Theorem gsumfzreidx 14138

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 3633	. . 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 2234	. . . . 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 13688	. . . 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 5619	. . . . . . . 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 5532	. . . . . . 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 13688	. . . . 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 3633	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) = .0. )
20	18, 19	eqtrd 2267	. . 3 |- ( ( ph /\ N < M ) -> ( G gsumg ( F o. H ) ) = .0. )
21	2, 11, 20	3eqtr4d 2277	. 2 |- ( ( ph /\ N < M ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
22	6	cmnmndd 14109	. . . . . 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 13720	. . . . 5 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
27	23, 24, 25, 26	syl3anc 1274	. . . 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 14103	. . . . 5 |- ( ( G e. CMnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
30	28, 24, 25, 29	syl3anc 1274	. . . 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 13721	. . . . 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 9718	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
37	35	zred 9718	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
38		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
39	36, 37, 38	nltled 8410	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
40		eluz2 9877	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
41	34, 35, 39, 40	syl3anbrc 1208	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
42		ssidd 3263	. . . 4 |- ( ( ph /\ -. N < M ) -> B C_ B )
43		plusgslid 13409	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
44	43	slotex 13323	. . . . . 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 5632	. . . . 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 5817	. . . 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 5619	. . . . . . . . 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 5817	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( `' H ` k ) e. ( M ... N ) )
58		fvco3 5753	. . . . . 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 5957	. . . . . . 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 5679	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` ( H ` ( `' H ` k ) ) ) = ( F ` k ) )
63	59, 62	eqtr2d 2268	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` k ) = ( ( F o. H ) ` ( `' H ` k ) ) )
64	7, 8	fzfigd 10817	. . . . . . 7 |- ( ph -> ( M ... N ) e. Fin )
65	9, 64	fexd 5921	. . . . . 6 |- ( ph -> F e. _V )
66	14, 64	fexd 5921	. . . . . 6 |- ( ph -> H e. _V )
67		coexg 5312	. . . . . 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 5921	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
73	27, 30, 33, 41, 42, 46, 49, 51, 63, 69, 72	seqf1og 10907	. . 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 3636	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
76	74, 75	eqtrd 2267	. . 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 3636	. . . 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 2267	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gsumg ( F o. H ) ) = ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) )
80	73, 76, 79	3eqtr4d 2277	. 2 |- ( ( ph /\ -. N < M ) -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
81		zdclt 9672	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
82	8, 7, 81	syl2anc 411	. . 3 |- ( ph -> DECID N < M )
83		exmiddc 844	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
84	82, 83	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
85	21, 80, 84	mpjaodan 806	1 |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   ifcif 3624   class class class wbr 4114   `'ccnv 4753   o. ccom 4758   -->wf 5353   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   Fincfn 6988   < clt 8324   <_ cle 8325   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   Basecbs 13296   +g cplusg 13374   0gc0g 13553   gsum_g cgsu 13554   Mndcmnd 13713  CMndccmn 14085

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-igsum 13556  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-cmn 14087

This theorem is referenced by:  lgseisenlem3  16057