Theorem gsumgfsum 14111

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 9881	. . . . 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 2234	. . . 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	gsumshift 14110	. . 3 |- ( ( ph /\ M <_ N ) -> ( G gsumg F ) = ( G gsumg ( F o. ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) ) ) )
15	4, 6	fzfigd 10821	. . . . 5 |- ( ph -> ( M ... N ) e. Fin )
16	15	adantr 276	. . . 4 |- ( ( ph /\ M <_ N ) -> ( M ... N ) e. Fin )
17		1zzd 9625	. . . . . . . 8 |- ( ph -> 1 e. ZZ )
18	17, 4	zsubcld 9727	. . . . . . 7 |- ( ph -> ( 1 - M ) e. ZZ )
19	18, 4, 6	mptfzshft 12158	. . . . . 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 9723	. . . . . . . . . 10 |- ( ph -> M e. CC )
22		1cnd 8307	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
23	21, 22	pncan3d 8605	. . . . . . . . 9 |- ( ph -> ( M + ( 1 - M ) ) = 1 )
24	23	oveq1d 6074	. . . . . . . 8 |- ( ph -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... ( N + ( 1 - M ) ) ) )
25	24	mpteq1d 4201	. . . . . . 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 11215	. . . . . . . . 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 9723	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> N e. CC )
31	21	adantr 276	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> M e. CC )
32		1cnd 8307	. . . . . . . . 9 |- ( ( ph /\ M <_ N ) -> 1 e. CC )
33	30, 31, 32	subadd23d 8624	. . . . . . . 8 |- ( ( ph /\ M <_ N ) -> ( ( N - M ) + 1 ) = ( N + ( 1 - M ) ) )
34	29, 33	eqtr2d 2268	. . . . . . 7 |- ( ( ph /\ M <_ N ) -> ( N + ( 1 - M ) ) = (♯ ` ( M ... N ) ) )
35	27, 34	oveq12d 6077	. . . . . 6 |- ( ( ph /\ M <_ N ) -> ( ( M + ( 1 - M ) ) ... ( N + ( 1 - M ) ) ) = ( 1 ... (♯ ` ( M ... N ) ) ) )
36		eqidd 2235	. . . . . 6 |- ( ( ph /\ M <_ N ) -> ( M ... N ) = ( M ... N ) )
37	26, 35, 36	f1oeq123d 5614	. . . . 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 14107	. . 3 |- ( ( ph /\ M <_ N ) -> ( G gfsumgf F ) = ( G gsumg ( F o. ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) ) ) ) )
40	14, 39	eqtr4d 2270	. 2 |- ( ( ph /\ M <_ N ) -> ( G gsumg F ) = ( G gfsumgf F ) )
41	2	adantr 276	. . . 4 |- ( ( ph /\ -. M <_ N ) -> G e. CMnd )
42		gfsum0 14109	. . . 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 9644	. . . . . . . . . 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 10400	. . . . . . . . 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 5502	. . . . . 6 |- ( ( ph /\ -. M <_ N ) -> ( F : ( M ... N ) --> B <-> F : (/) --> B ) )
55	44, 54	mpbid 147	. . . . 5 |- ( ( ph /\ -. M <_ N ) -> F : (/) --> B )
56		f0bi 5566	. . . . 5 |- ( F : (/) --> B <-> F = (/) )
57	55, 56	sylib 122	. . . 4 |- ( ( ph /\ -. M <_ N ) -> F = (/) )
58	57	oveq2d 6075	. . 3 |- ( ( ph /\ -. M <_ N ) -> ( G gfsumgf F ) = ( G gfsumgf (/) ) )
59	57	oveq2d 6075	. . . 4 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( G gsumg (/) ) )
60		eqid 2234	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
61	60	gsum0g 13664	. . . . 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 2267	. . 3 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( 0g ` G ) )
64	43, 58, 63	3eqtr4rd 2278	. 2 |- ( ( ph /\ -. M <_ N ) -> ( G gsumg F ) = ( G gfsumgf F ) )
65		zdcle 9675	. . . 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 2205   (/)c0 3512   class class class wbr 4115   |-> cmpt 4177   o. ccom 4759   -->wf 5354   -1-1-onto->wf1o 5357   ` cfv 5358  (class class class)co 6059   Fincfn 6989   CCcc 8142   1c1 8145   + caddc 8147   < clt 8325   <_ cle 8326   - cmin 8462   ZZcz 9598   ZZ>=cuz 9875   ...cfz 10365  ♯chash 11167   Basecbs 13301   0gc0g 13558   gsum_g cgsu 13559  CMndccmn 14042   gfsum_gf cgfsu 14105

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-mulrcl 8243  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-precex 8254  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-apti 8259  ax-pre-ltadd 8260  ax-pre-mulgt0 8261

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-frec 6636  df-1o 6661  df-er 6781  df-en 6990  df-dom 6991  df-fin 6992  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-reap 8868  df-ap 8875  df-inn 9259  df-2 9317  df-n0 9518  df-z 9599  df-uz 9876  df-fz 10366  df-fzo 10503  df-seqfrec 10838  df-ihash 11168  df-ndx 13304  df-slot 13305  df-base 13307  df-plusg 13392  df-0g 13560  df-igsum 13561  df-mgm 13624  df-sgrp 13670  df-mnd 13683  df-cmn 14044  df-gfsum 14106

This theorem is referenced by: (None)