Theorem gsumfzreidx 13929

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 3612	. . 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 2231	. . . . 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 13479	. . . 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 5583	. . . . . . . 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 5500	. . . . . . 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 13479	. . . . 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 3612	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) = .0. )
20	18, 19	eqtrd 2264	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg ( F o. H ) ) = .0. )
21	2, 11, 20	3eqtr4d 2274	. 2 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
22	6	cmnmndd 13900	. . . . . 6 |- ( ph -> G e. Mnd )
23	22	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> G e. Mnd )
24		simprl 531	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
25		simprr 533	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
26	3, 5	mndcl 13511	. . . . 5 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
27	23, 24, 25, 26	syl3anc 1273	. . . 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 13894	. . . . 5 |- ( ( G e. CMnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
30	28, 24, 25, 29	syl3anc 1273	. . . 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 13512	. . . . 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 9602	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
37	35	zred 9602	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
38		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
39	36, 37, 38	nltled 8300	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
40		eluz2 9761	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
41	34, 35, 39, 40	syl3anbrc 1207	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
42		ssidd 3248	. . . 4 |- ( ( ph /\ -. N < M ) -> B C_ B )
43		plusgslid 13200	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
44	43	slotex 13114	. . . . . 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 5596	. . . . 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 5782	. . . 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 5583	. . . . . . . . 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 5782	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( `' H ` k ) e. ( M ... N ) )
58		fvco3 5717	. . . . . 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 5919	. . . . . . 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 5643	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` ( H ` ( `' H ` k ) ) ) = ( F ` k ) )
63	59, 62	eqtr2d 2265	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` k ) = ( ( F o. H ) ` ( `' H ` k ) ) )
64	7, 8	fzfigd 10694	. . . . . . 7 |- ( ph -> ( M ... N ) e. Fin )
65	9, 64	fexd 5884	. . . . . 6 |- ( ph -> F e. _V )
66	14, 64	fexd 5884	. . . . . 6 |- ( ph -> H e. _V )
67		coexg 5281	. . . . . 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 5884	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
73	27, 30, 33, 41, 42, 46, 49, 51, 63, 69, 72	seqf1og 10784	. . 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 3615	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
76	74, 75	eqtrd 2264	. . 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 3615	. . . 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 2264	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gsumg ( F o. H ) ) = ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) )
80	73, 76, 79	3eqtr4d 2274	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
81		zdclt 9557	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
82	8, 7, 81	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
83		exmiddc 843	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
84	82, 83	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
85	21, 80, 84	mpjaodan 805	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   ifcif 3605   class class class wbr 4088   `'ccnv 4724   o. ccom 4729   -->wf 5322   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6018   Fincfn 6909   < clt 8214   <_ cle 8215   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243   seqcseq 10710   Basecbs 13087   +g cplusg 13165   0gc0g 13344   gsum_g cgsu 13345   Mndcmnd 13504  CMndccmn 13876

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-en 6910  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-ndx 13090  df-slot 13091  df-base 13093  df-plusg 13178  df-0g 13346  df-igsum 13347  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-cmn 13878

This theorem is referenced by:  lgseisenlem3  15807