Theorem gsumfzval 13010

Description: An expression for gsum_g when summing over a finite set of sequential integers. (Contributed by Jim Kingdon, 14-Aug-2025.)

Hypotheses

Ref	Expression
gsumval.b	|- B = ( Base ` G )
gsumval.z	|- .0. = ( 0g ` G )
gsumval.p	|- .+ = ( +g ` G )
gsumval.g	|- ( ph -> G e. V )
gsumfzval.m	|- ( ph -> M e. ZZ )
gsumfzval.n	|- ( ph -> N e. ZZ )
gsumfzval.f	|- ( ph -> F : ( M ... N ) --> B )

Assertion

Ref	Expression
gsumfzval	|- ( ph -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )

Proof of Theorem gsumfzval

Dummy variables m n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval.b	. . 3 |- B = ( Base ` G )
2		gsumval.z	. . 3 |- .0. = ( 0g ` G )
3		gsumval.p	. . 3 |- .+ = ( +g ` G )
4		gsumval.g	. . 3 |- ( ph -> G e. V )
5		gsumfzval.m	. . . 4 |- ( ph -> M e. ZZ )
6		gsumfzval.n	. . . 4 |- ( ph -> N e. ZZ )
7	5, 6	fzfigd 10508	. . 3 |- ( ph -> ( M ... N ) e. Fin )
8		gsumfzval.f	. . 3 |- ( ph -> F : ( M ... N ) --> B )
9	1, 2, 3, 4, 7, 8	igsumval 13009	. 2 |- ( ph -> ( G gsumg F ) = ( iota x ( ( ( M ... N ) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )
10		fn0g 12994	. . . . . 6 |- 0g Fn _V
11	4	elexd 2776	. . . . . 6 |- ( ph -> G e. _V )
12		funfvex 5575	. . . . . . 7 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
13	12	funfni 5358	. . . . . 6 |- ( ( 0g Fn _V /\ G e. _V ) -> ( 0g ` G ) e. _V )
14	10, 11, 13	sylancr 414	. . . . 5 |- ( ph -> ( 0g ` G ) e. _V )
15	2, 14	eqeltrid 2283	. . . 4 |- ( ph -> .0. e. _V )
16		seqex 10526	. . . . 5 |- seq M ( .+ , F ) e. _V
17		fvexg 5577	. . . . 5 |- ( ( seq M ( .+ , F ) e. _V /\ N e. ZZ ) -> ( seq M ( .+ , F ) ` N ) e. _V )
18	16, 6, 17	sylancr 414	. . . 4 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. _V )
19	15, 18	ifexd 4519	. . 3 |- ( ph -> if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) e. _V )
20		zdclt 9400	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
21	6, 5, 20	syl2anc 411	. . . . . . 7 |- ( ph -> DECID N < M )
22		eqifdc 3596	. . . . . . 7 |- (DECID N < M -> ( x = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) <-> ( ( N < M /\ x = .0. ) \/ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) ) )
23	21, 22	syl 14	. . . . . 6 |- ( ph -> ( x = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) <-> ( ( N < M /\ x = .0. ) \/ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) ) )
24		fzn 10114	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> ( M ... N ) = (/) ) )
25	5, 6, 24	syl2anc 411	. . . . . . . 8 |- ( ph -> ( N < M <-> ( M ... N ) = (/) ) )
26	25	anbi1d 465	. . . . . . 7 |- ( ph -> ( ( N < M /\ x = .0. ) <-> ( ( M ... N ) = (/) /\ x = .0. ) ) )
27	5	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> M e. ZZ )
28	27	zred 9445	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> M e. RR )
29	6	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> N e. ZZ )
30	29	zred 9445	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> N e. RR )
31		simprl 529	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> -. N < M )
32	28, 30, 31	nltled 8145	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> M <_ N )
33		eluz 9611	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) <-> M <_ N ) )
34	27, 29, 33	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> ( N e. ( ZZ>= ` M ) <-> M <_ N ) )
35	32, 34	mpbird 167	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> N e. ( ZZ>= ` M ) )
36		oveq2 5930	. . . . . . . . . . . . . 14 |- ( n = N -> ( M ... n ) = ( M ... N ) )
37	36	eqeq2d 2208	. . . . . . . . . . . . 13 |- ( n = N -> ( ( M ... N ) = ( M ... n ) <-> ( M ... N ) = ( M ... N ) ) )
38		fveq2 5558	. . . . . . . . . . . . . 14 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
39	38	eqeq2d 2208	. . . . . . . . . . . . 13 |- ( n = N -> ( x = ( seq M ( .+ , F ) ` n ) <-> x = ( seq M ( .+ , F ) ` N ) ) )
40	37, 39	anbi12d 473	. . . . . . . . . . . 12 |- ( n = N -> ( ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) ) )
41	40	adantl 277	. . . . . . . . . . 11 |- ( ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) /\ n = N ) -> ( ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) ) )
42		eqidd 2197	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> ( M ... N ) = ( M ... N ) )
43		simprr 531	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> x = ( seq M ( .+ , F ) ` N ) )
44	42, 43	jca 306	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) )
45	35, 41, 44	rspcedvd 2874	. . . . . . . . . 10 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) )
46		fveq2 5558	. . . . . . . . . . . 12 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
47		oveq1 5929	. . . . . . . . . . . . . 14 |- ( m = M -> ( m ... n ) = ( M ... n ) )
48	47	eqeq2d 2208	. . . . . . . . . . . . 13 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
49		seqeq1 10527	. . . . . . . . . . . . . . 15 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
50	49	fveq1d 5560	. . . . . . . . . . . . . 14 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
51	50	eqeq2d 2208	. . . . . . . . . . . . 13 |- ( m = M -> ( x = ( seq m ( .+ , F ) ` n ) <-> x = ( seq M ( .+ , F ) ` n ) ) )
52	48, 51	anbi12d 473	. . . . . . . . . . . 12 |- ( m = M -> ( ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) ) )
53	46, 52	rexeqbidv 2710	. . . . . . . . . . 11 |- ( m = M -> ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) ) )
54	53	spcegv 2852	. . . . . . . . . 10 |- ( M e. ZZ -> ( E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) -> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
55	27, 45, 54	sylc 62	. . . . . . . . 9 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) )
56	55	ex 115	. . . . . . . 8 |- ( ph -> ( ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) -> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
57		eluzel2 9603	. . . . . . . . . . . . . . 15 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
58	57	ad2antlr 489	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m e. ZZ )
59	58	zred 9445	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m e. RR )
60		eluzelre 9608	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> n e. RR )
61	60	ad2antlr 489	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> n e. RR )
62		eluzle 9610	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> m <_ n )
63	62	ad2antlr 489	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m <_ n )
64	59, 61, 63	lensymd 8146	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> -. n < m )
65		simprl 529	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( M ... N ) = ( m ... n ) )
66	65	eqcomd 2202	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( m ... n ) = ( M ... N ) )
67		fzopth 10133	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> ( ( m ... n ) = ( M ... N ) <-> ( m = M /\ n = N ) ) )
68	67	ad2antlr 489	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( ( m ... n ) = ( M ... N ) <-> ( m = M /\ n = N ) ) )
69	66, 68	mpbid 147	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( m = M /\ n = N ) )
70	69	simprd 114	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> n = N )
71	69	simpld 112	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m = M )
72	70, 71	breq12d 4046	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( n < m <-> N < M ) )
73	64, 72	mtbid 673	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> -. N < M )
74		simprr 531	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq m ( .+ , F ) ` n ) )
75	71	seqeq1d 10530	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> seq m ( .+ , F ) = seq M ( .+ , F ) )
76	75, 70	fveq12d 5565	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
77	74, 76	eqtrd 2229	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq M ( .+ , F ) ` N ) )
78	73, 77	jca 306	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) )
79	78	rexlimdva2 2617	. . . . . . . . 9 |- ( ph -> ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) )
80	79	exlimdv 1833	. . . . . . . 8 |- ( ph -> ( E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) )
81	56, 80	impbid 129	. . . . . . 7 |- ( ph -> ( ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) <-> E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) )
82	26, 81	orbi12d 794	. . . . . 6 |- ( ph -> ( ( ( N < M /\ x = .0. ) \/ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) <-> ( ( ( M ... N ) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) )
83	23, 82	bitr2d 189	. . . . 5 |- ( ph -> ( ( ( ( M ... N ) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) <-> x = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) ) )
84	83	adantr 276	. . . 4 |- ( ( ph /\ if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) e. _V ) -> ( ( ( ( M ... N ) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) <-> x = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) ) )
85	84	iota5 5240	. . 3 |- ( ( ph /\ if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) e. _V ) -> ( iota x ( ( ( M ... N ) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )
86	19, 85	mpdan 421	. 2 |- ( ph -> ( iota x ( ( ( M ... N ) = (/) /\ x = .0. ) \/ E. m E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )
87	9, 86	eqtrd 2229	1 |- ( ph -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   _Vcvv 2763   (/)c0 3450   ifcif 3561   class class class wbr 4033   iotacio 5217   Fn wfn 5253   -->wf 5254   ` cfv 5258  (class class class)co 5922   Fincfn 6799   RRcr 7876   < clt 8059   <_ cle 8060   ZZcz 9323   ZZ>=cuz 9598   ...cfz 10080   seqcseq 10524   Basecbs 12654   +g cplusg 12731   0gc0g 12903   gsum_g cgsu 12904

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7968  ax-resscn 7969  ax-1cn 7970  ax-1re 7971  ax-icn 7972  ax-addcl 7973  ax-addrcl 7974  ax-mulcl 7975  ax-addcom 7977  ax-addass 7979  ax-distr 7981  ax-i2m1 7982  ax-0lt1 7983  ax-0id 7985  ax-rnegex 7986  ax-cnre 7988  ax-pre-ltirr 7989  ax-pre-ltwlin 7990  ax-pre-lttrn 7991  ax-pre-apti 7992  ax-pre-ltadd 7993

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-er 6592  df-en 6800  df-fin 6802  df-pnf 8061  df-mnf 8062  df-xr 8063  df-ltxr 8064  df-le 8065  df-sub 8197  df-neg 8198  df-inn 8988  df-n0 9247  df-z 9324  df-uz 9599  df-fz 10081  df-seqfrec 10525  df-ndx 12657  df-slot 12658  df-base 12660  df-0g 12905  df-igsum 12906

This theorem is referenced by:  gsumfzz  13103  gsumfzcl  13107  gsumfzreidx  13443  gsumfzsubmcl  13444  gsumfzmptfidmadd  13445  gsumfzmhm  13449