Theorem gsumfzreidx 13407

Description: Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with M = 1. (Contributed by AV, 26-Dec-2023.)

Hypotheses

Ref	Expression
gsumreidx.b	|- B = ( Base ` G )
gsumreidx.z	|- .0. = ( 0g ` G )
gsumreidx.g	|- ( ph -> G e. CMnd )
gsumfzreidx.m	|- ( ph -> M e. ZZ )
gsumfzreidx.n	|- ( ph -> N e. ZZ )
gsumreidx.f	|- ( ph -> F : ( M ... N ) --> B )
gsumreidx.h	|- ( ph -> H : ( M ... N ) -1-1-onto-> ( M ... N ) )

Assertion

Ref	Expression
gsumfzreidx	|- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Proof of Theorem gsumfzreidx

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

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 |- ( ( ph /\ N < M ) -> N < M )
2	1	iftrued 3564	. . 3 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
3		gsumreidx.b	. . . . 5 |- B = ( Base ` G )
4		gsumreidx.z	. . . . 5 |- .0. = ( 0g ` G )
5		eqid 2193	. . . . 5 |- ( +g ` G ) = ( +g ` G )
6		gsumreidx.g	. . . . 5 |- ( ph -> G e. CMnd )
7		gsumfzreidx.m	. . . . 5 |- ( ph -> M e. ZZ )
8		gsumfzreidx.n	. . . . 5 |- ( ph -> N e. ZZ )
9		gsumreidx.f	. . . . 5 |- ( ph -> F : ( M ... N ) --> B )
10	3, 4, 5, 6, 7, 8, 9	gsumfzval 12974	. . . 4 |- ( ph -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
11	10	adantr 276	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
12		gsumreidx.h	. . . . . . . 8 |- ( ph -> H : ( M ... N ) -1-1-onto-> ( M ... N ) )
13		f1of 5500	. . . . . . . 8 |- ( H : ( M ... N ) -1-1-onto-> ( M ... N ) -> H : ( M ... N ) --> ( M ... N ) )
14	12, 13	syl 14	. . . . . . 7 |- ( ph -> H : ( M ... N ) --> ( M ... N ) )
15		fco 5419	. . . . . . 7 |- ( ( F : ( M ... N ) --> B /\ H : ( M ... N ) --> ( M ... N ) ) -> ( F o. H ) : ( M ... N ) --> B )
16	9, 14, 15	syl2anc 411	. . . . . 6 |- ( ph -> ( F o. H ) : ( M ... N ) --> B )
17	3, 4, 5, 6, 7, 8, 16	gsumfzval 12974	. . . . 5 |- ( ph -> ( G gsumg ( F o. H ) ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) )
18	17	adantr 276	. . . 4 |- ( ( ph /\ N < M ) -> ( G gsumg ( F o. H ) ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) )
19	1	iftrued 3564	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) = .0. )
20	18, 19	eqtrd 2226	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg ( F o. H ) ) = .0. )
21	2, 11, 20	3eqtr4d 2236	. 2 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
22	6	cmnmndd 13378	. . . . . 6 |- ( ph -> G e. Mnd )
23	22	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> G e. Mnd )
24		simprl 529	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
25		simprr 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
26	3, 5	mndcl 13004	. . . . 5 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
27	23, 24, 25, 26	syl3anc 1249	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
28	6	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> G e. CMnd )
29	3, 5	cmncom 13372	. . . . 5 |- ( ( G e. CMnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
30	28, 24, 25, 29	syl3anc 1249	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
31	22	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B /\ z e. B ) ) -> G e. Mnd )
32	3, 5	mndass 13005	. . . . 5 |- ( ( G e. Mnd /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x ( +g ` G ) y ) ( +g ` G ) z ) = ( x ( +g ` G ) ( y ( +g ` G ) z ) ) )
33	31, 32	sylancom 420	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x ( +g ` G ) y ) ( +g ` G ) z ) = ( x ( +g ` G ) ( y ( +g ` G ) z ) ) )
34	7	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> M e. ZZ )
35	8	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> N e. ZZ )
36	34	zred 9439	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
37	35	zred 9439	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
38		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
39	36, 37, 38	nltled 8140	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
40		eluz2 9598	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
41	34, 35, 39, 40	syl3anbrc 1183	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
42		ssidd 3200	. . . 4 |- ( ( ph /\ -. N < M ) -> B C_ B )
43		plusgslid 12730	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
44	43	slotex 12645	. . . . . 6 |- ( G e. CMnd -> ( +g ` G ) e. _V )
45	6, 44	syl 14	. . . . 5 |- ( ph -> ( +g ` G ) e. _V )
46	45	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( +g ` G ) e. _V )
47	12	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> H : ( M ... N ) -1-1-onto-> ( M ... N ) )
48		f1ocnv 5513	. . . . 5 |- ( H : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' H : ( M ... N ) -1-1-onto-> ( M ... N ) )
49	47, 48	syl 14	. . . 4 |- ( ( ph /\ -. N < M ) -> `' H : ( M ... N ) -1-1-onto-> ( M ... N ) )
50	16	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( F o. H ) : ( M ... N ) --> B )
51	50	ffvelcdmda 5693	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( ( F o. H ) ` x ) e. B )
52	14	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> H : ( M ... N ) --> ( M ... N ) )
53	12, 48	syl 14	. . . . . . . . 9 |- ( ph -> `' H : ( M ... N ) -1-1-onto-> ( M ... N ) )
54		f1of 5500	. . . . . . . . 9 |- ( `' H : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' H : ( M ... N ) --> ( M ... N ) )
55	53, 54	syl 14	. . . . . . . 8 |- ( ph -> `' H : ( M ... N ) --> ( M ... N ) )
56	55	adantr 276	. . . . . . 7 |- ( ( ph /\ -. N < M ) -> `' H : ( M ... N ) --> ( M ... N ) )
57	56	ffvelcdmda 5693	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( `' H ` k ) e. ( M ... N ) )
58		fvco3 5628	. . . . . 6 |- ( ( H : ( M ... N ) --> ( M ... N ) /\ ( `' H ` k ) e. ( M ... N ) ) -> ( ( F o. H ) ` ( `' H ` k ) ) = ( F ` ( H ` ( `' H ` k ) ) ) )
59	52, 57, 58	syl2anc 411	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( ( F o. H ) ` ( `' H ` k ) ) = ( F ` ( H ` ( `' H ` k ) ) ) )
60		f1ocnvfv2 5821	. . . . . . 7 |- ( ( H : ( M ... N ) -1-1-onto-> ( M ... N ) /\ k e. ( M ... N ) ) -> ( H ` ( `' H ` k ) ) = k )
61	47, 60	sylan 283	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( H ` ( `' H ` k ) ) = k )
62	61	fveq2d 5558	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` ( H ` ( `' H ` k ) ) ) = ( F ` k ) )
63	59, 62	eqtr2d 2227	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` k ) = ( ( F o. H ) ` ( `' H ` k ) ) )
64	7, 8	fzfigd 10502	. . . . . . 7 |- ( ph -> ( M ... N ) e. Fin )
65	9, 64	fexd 5788	. . . . . 6 |- ( ph -> F e. _V )
66	14, 64	fexd 5788	. . . . . 6 |- ( ph -> H e. _V )
67		coexg 5210	. . . . . 6 |- ( ( F e. _V /\ H e. _V ) -> ( F o. H ) e. _V )
68	65, 66, 67	syl2anc 411	. . . . 5 |- ( ph -> ( F o. H ) e. _V )
69	68	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( F o. H ) e. _V )
70	9	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> F : ( M ... N ) --> B )
71	64	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( M ... N ) e. Fin )
72	70, 71	fexd 5788	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
73	27, 30, 33, 41, 42, 46, 49, 51, 63, 69, 72	seqf1og 10592	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( ( +g ` G ) , F ) ` N ) = ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) )
74	10	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
75	38	iffalsed 3567	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
76	74, 75	eqtrd 2226	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
77	17	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gsumg ( F o. H ) ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) )
78	38	iffalsed 3567	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) = ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) )
79	77, 78	eqtrd 2226	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gsumg ( F o. H ) ) = ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) )
80	73, 76, 79	3eqtr4d 2236	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
81		zdclt 9394	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
82	8, 7, 81	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
83		exmiddc 837	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
84	82, 83	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
85	21, 80, 84	mpjaodan 799	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2164   _Vcvv 2760   ifcif 3557   class class class wbr 4029   `'ccnv 4658   o. ccom 4663   -->wf 5250   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   Fincfn 6794   < clt 8054   <_ cle 8055   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518   Basecbs 12618   +g cplusg 12695   0gc0g 12867   gsum_g cgsu 12868   Mndcmnd 12997  CMndccmn 13354

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-igsum 12870  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-cmn 13356

This theorem is referenced by:  lgseisenlem3  15188