Theorem fngzsum 13685

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

Assertion

Ref	Expression
fngzsum	|- gzsumgz Fn ( _V X. _V )

Proof of Theorem fngzsum

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

Step	Hyp	Ref	Expression
1		df-gzsum 13590	. 2 |- gzsumgz = ( 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 3498	. . . 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 3711	. . . . . . 7 |- { ( 0g ` w ) } = { x | x = ( 0g ` w ) }
4		fn0g 13672	. . . . . . . . 9 |- 0g Fn _V
5		vex 2824	. . . . . . . . 9 |- w e. _V
6		funfvex 5707	. . . . . . . . . 10 |- ( ( Fun 0g /\ w e. dom 0g ) -> ( 0g ` w ) e. _V )
7	6	funfni 5478	. . . . . . . . 9 |- ( ( 0g Fn _V /\ w e. _V ) -> ( 0g ` w ) e. _V )
8	4, 5, 7	mp2an 430	. . . . . . . 8 |- ( 0g ` w ) e. _V
9	8	snex 4317	. . . . . . 7 |- { ( 0g ` w ) } e. _V
10	3, 9	eqeltrri 2312	. . . . . 6 |- { x | x = ( 0g ` w ) } e. _V
11		simpr 110	. . . . . . 7 |- ( ( dom f = (/) /\ x = ( 0g ` w ) ) -> x = ( 0g ` w ) )
12	11	ss2abi 3320	. . . . . 6 |- { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } C_ { x | x = ( 0g ` w ) }
13	10, 12	ssexi 4266	. . . . 5 |- { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } e. _V
14		zex 9632	. . . . . . 7 |- ZZ e. _V
15	14, 14	ab2rexex 6354	. . . . . 6 |- { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) } e. _V
16		df-rex 2534	. . . . . . . . . . . 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 9905	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
18		eluzelz 9910	. . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . 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 1653	. . . . . . . . . . . 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 1962	. . . . . . . . . . 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 2534	. . . . . . . . . . 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 461	. . . . . . . . . 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 1653	. . . . . . . 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 2534	. . . . . . . 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 3320	. . . . . 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 4266	. . . . 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 4582	. . . 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 2312	. . 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 5351	. . 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 6429	1 |- gzsumgz Fn ( _V X. _V )

Syntax hints:   /\ wa 104   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   E.wrex 2529   _Vcvv 2821   u. cun 3218   (/)c0 3520   {csn 3705   X. cxp 4767   dom cdm 4769   iotacio 5330   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   seqcseq 10862   +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-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-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  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-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  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-id 4433  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-neg 8490  df-inn 9284  df-z 9624  df-uz 9901  df-ndx 13333  df-slot 13334  df-base 13336  df-0g 13589  df-gzsum 13590

This theorem is referenced by:  gsumvalfi  14129