Theorem gsumfzreidx 13743

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 3582	. . 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 2206	. . . . 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 13293	. . . 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 5533	. . . . . . . 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 5450	. . . . . . 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 13293	. . . . 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 3582	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) = .0. )
20	18, 19	eqtrd 2239	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg ( F o. H ) ) = .0. )
21	2, 11, 20	3eqtr4d 2249	. 2 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
22	6	cmnmndd 13714	. . . . . 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 13325	. . . . 5 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
27	23, 24, 25, 26	syl3anc 1250	. . . 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 13708	. . . . 5 |- ( ( G e. CMnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
30	28, 24, 25, 29	syl3anc 1250	. . . 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 13326	. . . . 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 9510	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
37	35	zred 9510	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
38		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
39	36, 37, 38	nltled 8208	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
40		eluz2 9669	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
41	34, 35, 39, 40	syl3anbrc 1184	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
42		ssidd 3218	. . . 4 |- ( ( ph /\ -. N < M ) -> B C_ B )
43		plusgslid 13014	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
44	43	slotex 12929	. . . . . 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 5546	. . . . 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 5727	. . . 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 5533	. . . . . . . . 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 5727	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( `' H ` k ) e. ( M ... N ) )
58		fvco3 5662	. . . . . 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 5859	. . . . . . 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 5592	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` ( H ` ( `' H ` k ) ) ) = ( F ` k ) )
63	59, 62	eqtr2d 2240	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` k ) = ( ( F o. H ) ` ( `' H ` k ) ) )
64	7, 8	fzfigd 10593	. . . . . . 7 |- ( ph -> ( M ... N ) e. Fin )
65	9, 64	fexd 5826	. . . . . 6 |- ( ph -> F e. _V )
66	14, 64	fexd 5826	. . . . . 6 |- ( ph -> H e. _V )
67		coexg 5235	. . . . . 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 5826	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
73	27, 30, 33, 41, 42, 46, 49, 51, 63, 69, 72	seqf1og 10683	. . 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 3585	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
76	74, 75	eqtrd 2239	. . 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 3585	. . . 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 2239	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gsumg ( F o. H ) ) = ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) )
80	73, 76, 79	3eqtr4d 2249	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
81		zdclt 9465	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
82	8, 7, 81	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
83		exmiddc 838	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
84	82, 83	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
85	21, 80, 84	mpjaodan 800	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2177   _Vcvv 2773   ifcif 3575   class class class wbr 4050   `'ccnv 4681   o. ccom 4686   -->wf 5275   -1-1-onto->wf1o 5278   ` cfv 5279  (class class class)co 5956   Fincfn 6839   < clt 8122   <_ cle 8123   ZZcz 9387   ZZ>=cuz 9663   ...cfz 10145   seqcseq 10609   Basecbs 12902   +g cplusg 12979   0gc0g 13158   gsum_g cgsu 13159   Mndcmnd 13318  CMndccmn 13690

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-mulrcl 8039  ax-addcom 8040  ax-mulcom 8041  ax-addass 8042  ax-mulass 8043  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-1rid 8047  ax-0id 8048  ax-rnegex 8049  ax-precex 8050  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-apti 8055  ax-pre-ltadd 8056  ax-pre-mulgt0 8057

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-frec 6489  df-1o 6514  df-er 6632  df-en 6840  df-fin 6842  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-reap 8663  df-ap 8670  df-inn 9052  df-2 9110  df-n0 9311  df-z 9388  df-uz 9664  df-fz 10146  df-fzo 10280  df-seqfrec 10610  df-ndx 12905  df-slot 12906  df-base 12908  df-plusg 12992  df-0g 13160  df-igsum 13161  df-mgm 13258  df-sgrp 13304  df-mnd 13319  df-cmn 13692

This theorem is referenced by:  lgseisenlem3  15619