Theorem gzsumfzval 13688

Description: An expression for gzsum_gz 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 )
gzsumfzval.m	|- ( ph -> M e. ZZ )
gzsumfzval.n	|- ( ph -> N e. ZZ )
gzsumfzval.f	|- ( ph -> F : ( M ... N ) --> B )

Assertion

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

Proof of Theorem gzsumfzval

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		gzsumfzval.m	. . . 4 |- ( ph -> M e. ZZ )
6		gzsumfzval.n	. . . 4 |- ( ph -> N e. ZZ )
7	5, 6	fzfigd 10846	. . 3 |- ( ph -> ( M ... N ) e. Fin )
8		gzsumfzval.f	. . 3 |- ( ph -> F : ( M ... N ) --> B )
9	1, 2, 3, 4, 7, 8	gzsumval 13687	. 2 |- ( ph -> ( G gzsumgz 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 2835	. . . . . 6 |- ( ph -> G e. _V )
12		funfvex 5707	. . . . . . 7 |- ( ( Fun 0g /\ G e. dom 0g ) -> ( 0g ` G ) e. _V )
13	12	funfni 5478	. . . . . 6 |- ( ( 0g Fn _V /\ G e. _V ) -> ( 0g ` G ) e. _V )
14	10, 11, 13	sylancr 418	. . . . 5 |- ( ph -> ( 0g ` G ) e. _V )
15	2, 14	eqeltrid 2325	. . . 4 |- ( ph -> .0. e. _V )
16		seqex 10864	. . . . 5 |- seq M ( .+ , F ) e. _V
17		fvexg 5709	. . . . 5 |- ( ( seq M ( .+ , F ) e. _V /\ N e. ZZ ) -> ( seq M ( .+ , F ) ` N ) e. _V )
18	16, 6, 17	sylancr 418	. . . 4 |- ( ph -> ( seq M ( .+ , F ) ` N ) e. _V )
19	15, 18	ifexd 4625	. . 3 |- ( ph -> if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) e. _V )
20		zdclt 9701	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. ZZ ) -> DECID N < M )
21	6, 5, 20	syl2anc 415	. . . . . . 7 |- ( ph -> DECID N < M )
22		eqifdc 3674	. . . . . . 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 10425	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> ( M ... N ) = (/) ) )
25	5, 6, 24	syl2anc 415	. . . . . . . 8 |- ( ph -> ( N < M <-> ( M ... N ) = (/) ) )
26	25	anbi1d 469	. . . . . . 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		oveq2 6083	. . . . . . . . . . . . 13 |- ( n = N -> ( M ... n ) = ( M ... N ) )
29	28	eqeq2d 2250	. . . . . . . . . . . 12 |- ( n = N -> ( ( M ... N ) = ( M ... n ) <-> ( M ... N ) = ( M ... N ) ) )
30		fveq2 5690	. . . . . . . . . . . . 13 |- ( n = N -> ( seq M ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
31	30	eqeq2d 2250	. . . . . . . . . . . 12 |- ( n = N -> ( x = ( seq M ( .+ , F ) ` n ) <-> x = ( seq M ( .+ , F ) ` N ) ) )
32	29, 31	anbi12d 477	. . . . . . . . . . 11 |- ( n = N -> ( ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) ) )
33	27	zred 9747	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> M e. RR )
34	6	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> N e. ZZ )
35	34	zred 9747	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> N e. RR )
36		simprl 535	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> -. N < M )
37	33, 35, 36	nltled 8437	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> M <_ N )
38		eluz 9914	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) <-> M <_ N ) )
39	27, 34, 38	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> ( N e. ( ZZ>= ` M ) <-> M <_ N ) )
40	37, 39	mpbird 167	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> N e. ( ZZ>= ` M ) )
41		eqidd 2239	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> ( M ... N ) = ( M ... N ) )
42		simprr 537	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> x = ( seq M ( .+ , F ) ` N ) )
43	41, 42	jca 306	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> ( ( M ... N ) = ( M ... N ) /\ x = ( seq M ( .+ , F ) ` N ) ) )
44	32, 40, 43	rspcedvdw 2936	. . . . . . . . . 10 |- ( ( ph /\ ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) -> E. n e. ( ZZ>= ` M ) ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) )
45		fveq2 5690	. . . . . . . . . . . 12 |- ( m = M -> ( ZZ>= ` m ) = ( ZZ>= ` M ) )
46		oveq1 6082	. . . . . . . . . . . . . 14 |- ( m = M -> ( m ... n ) = ( M ... n ) )
47	46	eqeq2d 2250	. . . . . . . . . . . . 13 |- ( m = M -> ( ( M ... N ) = ( m ... n ) <-> ( M ... N ) = ( M ... n ) ) )
48		seqeq1 10865	. . . . . . . . . . . . . . 15 |- ( m = M -> seq m ( .+ , F ) = seq M ( .+ , F ) )
49	48	fveq1d 5692	. . . . . . . . . . . . . 14 |- ( m = M -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` n ) )
50	49	eqeq2d 2250	. . . . . . . . . . . . 13 |- ( m = M -> ( x = ( seq m ( .+ , F ) ` n ) <-> x = ( seq M ( .+ , F ) ` n ) ) )
51	47, 50	anbi12d 477	. . . . . . . . . . . 12 |- ( m = M -> ( ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) <-> ( ( M ... N ) = ( M ... n ) /\ x = ( seq M ( .+ , F ) ` n ) ) ) )
52	45, 51	rexeqbidv 2766	. . . . . . . . . . 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 ) ) ) )
53	52	spcegv 2913	. . . . . . . . . 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 ) ) ) )
54	27, 44, 53	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 ) ) )
55	54	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 ) ) ) )
56		eluzel2 9905	. . . . . . . . . . . . . . 15 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
57	56	ad2antlr 493	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m e. ZZ )
58	57	zred 9747	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m e. RR )
59		eluzelre 9911	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> n e. RR )
60	59	ad2antlr 493	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> n e. RR )
61		eluzle 9913	. . . . . . . . . . . . . 14 |- ( n e. ( ZZ>= ` m ) -> m <_ n )
62	61	ad2antlr 493	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m <_ n )
63	58, 60, 62	lensymd 8438	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> -. n < m )
64		simprl 535	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( M ... N ) = ( m ... n ) )
65	64	eqcomd 2244	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( m ... n ) = ( M ... N ) )
66		fzopth 10445	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> ( ( m ... n ) = ( M ... N ) <-> ( m = M /\ n = N ) ) )
67	66	ad2antlr 493	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( ( m ... n ) = ( M ... N ) <-> ( m = M /\ n = N ) ) )
68	65, 67	mpbid 147	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( m = M /\ n = N ) )
69	68	simprd 114	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> n = N )
70	68	simpld 112	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> m = M )
71	69, 70	breq12d 4138	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( n < m <-> N < M ) )
72	63, 71	mtbid 683	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> -. N < M )
73		simprr 537	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq m ( .+ , F ) ` n ) )
74	70	seqeq1d 10868	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> seq m ( .+ , F ) = seq M ( .+ , F ) )
75	74, 69	fveq12d 5697	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( seq m ( .+ , F ) ` n ) = ( seq M ( .+ , F ) ` N ) )
76	73, 75	eqtrd 2271	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> x = ( seq M ( .+ , F ) ` N ) )
77	72, 76	jca 306	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( ZZ>= ` m ) ) /\ ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) ) -> ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) )
78	77	rexlimdva2 2671	. . . . . . . . 9 |- ( ph -> ( E. n e. ( ZZ>= ` m ) ( ( M ... N ) = ( m ... n ) /\ x = ( seq m ( .+ , F ) ` n ) ) -> ( -. N < M /\ x = ( seq M ( .+ , F ) ` N ) ) ) )
79	78	exlimdv 1872	. . . . . . . 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 ) ) ) )
80	55, 79	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 ) ) ) )
81	26, 80	orbi12d 805	. . . . . 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 ) ) ) ) )
82	23, 81	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 ) ) ) )
83	82	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 ) ) ) )
84	83	iota5 5354	. . 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 ) ) )
85	19, 84	mpdan 425	. 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 ) ) )
86	9, 85	eqtrd 2271	1 |- ( ph -> ( G gzsumgz F ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   _Vcvv 2821   (/)c0 3520   ifcif 3635   class class class wbr 4125   iotacio 5330   Fn wfn 5367   -->wf 5368   ` cfv 5372  (class class class)co 6075   Fincfn 7012   RRcr 8168   < clt 8350   <_ cle 8351   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   Basecbs 13330   +g cplusg 13408   0gc0g 13587   gzsum_gz cgzsu 13588

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-0g 13589  df-gzsum 13590

This theorem is referenced by:  gzsumcl  13781  gzsumreidx  14118  gzsumsubmcl  14119  gzsummhm  14122