Theorem fngsum 12971

Description: Iterated sum has a universal domain. (Contributed by Jim Kingdon, 28-Jun-2025.)

Assertion

Ref	Expression
fngsum	|- gsumg Fn ( _V X. _V )

Proof of Theorem fngsum

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

Step	Hyp	Ref	Expression
1		df-igsum 12870	. 2 |- gsumg = ( w e. _V , f e. _V |-> ( iota x ( ( dom f = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) ) )
2		unab 3426	. . . 4 |- ( { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } u. { x | E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) } ) = { x | ( ( dom f = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) }
3		df-sn 3624	. . . . . . 7 |- { ( 0g ` w ) } = { x | x = ( 0g ` w ) }
4		fn0g 12958	. . . . . . . . 9 |- 0g Fn _V
5		vex 2763	. . . . . . . . 9 |- w e. _V
6		funfvex 5571	. . . . . . . . . 10 |- ( ( Fun 0g /\ w e. dom 0g ) -> ( 0g ` w ) e. _V )
7	6	funfni 5354	. . . . . . . . 9 |- ( ( 0g Fn _V /\ w e. _V ) -> ( 0g ` w ) e. _V )
8	4, 5, 7	mp2an 426	. . . . . . . 8 |- ( 0g ` w ) e. _V
9	8	snex 4214	. . . . . . 7 |- { ( 0g ` w ) } e. _V
10	3, 9	eqeltrri 2267	. . . . . 6 |- { x | x = ( 0g ` w ) } e. _V
11		simpr 110	. . . . . . 7 |- ( ( dom f = (/) /\ x = ( 0g ` w ) ) -> x = ( 0g ` w ) )
12	11	ss2abi 3251	. . . . . 6 |- { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } C_ { x | x = ( 0g ` w ) }
13	10, 12	ssexi 4167	. . . . 5 |- { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } e. _V
14		zex 9326	. . . . . . 7 |- ZZ e. _V
15	14, 14	ab2rexex 6183	. . . . . 6 |- { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) } e. _V
16		df-rex 2478	. . . . . . . . . . . 12 |- ( E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) <-> E. n ( n e. ( ZZ>= ` m ) /\ ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) )
17		eluzel2 9597	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
18		eluzelz 9601	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> n e. ZZ )
19	17, 18	jca 306	. . . . . . . . . . . . . . 15 |- ( n e. ( ZZ>= ` m ) -> ( m e. ZZ /\ n e. ZZ ) )
20		simpr 110	. . . . . . . . . . . . . . 15 |- ( ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) -> x = ( seq m ( ( +g ` w ) , f ) ` n ) )
21	19, 20	anim12i 338	. . . . . . . . . . . . . 14 |- ( ( n e. ( ZZ>= ` m ) /\ ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) -> ( ( m e. ZZ /\ n e. ZZ ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) )
22		anass 401	. . . . . . . . . . . . . 14 |- ( ( ( m e. ZZ /\ n e. ZZ ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) <-> ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) )
23	21, 22	sylib 122	. . . . . . . . . . . . 13 |- ( ( n e. ( ZZ>= ` m ) /\ ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) -> ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) )
24	23	eximi 1611	. . . . . . . . . . . 12 |- ( E. n ( n e. ( ZZ>= ` m ) /\ ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) -> E. n ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) )
25	16, 24	sylbi 121	. . . . . . . . . . 11 |- ( E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) -> E. n ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) )
26		19.42v 1918	. . . . . . . . . . 11 |- ( E. n ( m e. ZZ /\ ( n e. ZZ /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) <-> ( m e. ZZ /\ E. n ( n e. ZZ /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) )
27	25, 26	sylib 122	. . . . . . . . . 10 |- ( E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) -> ( m e. ZZ /\ E. n ( n e. ZZ /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) )
28		df-rex 2478	. . . . . . . . . . 11 |- ( E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) <-> E. n ( n e. ZZ /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) )
29	28	anbi2i 457	. . . . . . . . . 10 |- ( ( m e. ZZ /\ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) <-> ( m e. ZZ /\ E. n ( n e. ZZ /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) )
30	27, 29	sylibr 134	. . . . . . . . 9 |- ( E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) -> ( m e. ZZ /\ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) )
31	30	eximi 1611	. . . . . . . 8 |- ( E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) -> E. m ( m e. ZZ /\ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) )
32		df-rex 2478	. . . . . . . 8 |- ( E. m e. ZZ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) <-> E. m ( m e. ZZ /\ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) )
33	31, 32	sylibr 134	. . . . . . 7 |- ( E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) -> E. m e. ZZ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) )
34	33	ss2abi 3251	. . . . . 6 |- { x | E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) } C_ { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) }
35	15, 34	ssexi 4167	. . . . 5 |- { x | E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) } e. _V
36	13, 35	unex 4472	. . . 4 |- ( { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } u. { x | E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) } ) e. _V
37	2, 36	eqeltrri 2267	. . 3 |- { x | ( ( dom f = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) } e. _V
38		iotaexab 5233	. . 3 |- ( { x | ( ( dom f = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) } e. _V -> ( iota x ( ( dom f = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) ) e. _V )
39	37, 38	ax-mp 5	. 2 |- ( iota x ( ( dom f = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) ) e. _V
40	1, 39	fnmpoi 6257	1 |- gsumg Fn ( _V X. _V )

Syntax hints:   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   {cab 2179   E.wrex 2473   _Vcvv 2760   u. cun 3151   (/)c0 3446   {csn 3618   X. cxp 4657   dom cdm 4659   iotacio 5213   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518   +g cplusg 12695   0gc0g 12867   gsum_g cgsu 12868

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-neg 8193  df-inn 8983  df-z 9318  df-uz 9593  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869  df-igsum 12870

This theorem is referenced by: (None)