd )
4		gsumsnfd.p	. . . . . . . 8 |- F/ k ph
5		elsni 3727	. . . . . . . . 9 |- ( k e. { M } -> k = M )
6		gsumsnd.s	. . . . . . . . . 10 |- ( ( ph /\ k = M ) -> A = C )
7	6	ex 115	. . . . . . . . 9 |- ( ph -> ( k = M -> A = C ) )
8	5, 7	syl5 32	. . . . . . . 8 |- ( ph -> ( k e. { M } -> A = C ) )
9	4, 8	ralrimi 2621	. . . . . . 7 |- ( ph -> A. k e. { M } A = C )
10		nfcsb1v 3180	. . . . . . . . 9 |- F/_ k [_ q / k ]_ A
11		gsumsnfd.c	. . . . . . . . 9 |- F/_ k C
12	10, 11	nfeq 2400	. . . . . . . 8 |- F/ k [_ q / k ]_ A = C
13		csbeq1a 3156	. . . . . . . . 9 |- ( k = q -> A = [_ q / k ]_ A )
14	13	eqeq1d 2247	. . . . . . . 8 |- ( k = q -> ( A = C <-> [_ q / k ]_ A = C ) )
15	12, 14	rspc 2923	. . . . . . 7 |- ( q e. { M } -> ( A. k e. { M } A = C -> [_ q / k ]_ A = C ) )
16	9, 15	mpan9 281	. . . . . 6 |- ( ( ph /\ q e. { M } ) -> [_ q / k ]_ A = C )
17		gsumsnd.c	. . . . . . 7 |- ( ph -> C e. B )
18	17	adantr 276	. . . . . 6 |- ( ( ph /\ q e. { M } ) -> C e. B )
19	16, 18	eqeltrd 2315	. . . . 5 |- ( ( ph /\ q e. { M } ) -> [_ q / k ]_ A e. B )
20		nfcv 2392	. . . . . 6 |- F/_ q A
21	20, 10, 13	cbvmpt 4228	. . . . 5 |- ( k e. { M } |-> A ) = ( q e. { M } |-> [_ q / k ]_ A )
22	19, 21	fmptd 5865	. . . 4 |- ( ph -> ( k e. { M } |-> A ) : { M } --> B )
23		uncom 3373	. . . . . 6 |- ( (/) u. { M } ) = ( { M } u. (/) )
24		un0 3556	. . . . . 6 |- ( { M } u. (/) ) = { M }
25	23, 24	eqtri 2259	. . . . 5 |- ( (/) u. { M } ) = { M }
26	25	feq2i 5530	. . . 4 |- ( ( k e. { M } |-> A ) : ( (/) u. { M } ) --> B <-> ( k e. { M } |-> A ) : { M } --> B )
27	22, 26	sylibr 134	. . 3 |- ( ph -> ( k e. { M } |-> A ) : ( (/) u. { M } ) --> B )
28		0fi 7192	. . . 4 |- (/) e. Fin
29	28	a1i 9	. . 3 |- ( ph -> (/) e. Fin )
30		gsumsnd.m	. . 3 |- ( ph -> M e. V )
31		noel 3525	. . . 4 |- -. M e. (/)
32	31	a1i 9	. . 3 |- ( ph -> -. M e. (/) )
33	1, 2, 3, 27, 29, 30, 32	gsump1 14247	. 2 |- ( ph -> ( G gsumg ( k e. { M } |-> A ) ) = ( ( G gsumg ( ( k e. { M } |-> A ) |` (/) ) ) ( +g ` G ) ( ( k e. { M } |-> A ) ` M ) ) )
34		res0 5069	. . . . 5 |- ( ( k e. { M } |-> A ) |` (/) ) = (/)
35	34	oveq2i 6099	. . . 4 |- ( G gsumg ( ( k e. { M } |-> A ) |` (/) ) ) = ( G gsumg (/) )
36		gsum0cmn 14244	. . . . 5 |- ( G e. CMnd -> ( G gsumg (/) ) = ( 0g ` G ) )
37	3, 36	syl 14	. . . 4 |- ( ph -> ( G gsumg (/) ) = ( 0g ` G ) )
38	35, 37	eqtrid 2283	. . 3 |- ( ph -> ( G gsumg ( ( k e. { M } |-> A ) |` (/) ) ) = ( 0g ` G ) )
39		nfv 1581	. . . . . 6 |- F/ k q = M
40	4, 39	nfan 1618	. . . . 5 |- F/ k ( ph /\ q = M )
41	11	a1i 9	. . . . 5 |- ( ( ph /\ q = M ) -> F/_ k C )
42		vex 2824	. . . . . 6 |- q e. _V
43	42	a1i 9	. . . . 5 |- ( ( ph /\ q = M ) -> q e. _V )
44		simpll 531	. . . . . 6 |- ( ( ( ph /\ q = M ) /\ k = q ) -> ph )
45		simpr 110	. . . . . . 7 |- ( ( ( ph /\ q = M ) /\ k = q ) -> k = q )
46		simplr 533	. . . . . . 7 |- ( ( ( ph /\ q = M ) /\ k = q ) -> q = M )
47	45, 46	eqtrd 2271	. . . . . 6 |- ( ( ( ph /\ q = M ) /\ k = q ) -> k = M )
48	44, 47, 6	syl2anc 415	. . . . 5 |- ( ( ( ph /\ q = M ) /\ k = q ) -> A = C )
49	40, 41, 43, 48	csbiedf 3188	. . . 4 |- ( ( ph /\ q = M ) -> [_ q / k ]_ A = C )
50		snidg 3738	. . . . 5 |- ( M e. V -> M e. { M } )
51	30, 50	syl 14	. . . 4 |- ( ph -> M e. { M } )
52	21, 49, 51, 17	fvmptd2 5790	. . 3 |- ( ph -> ( ( k e. { M } |-> A ) ` M ) = C )
53	38, 52	oveq12d 6106	. 2 |- ( ph -> ( ( G gsumg ( ( k e. { M } |-> A ) |` (/) ) ) ( +g ` G ) ( ( k e. { M } |-> A ) ` M ) ) = ( ( 0g ` G ) ( +g ` G ) C ) )
54	3	cmnmndd 14201	. . 3 |- ( ph -> G e. Mnd )
55		eqid 2238	. . . 4 |- ( 0g ` G ) = ( 0g ` G )
56	1, 2, 55	mndlid 13807	. . 3 |- ( ( G e. Mnd /\ C e. B ) -> ( ( 0g ` G ) ( +g ` G ) C ) = C )
57	54, 17, 56	syl2anc 415	. 2 |- ( ph -> ( ( 0g ` G ) ( +g ` G ) C ) = C )
58	33, 53, 57	3eqtrd 2275	1 |- ( ph -> ( G gsumg ( k e. { M } |-> A ) ) = C )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   F/wnf 1513   e. wcel 2209   F/_wnfc 2379   A.wral 2528   _Vcvv 2821   [_csb 3147   u. cun 3218   (/)c0 3520   {csn 3709   |-> cmpt 4193   |` cres 4778   -->wf 5376   ` cfv 5380  (class class class)co 6088   Fincfn 7026   Basecbs 13410   +g cplusg 13490   0gc0g 13669   Mndcmnd 13788  CMndccmn 14177   gsum_g cgsu 14240

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ihash 11237  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-minusg 13868  df-mulg 13982  df-cmn 14179  df-gsumfi 14241

This theorem is used by:  gsumsndcmn  14258  gsumunsnfd  14259