Theorem gsumfzval 13688

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 10817	. . 3 |- ( ph -> ( M ... N ) e. Fin )
8		gsumfzval.f	. . 3 |- ( ph -> F : ( M ... N ) --> B )
9	1, 2, 3, 4, 7, 8	igsumval 13687	. 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 13672	. . . . . 6 |- 0g Fn _V
11	4	elexd 2829	. . . . . 6 |- ( ph -> G e. _V )
12		funfvex 5692	. . . . . . 7 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
13	12	funfni 5463	. . . . . 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 2321	. . . 4 |- ( ph -> .0. e. _V )
16		seqex 10835	. . . . 5 |- seq M ( .+ , F ) e. _V
17		fvexg 5694	. . . . 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 4610	. . 3 |- ( ph -> if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) e. _V )
20		zdclt 9672	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
21	6, 5, 20	syl2anc 411	. . . . . . 7 |- ( ph -> DECID N < M )
22		eqifdc 3663	. . . . . . 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 10396	. . . . . . . . 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 9718	. . . . . . . . . . . . 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 9718	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> N e. RR )
31		simprl 531	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> -. N < M )
32	28, 30, 31	nltled 8410	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> M <_ N )
33		eluz 9885	. . . . . . . . . . . . 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 6066	. . . . . . . . . . . . . 14 |- ( n = N -> ( M ... n ) = ( M ... N ) )
37	36	eqeq2d 2246	. . . . . . . . . . . . 13 |- ( n = N -> ( ( M ... N ) = ( M ... n ) <-> ( M ... N ) = ( M ... N ) ) )
38		fveq2 5675	. . . . . . . . . . . . . 14 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
39	38	eqeq2d 2246	. . . . . . . . . . . . 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 2235	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> ( M ... N ) = ( M ... N ) )
43		simprr 533	. . . . . . . . . . . 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 2929	. . . . . . . . . 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 5675	. . . . . . . . . . . 12 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
47		oveq1 6065	. . . . . . . . . . . . . 14 |- ( m = M -> ( m ... n ) = ( M ... n ) )
48	47	eqeq2d 2246	. . . . . . . . . . . . 13 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
49		seqeq1 10836	. . . . . . . . . . . . . . 15 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
50	49	fveq1d 5677	. . . . . . . . . . . . . 14 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
51	50	eqeq2d 2246	. . . . . . . . . . . . 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 2760	. . . . . . . . . . 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 2907	. . . . . . . . . 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 9876	. . . . . . . . . . . . . . 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 9718	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m e. RR )
60		eluzelre 9882	. . . . . . . . . . . . . 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 9884	. . . . . . . . . . . . . 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 8411	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> -. n < m )
65		simprl 531	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( M ... N ) = ( m ... n ) )
66	65	eqcomd 2240	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( m ... n ) = ( M ... N ) )
67		fzopth 10416	. . . . . . . . . . . . . . . 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 4127	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( n < m <-> N < M ) )
73	64, 72	mtbid 679	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> -. N < M )
74		simprr 533	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq m ( .+ , F ) ` n ) )
75	71	seqeq1d 10839	. . . . . . . . . . . . 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 5682	. . . . . . . . . . . 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 2267	. . . . . . . . . . 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 2665	. . . . . . . . 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 1868	. . . . . . . 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 801	. . . . . 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 5339	. . 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 2267	1 |- ( ph -> ( G gsumg F ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   _Vcvv 2815   (/)c0 3512   ifcif 3624   class class class wbr 4114   iotacio 5315   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   Fincfn 6988   RRcr 8142   < clt 8324   <_ cle 8325   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833   Basecbs 13296   +g cplusg 13374   0gc0g 13553   gsum_g cgsu 13554

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 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  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 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-0g 13555  df-igsum 13556

This theorem is referenced by:  gsumfzz  13792  gsumfzcl  13796  gsumfzreidx  14138  gsumfzsubmcl  14139  gsumfzmptfidmadd  14140  gsumfzmhm  14144