Theorem gzsumreidx 14231

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 3647	. . 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 13770	. . . 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 5642	. . . . . . . 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 5556	. . . . . 6 |- ( ph -> ( F o. H ) : ( M ... N ) --> B )
16	3, 4, 5, 6, 7, 8, 15	gzsumfzval 13770	. . . . 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 3647	. . . 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 14201	. . . . . 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 13795	. . . . 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 14195	. . . . 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 13796	. . . . 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 9779	. . . . . 6 |- ( ( ph /\ -. N < M ) -> M e. RR )
36	34	zred 9779	. . . . . 6 |- ( ( ph /\ -. N < M ) -> N e. RR )
37		simpr 110	. . . . . 6 |- ( ( ph /\ -. N < M ) -> -. N < M )
38	35, 36, 37	nltled 8454	. . . . 5 |- ( ( ph /\ -. N < M ) -> M <_ N )
39		eluz2 9943	. . . . 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 13525	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
43	42	slotex 13437	. . . . . 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 5655	. . . . 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 5846	. . . 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 5642	. . . . . . . . 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 5846	. . . . . 6 |- ( ( ( ph /\ -. N < M ) /\ k e. ( M ... N ) ) -> ( `' H ` k ) e. ( M ... N ) )
57		fvco3 5779	. . . . . 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 5987	. . . . . . 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 5702	. . . . 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 10889	. . . . . . 7 |- ( ph -> ( M ... N ) e. Fin )
64	9, 63	fexd 5951	. . . . . 6 |- ( ph -> F e. _V )
65	14, 63	fexd 5951	. . . . . 6 |- ( ph -> H e. _V )
66		coexg 5335	. . . . . 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 5951	. . . 4 |- ( ( ph /\ -. N < M ) -> F e. _V )
72	26, 29, 32, 40, 41, 45, 48, 50, 62, 68, 71	seqf1og 10979	. . 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 3650	. . . 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 3650	. . . 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 9733	. . . 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 ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   ifcif 3638   class class class wbr 4131   `'ccnv 4775   o. ccom 4780   -->wf 5376   -1-1-onto->wf1o 5379   ` cfv 5380  (class class class)co 6088   Fincfn 7026   < clt 8366   <_ cle 8367   ZZcz 9655   ZZ>=cuz 9937   ...cfz 10428   seqcseq 10905   Basecbs 13410   +g cplusg 13490   0gc0g 13669   gzsum_gz cgzsu 13670   Mndcmnd 13788  CMndccmn 14177

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-cmn 14179

This theorem is used by:  gsumf1ofi  14250