Theorem gsumgfsum 16790

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 9804	. . . . 5 |- ( N e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ N e. ZZ /\ M <_ N ) )
10	5, 7, 8, 9	syl3anbrc 1208	. . . 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 16789	. . 3 |- ( ( ph /\ M <_ N ) -> ( G gsumg F ) = ( G gsumg ( F o. ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) ) ) )
15	4, 6	fzfigd 10737	. . . . 5 |- ( ph -> ( M ... N ) e. Fin )
16	15	adantr 276	. . . 4 |- ( ( ph /\ M <_ N ) -> ( M ... N ) e. Fin )
17		1zzd 9549	. . . . . . . 8 |- ( ph -> 1 e. ZZ )
18	17, 4	zsubcld 9650	. . . . . . 7 |- ( ph -> ( 1 - M ) e. ZZ )
19	18, 4, 6	mptfzshft 12064	. . . . . 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 9646	. . . . . . . . . 10 |- ( ph -> M e. CC )
22		1cnd 8238	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
23	21, 22	pncan3d 8536	. . . . . . . . 9 |- ( ph -> ( M + ( 1 - M ) ) = 1 )
24	23	oveq1d 6043	. . . . . . . 8 |- ( ph -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... ( N + ( 1 - M ) ) ) )
25	24	mpteq1d 4179	. . . . . . 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 11129	. . . . . . . . 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 9646	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> N e. CC )
31	21	adantr 276	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> M e. CC )
32		1cnd 8238	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> 1 e. CC )
33	30, 31, 32	subadd23d 8555	. . . . . . . 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 6046	. . . . . 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 5586	. . . . 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 16786	. . 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 16788	. . . 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 9568	. . . . . . . . . 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 10320	. . . . . . . . 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 5477	. . . . . 6 |- ( ( ph /\ -. M <_ N ) -> ( F : ( M ... N ) --> B <-> F : (/) --> B ) )
55	44, 54	mpbid 147	. . . . 5 |- ( ( ph /\ -. M <_ N ) -> F : (/) --> B )
56		f0bi 5538	. . . . 5 |- ( F : (/) --> B <-> F = (/) )
57	55, 56	sylib 122	. . . 4 |- ( ( ph /\ -. M <_ N ) -> F = (/) )
58	57	oveq2d 6044	. . 3 |- ( ( ph /\ -. M <_ N ) -> ( G gfsumgf F ) = ( G gfsumgf (/) ) )
59	57	oveq2d 6044	. . . 4 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( G gsumg (/) ) )
60		eqid 2231	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
61	60	gsum0g 13540	. . . . 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 9599	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M <_ N )
66	4, 6, 65	syl2anc 411	. . 3 |- ( ph -> DECID M <_ N )
67		exmiddc 844	. . 3 |- (DECID M <_ N -> ( M <_ N \/ -. M <_ N ) )
68	66, 67	syl 14	. 2 |- ( ph -> ( M <_ N \/ -. M <_ N ) )
69	40, 64, 68	mpjaodan 806	1 |- ( ph -> ( G gsumg F ) = ( G gfsumgf F ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2202   (/)c0 3496   class class class wbr 4093   |-> cmpt 4155   o. ccom 4735   -->wf 5329   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   Fincfn 6952   CCcc 8073   1c1 8076   + caddc 8078   < clt 8257   <_ cle 8258   - cmin 8393   ZZcz 9522   ZZ>=cuz 9798   ...cfz 10286  ♯chash 11081   Basecbs 13143   0gc0g 13400   gsum_g cgsu 13401  CMndccmn 13932   gfsum_gf cgfsu 16784

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-inn 9187  df-2 9245  df-n0 9446  df-z 9523  df-uz 9799  df-fz 10287  df-fzo 10421  df-seqfrec 10754  df-ihash 11082  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-0g 13402  df-igsum 13403  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-cmn 13934  df-gfsum 16785

This theorem is referenced by: (None)