Theorem gzsumreidx 14118

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

Assertion

Ref	Expression
gzsumreidx	|- ( ph -> ( G gzsumgz F ) = ( G gzsumgz ( F o. H ) ) )

Proof of Theorem gzsumreidx

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 3644	. . 3 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = .0. )
3		gzsumreidx.b	. . . . 5 |- B = ( Base ` G )
4		gzsumreidx.z	. . . . 5 |- .0. = ( 0g ` G )
5		eqid 2238	. . . . 5 |- ( +g ` G ) = ( +g ` G )
6		gzsumreidx.g	. . . . 5 |- ( ph -> G e. CMnd )
7		gzsumreidx.m	. . . . 5 |- ( ph -> M e. ZZ )
8		gzsumreidx.n	. . . . 5 |- ( ph -> N e. ZZ )
9		gzsumreidx.f	. . . . 5 |- ( ph -> F : ( M ... N ) --> B )
10	3, 4, 5, 6, 7, 8, 9	gzsumfzval 13688	. . . 4 |- ( ph -> ( G gzsumgz F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
11	10	adantr 276	. . 3 |- ( ( ph /\ N < M ) -> ( G gzsumgz F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
12		gzsumreidx.h	. . . . . . . 8 |- ( ph -> H : ( M ... N ) -1-1-onto-> ( M ... N ) )
13		f1of 5634	. . . . . . . 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	9, 14	fcod 5548	. . . . . 6 |- ( ph -> ( F o. H ) : ( M ... N ) --> B )
16	3, 4, 5, 6, 7, 8, 15	gzsumfzval 13688	. . . . 5 |- ( ph -> ( G gzsumgz ( F o. H ) ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) )
17	16	adantr 276	. . . 4 |- ( ( ph /\ N < M ) -> ( G gzsumgz ( F o. H ) ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) )
18	1	iftrued 3644	. . . 4 |- ( ( ph /\ N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) = .0. )
19	17, 18	eqtrd 2271	. . 3 |- ( ( ph /\ N < M ) -> ( G gzsumgz ( F o. H ) ) = .0. )
20	2, 11, 19	3eqtr4d 2281	. 2 |- ( ( ph /\ N < M ) -> ( G gzsumgz F ) = ( G gzsumgz ( F o. H ) ) )
21	6	cmnmndd 14088	. . . . . 6 |- ( ph -> G e. Mnd )
22	21	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> G e. Mnd )
23		simprl 535	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> x e. B )
24		simprr 537	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> y e. B )
25	3, 5	mndcl 13713	. . . . 5 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
26	22, 23, 24, 25	syl3anc 1278	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
27	6	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> G e. CMnd )
28	3, 5	cmncom 14082	. . . . 5 |- ( ( G e. CMnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
29	27, 23, 24, 28	syl3anc 1278	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
30	21	ad2antrr 492	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ ( x e. B /\ y e. B /\ z e. B ) ) -> G e. Mnd )
31	3, 5	mndass 13714	. . . . 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 ) ) )
32	30, 31	sylancom 424	. . . 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 ) ) )
33	7	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> M e. ZZ )
34	8	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> N e. ZZ )
35	33	zred 9747	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
36	34	zred 9747	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
37		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
38	35, 36, 37	nltled 8437	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
39		eluz2 9906	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
40	33, 34, 38, 39	syl3anbrc 1212	. . . 4 |- ( ( ph /\ -. N < M ) -> N e. ( ZZ>= ` M ) )
41		ssidd 3269	. . . 4 |- ( ( ph /\ -. N < M ) -> B C_ B )
42		plusgslid 13443	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13357	. . . . . 6 |- ( G e. CMnd -> ( +g ` G ) e. _V )
44	6, 43	syl 14	. . . . 5 |- ( ph -> ( +g ` G ) e. _V )
45	44	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( +g ` G ) e. _V )
46	12	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> H : ( M ... N ) -1-1-onto-> ( M ... N ) )
47		f1ocnv 5647	. . . . 5 |- ( H : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' H : ( M ... N ) -1-1-onto-> ( M ... N ) )
48	46, 47	syl 14	. . . 4 |- ( ( ph /\ -. N < M ) -> `' H : ( M ... N ) -1-1-onto-> ( M ... N ) )
49	15	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( F o. H ) : ( M ... N ) --> B )
50	49	ffvelcdmda 5834	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ x e. ( M ... N ) ) -> ( ( F o. H ) ` x ) e. B )
51	14	ad2antrr 492	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> H : ( M ... N ) --> ( M ... N ) )
52	12, 47	syl 14	. . . . . . . . 9 |- ( ph -> `' H : ( M ... N ) -1-1-onto-> ( M ... N ) )
53		f1of 5634	. . . . . . . . 9 |- ( `' H : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' H : ( M ... N ) --> ( M ... N ) )
54	52, 53	syl 14	. . . . . . . 8 |- ( ph -> `' H : ( M ... N ) --> ( M ... N ) )
55	54	adantr 276	. . . . . . 7 |- ( ( ph /\ -. N < M ) -> `' H : ( M ... N ) --> ( M ... N ) )
56	55	ffvelcdmda 5834	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( `' H ` k ) e. ( M ... N ) )
57		fvco3 5770	. . . . . 6 |- ( ( H : ( M ... N ) --> ( M ... N ) /\ ( `' H ` k ) e. ( M ... N ) ) -> ( ( F o. H ) ` ( `' H ` k ) ) = ( F ` ( H ` ( `' H ` k ) ) ) )
58	51, 56, 57	syl2anc 415	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( ( F o. H ) ` ( `' H ` k ) ) = ( F ` ( H ` ( `' H ` k ) ) ) )
59		f1ocnvfv2 5974	. . . . . . 7 |- ( ( H : ( M ... N ) -1-1-onto-> ( M ... N ) /\ k e. ( M ... N ) ) -> ( H ` ( `' H ` k ) ) = k )
60	46, 59	sylan 283	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( H ` ( `' H ` k ) ) = k )
61	60	fveq2d 5694	. . . . 5 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` ( H ` ( `' H ` k ) ) ) = ( F ` k ) )
62	58, 61	eqtr2d 2272	. . . 4 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( F ` k ) = ( ( F o. H ) ` ( `' H ` k ) ) )
63	7, 8	fzfigd 10846	. . . . . . 7 |- ( ph -> ( M ... N ) e. Fin )
64	9, 63	fexd 5938	. . . . . 6 |- ( ph -> F e. _V )
65	14, 63	fexd 5938	. . . . . 6 |- ( ph -> H e. _V )
66		coexg 5327	. . . . . 6 |- ( ( F e. _V /\ H e. _V ) -> ( F o. H ) e. _V )
67	64, 65, 66	syl2anc 415	. . . . 5 |- ( ph -> ( F o. H ) e. _V )
68	67	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( F o. H ) e. _V )
69	9	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> F : ( M ... N ) --> B )
70	63	adantr 276	. . . . 5 |- ( ( ph /\ -. N < M ) -> ( M ... N ) e. Fin )
71	69, 70	fexd 5938	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
72	26, 29, 32, 40, 41, 45, 48, 50, 62, 68, 71	seqf1og 10936	. . 3 |- ( ( ph /\ -. N < M ) -> ( seq M ( ( +g ` G ) , F ) ` N ) = ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) )
73	10	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz F ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) )
74	37	iffalsed 3647	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , F ) ` N ) ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
75	73, 74	eqtrd 2271	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz F ) = ( seq M ( ( +g ` G ) , F ) ` N ) )
76	16	adantr 276	. . . 4 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz ( F o. H ) ) = if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) )
77	37	iffalsed 3647	. . . 4 |- ( ( ph /\ -. N < M ) -> if ( N < M , .0. , ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) ) = ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) )
78	76, 77	eqtrd 2271	. . 3 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz ( F o. H ) ) = ( seq M ( ( +g ` G ) , ( F o. H ) ) ` N ) )
79	72, 75, 78	3eqtr4d 2281	. 2 |- ( ( ph /\ -. N < M ) -> ( G gzsumgz F ) = ( G gzsumgz ( F o. H ) ) )
80		zdclt 9701	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
81	8, 7, 80	syl2anc 415	. . 3 |- ( ph -> DECID N < M )
82		exmiddc 848	. . 3 |- (DECID N < M -> ( N < M \/ -. N < M ) )
83	81, 82	syl 14	. 2 |- ( ph -> ( N < M \/ -. N < M ) )
84	20, 79, 83	mpjaodan 810	1 |- ( ph -> ( G gzsumgz F ) = ( G gzsumgz ( F o. H ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   ifcif 3635   class class class wbr 4125   `'ccnv 4768   o. ccom 4773   -->wf 5368   -1-1-onto->wf1o 5371   ` cfv 5372  (class class class)co 6075   Fincfn 7012   < clt 8350   <_ cle 8351   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   Basecbs 13330   +g cplusg 13408   0gc0g 13587   gzsum_gz cgzsu 13588   Mndcmnd 13706  CMndccmn 14064

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-cmn 14066

This theorem is referenced by:  gsumf1ofi  14137