Theorem gsumfzval 13410

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 10640	. . 3 |- ( ph -> ( M ... N ) e. Fin )
8		gsumfzval.f	. . 3 |- ( ph -> F : ( M ... N ) --> B )
9	1, 2, 3, 4, 7, 8	igsumval 13409	. 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 13394	. . . . . 6 |- 0g Fn _V
11	4	elexd 2813	. . . . . 6 |- ( ph -> G e. _V )
12		funfvex 5640	. . . . . . 7 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
13	12	funfni 5419	. . . . . 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 2316	. . . 4 |- ( ph -> .0. e. _V )
16		seqex 10658	. . . . 5 |- seq M ( .+ , F ) e. _V
17		fvexg 5642	. . . . 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 4572	. . 3 |- ( ph -> if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) e. _V )
20		zdclt 9512	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
21	6, 5, 20	syl2anc 411	. . . . . . 7 |- ( ph -> DECID N < M )
22		eqifdc 3639	. . . . . . 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 10226	. . . . . . . . 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 9557	. . . . . . . . . . . . 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 9557	. . . . . . . . . . . . 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 8255	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> M <_ N )
33		eluz 9723	. . . . . . . . . . . . 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 6002	. . . . . . . . . . . . . 14 |- ( n = N -> ( M ... n ) = ( M ... N ) )
37	36	eqeq2d 2241	. . . . . . . . . . . . 13 |- ( n = N -> ( ( M ... N ) = ( M ... n ) <-> ( M ... N ) = ( M ... N ) ) )
38		fveq2 5623	. . . . . . . . . . . . . 14 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
39	38	eqeq2d 2241	. . . . . . . . . . . . 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 2230	. . . . . . . . . . . 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 2913	. . . . . . . . . 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 5623	. . . . . . . . . . . 12 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
47		oveq1 6001	. . . . . . . . . . . . . 14 |- ( m = M -> ( m ... n ) = ( M ... n ) )
48	47	eqeq2d 2241	. . . . . . . . . . . . 13 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
49		seqeq1 10659	. . . . . . . . . . . . . . 15 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
50	49	fveq1d 5625	. . . . . . . . . . . . . 14 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
51	50	eqeq2d 2241	. . . . . . . . . . . . 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 2745	. . . . . . . . . . 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 2891	. . . . . . . . . 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 9715	. . . . . . . . . . . . . . 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 9557	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m e. RR )
60		eluzelre 9720	. . . . . . . . . . . . . 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 9722	. . . . . . . . . . . . . 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 8256	. . . . . . . . . . . 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 2235	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( m ... n ) = ( M ... N ) )
67		fzopth 10245	. . . . . . . . . . . . . . . 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 4095	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( n < m <-> N < M ) )
73	64, 72	mtbid 676	. . . . . . . . . . 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 10662	. . . . . . . . . . . . 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 5630	. . . . . . . . . . . 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 2262	. . . . . . . . . . 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 2651	. . . . . . . . 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 1865	. . . . . . . 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 798	. . . . . 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 5296	. . 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 2262	1 |- ( ph -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   _Vcvv 2799   (/)c0 3491   ifcif 3602   class class class wbr 4082   iotacio 5272   Fn wfn 5309   -->wf 5310   ` cfv 5314  (class class class)co 5994   Fincfn 6877   RRcr 7986   < clt 8169   <_ cle 8170   ZZcz 9434   ZZ>=cuz 9710   ...cfz 10192   seqcseq 10656   Basecbs 13018   +g cplusg 13096   0gc0g 13275   gsum_g cgsu 13276

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-addass 8089  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-er 6670  df-en 6878  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-inn 9099  df-n0 9358  df-z 9435  df-uz 9711  df-fz 10193  df-seqfrec 10657  df-ndx 13021  df-slot 13022  df-base 13024  df-0g 13277  df-igsum 13278

This theorem is referenced by:  gsumfzz  13514  gsumfzcl  13518  gsumfzreidx  13860  gsumfzsubmcl  13861  gsumfzmptfidmadd  13862  gsumfzmhm  13866