Theorem gsummptshft 18336

Description: Index shift of a finite group sum over a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)

Hypotheses

Ref	Expression
gsummptshft.b	|- B = ( Base ` G )
gsummptshft.z	|- .0. = ( 0g ` G )
gsummptshft.g	|- ( ph -> G e. CMnd )
gsummptshft.k	|- ( ph -> K e. ZZ )
gsummptshft.m	|- ( ph -> M e. ZZ )
gsummptshft.n	|- ( ph -> N e. ZZ )
gsummptshft.a	|- ( ( ph /\ j e. ( M ... N ) ) -> A e. B )
gsummptshft.c	|- ( j = ( k - K ) -> A = C )

Assertion

Ref	Expression
gsummptshft	|- ( ph -> ( G gsumg ( j e. ( M ... N ) |-> A ) ) = ( G gsumg ( k e. ( ( M + K ) ... ( N + K ) ) |-> C ) ) )

Distinct variable groups:   A, k   B, j   C, j   j, k, K   j, M, k   j, N, k   ph, j, k

Allowed substitution hints:   A( j)   B( k)   C( k)   G( j, k)   .0. ( j, k)

Proof of Theorem gsummptshft

Step	Hyp	Ref	Expression
1		gsummptshft.b	. . 3 |- B = ( Base ` G )
2		gsummptshft.z	. . 3 |- .0. = ( 0g ` G )
3		gsummptshft.g	. . 3 |- ( ph -> G e. CMnd )
4		ovexd 6680	. . 3 |- ( ph -> ( M ... N ) e. _V )
5		gsummptshft.a	. . . 4 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. B )
6		eqid 2622	. . . 4 |- ( j e. ( M ... N ) |-> A ) = ( j e. ( M ... N ) |-> A )
7	5, 6	fmptd 6385	. . 3 |- ( ph -> ( j e. ( M ... N ) |-> A ) : ( M ... N ) --> B )
8		fzfid 12772	. . . 4 |- ( ph -> ( M ... N ) e. Fin )
9		fvex 6201	. . . . . 6 |- ( 0g ` G ) e. _V
10	2, 9	eqeltri 2697	. . . . 5 |- .0. e. _V
11	10	a1i 11	. . . 4 |- ( ph -> .0. e. _V )
12	6, 8, 5, 11	fsuppmptdm 8286	. . 3 |- ( ph -> ( j e. ( M ... N ) |-> A ) finSupp .0. )
13		gsummptshft.k	. . . 4 |- ( ph -> K e. ZZ )
14		gsummptshft.m	. . . 4 |- ( ph -> M e. ZZ )
15		gsummptshft.n	. . . 4 |- ( ph -> N e. ZZ )
16	13, 14, 15	mptfzshft 14510	. . 3 |- ( ph -> ( k e. ( ( M + K ) ... ( N + K ) ) |-> ( k - K ) ) : ( ( M + K ) ... ( N + K ) ) -1-1-onto-> ( M ... N ) )
17	1, 2, 3, 4, 7, 12, 16	gsumf1o 18317	. 2 |- ( ph -> ( G gsumg ( j e. ( M ... N ) |-> A ) ) = ( G gsumg ( ( j e. ( M ... N ) |-> A ) o. ( k e. ( ( M + K ) ... ( N + K ) ) |-> ( k - K ) ) ) ) )
18		elfzelz 12342	. . . . . . . 8 |- ( k e. ( ( M + K ) ... ( N + K ) ) -> k e. ZZ )
19	18	zcnd 11483	. . . . . . 7 |- ( k e. ( ( M + K ) ... ( N + K ) ) -> k e. CC )
20	13	zcnd 11483	. . . . . . 7 |- ( ph -> K e. CC )
21		npcan 10290	. . . . . . 7 |- ( ( k e. CC /\ K e. CC ) -> ( ( k - K ) + K ) = k )
22	19, 20, 21	syl2anr 495	. . . . . 6 |- ( ( ph /\ k e. ( ( M + K ) ... ( N + K ) ) ) -> ( ( k - K ) + K ) = k )
23		simpr 477	. . . . . 6 |- ( ( ph /\ k e. ( ( M + K ) ... ( N + K ) ) ) -> k e. ( ( M + K ) ... ( N + K ) ) )
24	22, 23	eqeltrd 2701	. . . . 5 |- ( ( ph /\ k e. ( ( M + K ) ... ( N + K ) ) ) -> ( ( k - K ) + K ) e. ( ( M + K ) ... ( N + K ) ) )
25	14, 15	jca 554	. . . . . . 7 |- ( ph -> ( M e. ZZ /\ N e. ZZ ) )
26	25	adantr 481	. . . . . 6 |- ( ( ph /\ k e. ( ( M + K ) ... ( N + K ) ) ) -> ( M e. ZZ /\ N e. ZZ ) )
27	18	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( ( M + K ) ... ( N + K ) ) ) -> k e. ZZ )
28	13	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. ( ( M + K ) ... ( N + K ) ) ) -> K e. ZZ )
29	27, 28	zsubcld 11487	. . . . . 6 |- ( ( ph /\ k e. ( ( M + K ) ... ( N + K ) ) ) -> ( k - K ) e. ZZ )
30		fzaddel 12375	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( ( k - K ) e. ZZ /\ K e. ZZ ) ) -> ( ( k - K ) e. ( M ... N ) <-> ( ( k - K ) + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
31	26, 29, 28, 30	syl12anc 1324	. . . . 5 |- ( ( ph /\ k e. ( ( M + K ) ... ( N + K ) ) ) -> ( ( k - K ) e. ( M ... N ) <-> ( ( k - K ) + K ) e. ( ( M + K ) ... ( N + K ) ) ) )
32	24, 31	mpbird 247	. . . 4 |- ( ( ph /\ k e. ( ( M + K ) ... ( N + K ) ) ) -> ( k - K ) e. ( M ... N ) )
33		eqidd 2623	. . . 4 |- ( ph -> ( k e. ( ( M + K ) ... ( N + K ) ) |-> ( k - K ) ) = ( k e. ( ( M + K ) ... ( N + K ) ) |-> ( k - K ) ) )
34		eqidd 2623	. . . 4 |- ( ph -> ( j e. ( M ... N ) |-> A ) = ( j e. ( M ... N ) |-> A ) )
35		gsummptshft.c	. . . 4 |- ( j = ( k - K ) -> A = C )
36	32, 33, 34, 35	fmptco 6396	. . 3 |- ( ph -> ( ( j e. ( M ... N ) |-> A ) o. ( k e. ( ( M + K ) ... ( N + K ) ) |-> ( k - K ) ) ) = ( k e. ( ( M + K ) ... ( N + K ) ) |-> C ) )
37	36	oveq2d 6666	. 2 |- ( ph -> ( G gsumg ( ( j e. ( M ... N ) |-> A ) o. ( k e. ( ( M + K ) ... ( N + K ) ) |-> ( k - K ) ) ) ) = ( G gsumg ( k e. ( ( M + K ) ... ( N + K ) ) |-> C ) ) )
38	17, 37	eqtrd 2656	1 |- ( ph -> ( G gsumg ( j e. ( M ... N ) |-> A ) ) = ( G gsumg ( k e. ( ( M + K ) ... ( N + K ) ) |-> C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   o. ccom 5118   ` cfv 5888  (class class class)co 6650   CCcc 9934   + caddc 9939   - cmin 10266   ZZcz 11377   ...cfz 12326   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101  CMndccmn 18193

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  srgbinomlem4  18543  cpmadugsumlemF  20681