Theorem gsumgfsum 16857

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 9858	. . . . 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 2232	. . . 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 16856	. . 3 |- ( ( ph /\ M <_ N ) -> ( G gsumg F ) = ( G gsumg ( F o. ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) ) ) )
15	4, 6	fzfigd 10792	. . . . 5 |- ( ph -> ( M ... N ) e. Fin )
16	15	adantr 276	. . . 4 |- ( ( ph /\ M <_ N ) -> ( M ... N ) e. Fin )
17		1zzd 9603	. . . . . . . 8 |- ( ph -> 1 e. ZZ )
18	17, 4	zsubcld 9704	. . . . . . 7 |- ( ph -> ( 1 - M ) e. ZZ )
19	18, 4, 6	mptfzshft 12124	. . . . . 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 9700	. . . . . . . . . 10 |- ( ph -> M e. CC )
22		1cnd 8289	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
23	21, 22	pncan3d 8586	. . . . . . . . 9 |- ( ph -> ( M + ( 1 - M ) ) = 1 )
24	23	oveq1d 6064	. . . . . . . 8 |- ( ph -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... ( N + ( 1 - M ) ) ) )
25	24	mpteq1d 4194	. . . . . . 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 11184	. . . . . . . . 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 9700	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> N e. CC )
31	21	adantr 276	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> M e. CC )
32		1cnd 8289	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> 1 e. CC )
33	30, 31, 32	subadd23d 8605	. . . . . . . 8 |- ( ( ph /\ M <_ N ) -> ( ( N - M ) + 1 ) = ( N + ( 1 - M ) ) )
34	29, 33	eqtr2d 2266	. . . . . . 7 |- ( ( ph /\ M <_ N ) -> ( N + ( 1 - M ) ) = (♯ ` ( M ... N ) ) )
35	27, 34	oveq12d 6067	. . . . . 6 |- ( ( ph /\ M <_ N ) -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... (♯ ` ( M ... N ) ) ) )
36		eqidd 2233	. . . . . 6 |- ( ( ph /\ M <_ N ) -> ( M ... N ) = ( M ... N ) )
37	26, 35, 36	f1oeq123d 5607	. . . . 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 16853	. . 3 |- ( ( ph /\ M <_ N ) -> ( G gfsumgf F ) = ( G gsumg ( F o. ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) ) ) )
40	14, 39	eqtr4d 2268	. 2 |- ( ( ph /\ M <_ N ) -> ( G gsumg F ) = ( G gfsumgf F ) )
41	2	adantr 276	. . . 4 |- ( ( ph /\ -. M <_ N ) -> G e. CMnd )
42		gfsum0 16855	. . . 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 9622	. . . . . . . . . 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 10375	. . . . . . . . 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 5495	. . . . . 6 |- ( ( ph /\ -. M <_ N ) -> ( F : ( M ... N ) --> B <-> F : (/) --> B ) )
55	44, 54	mpbid 147	. . . . 5 |- ( ( ph /\ -. M <_ N ) -> F : (/) --> B )
56		f0bi 5559	. . . . 5 |- ( F : (/) --> B <-> F = (/) )
57	55, 56	sylib 122	. . . 4 |- ( ( ph /\ -. M <_ N ) -> F = (/) )
58	57	oveq2d 6065	. . 3 |- ( ( ph /\ -. M <_ N ) -> ( G gfsumgf F ) = ( G gfsumgf (/) ) )
59	57	oveq2d 6065	. . . 4 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( G gsumg (/) ) )
60		eqid 2232	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
61	60	gsum0g 13601	. . . . 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 2265	. . 3 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( 0g ` G ) )
64	43, 58, 63	3eqtr4rd 2276	. 2 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( G gfsumgf F ) )
65		zdcle 9653	. . . 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 2203   (/)c0 3507   class class class wbr 4108   |-> cmpt 4170   o. ccom 4752   -->wf 5347   -1-1-onto->wf1o 5350   ` cfv 5351  (class class class)co 6049   Fincfn 6974   CCcc 8124   1c1 8127   + caddc 8129   < clt 8307   <_ cle 8308   - cmin 8443   ZZcz 9576   ZZ>=cuz 9852   ...cfz 10341  ♯chash 11136   Basecbs 13204   0gc0g 13461   gsum_g cgsu 13462  CMndccmn 13993   gfsum_gf cgfsu 16851

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-inn 9237  df-2 9295  df-n0 9496  df-z 9577  df-uz 9853  df-fz 10342  df-fzo 10476  df-seqfrec 10809  df-ihash 11137  df-ndx 13207  df-slot 13208  df-base 13210  df-plusg 13295  df-0g 13463  df-igsum 13464  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-cmn 13995  df-gfsum 16852

This theorem is referenced by: (None)