Theorem fngsum 13220

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 13091	. 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 3440	. . . 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 3639	. . . . . . 7 |- { ( 0g ` w ) } = { x | x = ( 0g ` w ) }
4		fn0g 13207	. . . . . . . . 9 |- 0g Fn _V
5		vex 2775	. . . . . . . . 9 |- w e. _V
6		funfvex 5593	. . . . . . . . . 10 |- ( ( Fun 0g /\ w e. dom 0g ) -> ( 0g ` w ) e. _V )
7	6	funfni 5376	. . . . . . . . 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 4229	. . . . . . 7 |- { ( 0g ` w ) } e. _V
10	3, 9	eqeltrri 2279	. . . . . 6 |- { x | x = ( 0g ` w ) } e. _V
11		simpr 110	. . . . . . 7 |- ( ( dom f = (/) /\ x = ( 0g ` w ) ) -> x = ( 0g ` w ) )
12	11	ss2abi 3265	. . . . . 6 |- { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } C_ { x | x = ( 0g ` w ) }
13	10, 12	ssexi 4182	. . . . 5 |- { x | ( dom f = (/) /\ x = ( 0g ` w ) ) } e. _V
14		zex 9381	. . . . . . 7 |- ZZ e. _V
15	14, 14	ab2rexex 6216	. . . . . 6 |- { x | E. m e. ZZ E. n e. ZZ x = ( seq m ( ( +g ` w ) , f ) ` n ) } e. _V
16		df-rex 2490	. . . . . . . . . . . 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 9653	. . . . . . . . . . . . . . . 16 |- ( n e. ( ZZ>= ` m ) -> m e. ZZ )
18		eluzelz 9657	. . . . . . . . . . . . . . . 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 1623	. . . . . . . . . . . 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 1930	. . . . . . . . . . 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 2490	. . . . . . . . . . 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 1623	. . . . . . . 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 2490	. . . . . . . 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 3265	. . . . . 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 4182	. . . . 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 4488	. . . 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 2279	. . 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 5250	. . 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 6289	1 |- gsumg Fn ( _V X. _V )

Syntax hints:   /\ wa 104   \/ wo 710   = wceq 1373   E.wex 1515   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772   u. cun 3164   (/)c0 3460   {csn 3633   X. cxp 4673   dom cdm 4675   iotacio 5230   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   seqcseq 10592   +g cplusg 12909   0gc0g 13088   gsum_g cgsu 13089

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-neg 8246  df-inn 9037  df-z 9373  df-uz 9649  df-ndx 12835  df-slot 12836  df-base 12838  df-0g 13090  df-igsum 13091

This theorem is referenced by: (None)