Theorem fngsum 13436

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 13307	. 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 3471	. . . 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 3672	. . . . . . 7 |- { ( 0g ` w ) } = { x | x = ( 0g ` w ) }
4		fn0g 13423	. . . . . . . . 9 |- 0g Fn _V
5		vex 2802	. . . . . . . . 9 |- w e. _V
6		funfvex 5646	. . . . . . . . . 10 |- ( ( Fun 0g /\ w e. dom 0g ) -> ( 0g ` w ) e. _V )
7	6	funfni 5423	. . . . . . . . 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 4269	. . . . . . 7 |- { ( 0g ` w ) } e. _V
10	3, 9	eqeltrri 2303	. . . . . 6 |- { x | x = ( 0g ` w ) } e. _V
11		simpr 110	. . . . . . 7 |- ( ( dom f = (/) /\ x = ( 0g ` w ) ) -> x = ( 0g ` w ) )
12	11	ss2abi 3296	. . . . . 6 |- { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } C_ { x | x = ( 0g ` w ) }
13	10, 12	ssexi 4222	. . . . 5 |- { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } e. _V
14		zex 9466	. . . . . . 7 |- ZZ e. _V
15	14, 14	ab2rexex 6282	. . . . . 6 |- { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) } e. _V
16		df-rex 2514	. . . . . . . . . . . 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 9738	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
18		eluzelz 9743	. . . . . . . . . . . . . . . 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 1646	. . . . . . . . . . . 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 1953	. . . . . . . . . . 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 2514	. . . . . . . . . . 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 1646	. . . . . . . 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 2514	. . . . . . . 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 3296	. . . . . 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 4222	. . . . 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 4532	. . . 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 2303	. . 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 5297	. . 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 6355	1 |- gsumg Fn ( _V X. _V )

Syntax hints:   /\ wa 104   \/ wo 713   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   E.wrex 2509   _Vcvv 2799   u. cun 3195   (/)c0 3491   {csn 3666   X. cxp 4717   dom cdm 4719   iotacio 5276   Fn wfn 5313   ` cfv 5318  (class class class)co 6007   ZZcz 9457   ZZ>=cuz 9733   ...cfz 10216   seqcseq 10681   +g cplusg 13125   0gc0g 13304   gsum_g cgsu 13305

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 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 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  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-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-neg 8331  df-inn 9122  df-z 9458  df-uz 9734  df-ndx 13050  df-slot 13051  df-base 13053  df-0g 13306  df-igsum 13307

This theorem is referenced by: (None)