Theorem gsumgfsum 16705

Description: On an integer range, gsum_g and gfsum_gf agree. (Contributed by Jim Kingdon, 25-Mar-2026.)

Hypotheses

Ref	Expression
gsumgfsum.b	|- B = ( Base ` G )
gsumgfsum.g	|- ( ph -> G e. CMnd )
gsumgfsum.m	|- ( ph -> M e. ZZ )
gsumgfsum.n	|- ( ph -> N e. ZZ )
gsumgfsum.f	|- ( ph -> F : ( M ... N ) --> B )

Assertion

Ref	Expression
gsumgfsum	|- ( ph -> ( G gsumg F ) = ( G gfsumgf F ) )

Proof of Theorem gsumgfsum

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumgfsum.b	. . . 4 |- B = ( Base ` G )
2		gsumgfsum.g	. . . . 5 |- ( ph -> G e. CMnd )
3	2	adantr 276	. . . 4 |- ( ( ph /\ M <_ N ) -> G e. CMnd )
4		gsumgfsum.m	. . . . . 6 |- ( ph -> M e. ZZ )
5	4	adantr 276	. . . . 5 |- ( ( ph /\ M <_ N ) -> M e. ZZ )
6		gsumgfsum.n	. . . . . 6 |- ( ph -> N e. ZZ )
7	6	adantr 276	. . . . 5 |- ( ( ph /\ M <_ N ) -> N e. ZZ )
8		simpr 110	. . . . 5 |- ( ( ph /\ M <_ N ) -> M <_ N )
9		eluz2 9761	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
10	5, 7, 8, 9	syl3anbrc 1207	. . . 4 |- ( ( ph /\ M <_ N ) -> N e. ( ZZ>= ` M ) )
11		gsumgfsum.f	. . . . 5 |- ( ph -> F : ( M ... N ) --> B )
12	11	adantr 276	. . . 4 |- ( ( ph /\ M <_ N ) -> F : ( M ... N ) --> B )
13		eqid 2231	. . . 4 |- ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) = ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) )
14	1, 3, 10, 12, 13	gsumgfsumlem 16704	. . 3 |- ( ( ph /\ M <_ N ) -> ( G gsumg F ) = ( G gsumg ( F o. ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) ) ) )
15	4, 6	fzfigd 10694	. . . . 5 |- ( ph -> ( M ... N ) e. Fin )
16	15	adantr 276	. . . 4 |- ( ( ph /\ M <_ N ) -> ( M ... N ) e. Fin )
17		1zzd 9506	. . . . . . . 8 |- ( ph -> 1 e. ZZ )
18	17, 4	zsubcld 9607	. . . . . . 7 |- ( ph -> ( 1 - M ) e. ZZ )
19	18, 4, 6	mptfzshft 12005	. . . . . 6 |- ( ph -> ( j e. ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) : ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) -1-1-onto-> ( M ... N ) )
20	19	adantr 276	. . . . 5 |- ( ( ph /\ M <_ N ) -> ( j e. ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) : ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) -1-1-onto-> ( M ... N ) )
21	4	zcnd 9603	. . . . . . . . . 10 |- ( ph -> M e. CC )
22		1cnd 8195	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
23	21, 22	pncan3d 8493	. . . . . . . . 9 |- ( ph -> ( M + ( 1 - M ) ) = 1 )
24	23	oveq1d 6033	. . . . . . . 8 |- ( ph -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... ( N + ( 1 - M ) ) ) )
25	24	mpteq1d 4174	. . . . . . 7 |- ( ph -> ( j e. ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) = ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) )
26	25	adantr 276	. . . . . 6 |- ( ( ph /\ M <_ N ) -> ( j e. ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) = ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) )
27	23	adantr 276	. . . . . . 7 |- ( ( ph /\ M <_ N ) -> ( M + ( 1 - M ) ) = 1 )
28		hashfz 11086	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> (♯ ` ( M ... N ) ) = ( ( N - M ) + 1 ) )
29	10, 28	syl 14	. . . . . . . 8 |- ( ( ph /\ M <_ N ) -> (♯ ` ( M ... N ) ) = ( ( N - M ) + 1 ) )
30	7	zcnd 9603	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> N e. CC )
31	21	adantr 276	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> M e. CC )
32		1cnd 8195	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> 1 e. CC )
33	30, 31, 32	subadd23d 8512	. . . . . . . 8 |- ( ( ph /\ M <_ N ) -> ( ( N - M ) + 1 ) = ( N + ( 1 - M ) ) )
34	29, 33	eqtr2d 2265	. . . . . . 7 |- ( ( ph /\ M <_ N ) -> ( N + ( 1 - M ) ) = (♯ ` ( M ... N ) ) )
35	27, 34	oveq12d 6036	. . . . . 6 |- ( ( ph /\ M <_ N ) -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... (♯ ` ( M ... N ) ) ) )
36		eqidd 2232	. . . . . 6 |- ( ( ph /\ M <_ N ) -> ( M ... N ) = ( M ... N ) )
37	26, 35, 36	f1oeq123d 5577	. . . . 5 |- ( ( ph /\ M <_ N ) -> ( ( j e. ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) : ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) -1-1-onto-> ( M ... N ) <-> ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) : ( 1 ... (♯ ` ( M ... N ) ) ) -1-1-onto-> ( M ... N ) ) )
38	20, 37	mpbid 147	. . . 4 |- ( ( ph /\ M <_ N ) -> ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) : ( 1 ... (♯ ` ( M ... N ) ) ) -1-1-onto-> ( M ... N ) )
39	1, 3, 12, 16, 38	gfsumval 16701	. . 3 |- ( ( ph /\ M <_ N ) -> ( G gfsumgf F ) = ( G gsumg ( F o. ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) ) ) )
40	14, 39	eqtr4d 2267	. 2 |- ( ( ph /\ M <_ N ) -> ( G gsumg F ) = ( G gfsumgf F ) )
41	2	adantr 276	. . . 4 |- ( ( ph /\ -. M <_ N ) -> G e. CMnd )
42		gfsum0 16703	. . . 4 |- ( G e. CMnd -> ( G gfsumgf (/) ) = ( 0g ` G ) )
43	41, 42	syl 14	. . 3 |- ( ( ph /\ -. M <_ N ) -> ( G gfsumgf (/) ) = ( 0g ` G ) )
44	11	adantr 276	. . . . . 6 |- ( ( ph /\ -. M <_ N ) -> F : ( M ... N ) --> B )
45		simpr 110	. . . . . . . . 9 |- ( ( ph /\ -. M <_ N ) -> -. M <_ N )
46	6	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ -. M <_ N ) -> N e. ZZ )
47	4	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ -. M <_ N ) -> M e. ZZ )
48		zltnle 9525	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N < M <-> -. M <_ N ) )
49	46, 47, 48	syl2anc 411	. . . . . . . . 9 |- ( ( ph /\ -. M <_ N ) -> ( N < M <-> -. M <_ N ) )
50	45, 49	mpbird 167	. . . . . . . 8 |- ( ( ph /\ -. M <_ N ) -> N < M )
51		fzn 10277	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> ( M ... N ) = (/) ) )
52	47, 46, 51	syl2anc 411	. . . . . . . 8 |- ( ( ph /\ -. M <_ N ) -> ( N < M <-> ( M ... N ) = (/) ) )
53	50, 52	mpbid 147	. . . . . . 7 |- ( ( ph /\ -. M <_ N ) -> ( M ... N ) = (/) )
54	53	feq2d 5470	. . . . . 6 |- ( ( ph /\ -. M <_ N ) -> ( F : ( M ... N ) --> B <-> F : (/) --> B ) )
55	44, 54	mpbid 147	. . . . 5 |- ( ( ph /\ -. M <_ N ) -> F : (/) --> B )
56		f0bi 5529	. . . . 5 |- ( F : (/) --> B <-> F = (/) )
57	55, 56	sylib 122	. . . 4 |- ( ( ph /\ -. M <_ N ) -> F = (/) )
58	57	oveq2d 6034	. . 3 |- ( ( ph /\ -. M <_ N ) -> ( G gfsumgf F ) = ( G gfsumgf (/) ) )
59	57	oveq2d 6034	. . . 4 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( G gsumg (/) ) )
60		eqid 2231	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
61	60	gsum0g 13481	. . . . 5 |- ( G e. CMnd -> ( G gsumg (/) ) = ( 0g ` G ) )
62	41, 61	syl 14	. . . 4 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg (/) ) = ( 0g ` G ) )
63	59, 62	eqtrd 2264	. . 3 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( 0g ` G ) )
64	43, 58, 63	3eqtr4rd 2275	. 2 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( G gfsumgf F ) )
65		zdcle 9556	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M <_ N )
66	4, 6, 65	syl2anc 411	. . 3 |- ( ph -> DECID M <_ N )
67		exmiddc 843	. . 3 |- (DECID M <_ N -> ( M <_ N \/ -. M <_ N ) )
68	66, 67	syl 14	. 2 |- ( ph -> ( M <_ N \/ -. M <_ N ) )
69	40, 64, 68	mpjaodan 805	1 |- ( ph -> ( G gsumg F ) = ( G gfsumgf F ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   (/)c0 3494   class class class wbr 4088   |-> cmpt 4150   o. ccom 4729   -->wf 5322   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6018   Fincfn 6909   CCcc 8030   1c1 8033   + caddc 8035   < clt 8214   <_ cle 8215   - cmin 8350   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243  ♯chash 11038   Basecbs 13084   0gc0g 13341   gsum_g cgsu 13342  CMndccmn 13873   gfsum_gf cgfsu 16699

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-dom 6911  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-ihash 11039  df-ndx 13087  df-slot 13088  df-base 13090  df-plusg 13175  df-0g 13343  df-igsum 13344  df-mgm 13441  df-sgrp 13487  df-mnd 13502  df-cmn 13875  df-gfsum 16700

This theorem is referenced by: (None)