Theorem gsummptnn0fz 17141

Description: A final group sum over a function over the nonnegative integers (given as mapping) is equal to a final group sum over a finite interval of nonnegative integers. (Contributed by AV, 10-Oct-2019.)

Hypotheses

Ref	Expression
gsummptnn0fz.k	|- F/ k ph
gsummptnn0fz.b	|- B = ( Base ` G )
gsummptnn0fz.0	|- .0. = ( 0g ` G )
gsummptnn0fz.g	|- ( ph -> G e. CMnd )
gsummptnn0fz.f	|- ( ph -> A. k e. NN0 C e. B )
gsummptnn0fz.s	|- ( ph -> S e. NN0 )
gsummptnn0fz.u	|- ( ph -> A. k e. NN0 ( S < k -> C = .0. ) )

Assertion

Ref	Expression
gsummptnn0fz	|- ( ph -> ( G gsumg ( k e. NN0 |-> C ) ) = ( G gsumg ( k e. ( 0 ... S ) |-> C ) ) )

Distinct variable groups:   B, k   S, k   .0. , k

Allowed substitution hints:   ph( k)   C( k)   G( k)

Proof of Theorem gsummptnn0fz

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsummptnn0fz.u	. . . 4 |- ( ph -> A. k e. NN0 ( S < k -> C = .0. ) )
2		nfv 1708	. . . . 5 |- F/ x ( S < k -> C = .0. )
3		nfv 1708	. . . . . 6 |- F/ k S < x
4		nfcsb1v 3446	. . . . . . 7 |- F/_ k [_ x / k ]_ C
5	4	nfeq1 2634	. . . . . 6 |- F/ k [_ x / k ]_ C = .0.
6	3, 5	nfim 1921	. . . . 5 |- F/ k ( S < x -> [_ x / k ]_ C = .0. )
7		breq2 4460	. . . . . 6 |- ( k = x -> ( S < k <-> S < x ) )
8		csbeq1a 3439	. . . . . . 7 |- ( k = x -> C = [_ x / k ]_ C )
9	8	eqeq1d 2459	. . . . . 6 |- ( k = x -> ( C = .0. <-> [_ x / k ]_ C = .0. ) )
10	7, 9	imbi12d 320	. . . . 5 |- ( k = x -> ( ( S < k -> C = .0. ) <-> ( S < x -> [_ x / k ]_ C = .0. ) ) )
11	2, 6, 10	cbvral 3080	. . . 4 |- ( A. k e. NN0 ( S < k -> C = .0. ) <-> A. x e. NN0 ( S < x -> [_ x / k ]_ C = .0. ) )
12	1, 11	sylib 196	. . 3 |- ( ph -> A. x e. NN0 ( S < x -> [_ x / k ]_ C = .0. ) )
13		simpr 461	. . . . . . . . . 10 |- ( ( ph /\ x e. NN0 ) -> x e. NN0 )
14		gsummptnn0fz.f	. . . . . . . . . . . . 13 |- ( ph -> A. k e. NN0 C e. B )
15	14	anim2i 569	. . . . . . . . . . . 12 |- ( ( x e. NN0 /\ ph ) -> ( x e. NN0 /\ A. k e. NN0 C e. B ) )
16	15	ancoms 453	. . . . . . . . . . 11 |- ( ( ph /\ x e. NN0 ) -> ( x e. NN0 /\ A. k e. NN0 C e. B ) )
17		rspcsbela 3858	. . . . . . . . . . 11 |- ( ( x e. NN0 /\ A. k e. NN0 C e. B ) -> [_ x / k ]_ C e. B )
18	16, 17	syl 16	. . . . . . . . . 10 |- ( ( ph /\ x e. NN0 ) -> [_ x / k ]_ C e. B )
19	13, 18	jca 532	. . . . . . . . 9 |- ( ( ph /\ x e. NN0 ) -> ( x e. NN0 /\ [_ x / k ]_ C e. B ) )
20	19	adantr 465	. . . . . . . 8 |- ( ( ( ph /\ x e. NN0 ) /\ [_ x / k ]_ C = .0. ) -> ( x e. NN0 /\ [_ x / k ]_ C e. B ) )
21		eqid 2457	. . . . . . . . 9 |- ( k e. NN0 |-> C ) = ( k e. NN0 |-> C )
22	21	fvmpts 5958	. . . . . . . 8 |- ( ( x e. NN0 /\ [_ x / k ]_ C e. B ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
23	20, 22	syl 16	. . . . . . 7 |- ( ( ( ph /\ x e. NN0 ) /\ [_ x / k ]_ C = .0. ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
24		simpr 461	. . . . . . 7 |- ( ( ( ph /\ x e. NN0 ) /\ [_ x / k ]_ C = .0. ) -> [_ x / k ]_ C = .0. )
25	23, 24	eqtrd 2498	. . . . . 6 |- ( ( ( ph /\ x e. NN0 ) /\ [_ x / k ]_ C = .0. ) -> ( ( k e. NN0 |-> C ) ` x ) = .0. )
26	25	ex 434	. . . . 5 |- ( ( ph /\ x e. NN0 ) -> ( [_ x / k ]_ C = .0. -> ( ( k e. NN0 |-> C ) ` x ) = .0. ) )
27	26	imim2d 52	. . . 4 |- ( ( ph /\ x e. NN0 ) -> ( ( S < x -> [_ x / k ]_ C = .0. ) -> ( S < x -> ( ( k e. NN0 |-> C ) ` x ) = .0. ) ) )
28	27	ralimdva 2865	. . 3 |- ( ph -> ( A. x e. NN0 ( S < x -> [_ x / k ]_ C = .0. ) -> A. x e. NN0 ( S < x -> ( ( k e. NN0 |-> C ) ` x ) = .0. ) ) )
29	12, 28	mpd 15	. 2 |- ( ph -> A. x e. NN0 ( S < x -> ( ( k e. NN0 |-> C ) ` x ) = .0. ) )
30		gsummptnn0fz.b	. . 3 |- B = ( Base ` G )
31		gsummptnn0fz.0	. . 3 |- .0. = ( 0g ` G )
32		gsummptnn0fz.g	. . 3 |- ( ph -> G e. CMnd )
33	21	fmpt 6053	. . . . 5 |- ( A. k e. NN0 C e. B <-> ( k e. NN0 |-> C ) : NN0 --> B )
34	14, 33	sylib 196	. . . 4 |- ( ph -> ( k e. NN0 |-> C ) : NN0 --> B )
35		fvex 5882	. . . . . . 7 |- ( Base ` G ) e. _V
36	30, 35	eqeltri 2541	. . . . . 6 |- B e. _V
37		nn0ex 10822	. . . . . 6 |- NN0 e. _V
38	36, 37	pm3.2i 455	. . . . 5 |- ( B e. _V /\ NN0 e. _V )
39		elmapg 7451	. . . . 5 |- ( ( B e. _V /\ NN0 e. _V ) -> ( ( k e. NN0 |-> C ) e. ( B ^m NN0 ) <-> ( k e. NN0 |-> C ) : NN0 --> B ) )
40	38, 39	mp1i 12	. . . 4 |- ( ph -> ( ( k e. NN0 |-> C ) e. ( B ^m NN0 ) <-> ( k e. NN0 |-> C ) : NN0 --> B ) )
41	34, 40	mpbird 232	. . 3 |- ( ph -> ( k e. NN0 |-> C ) e. ( B ^m NN0 ) )
42		gsummptnn0fz.s	. . 3 |- ( ph -> S e. NN0 )
43		elfznn0 11797	. . . . . 6 |- ( k e. ( 0 ... S ) -> k e. NN0 )
44	43	ssriv 3503	. . . . 5 |- ( 0 ... S ) C_ NN0
45		resmpt 5333	. . . . 5 |- ( ( 0 ... S ) C_ NN0 -> ( ( k e. NN0 |-> C ) |` ( 0 ... S ) ) = ( k e. ( 0 ... S ) |-> C ) )
46	44, 45	ax-mp 5	. . . 4 |- ( ( k e. NN0 |-> C ) |` ( 0 ... S ) ) = ( k e. ( 0 ... S ) |-> C )
47	46	eqcomi 2470	. . 3 |- ( k e. ( 0 ... S ) |-> C ) = ( ( k e. NN0 |-> C ) |` ( 0 ... S ) )
48	30, 31, 32, 41, 42, 47	fsfnn0gsumfsffz 17138	. 2 |- ( ph -> ( A. x e. NN0 ( S < x -> ( ( k e. NN0 |-> C ) ` x ) = .0. ) -> ( G gsumg ( k e. NN0 |-> C ) ) = ( G gsumg ( k e. ( 0 ... S ) |-> C ) ) ) )
49	29, 48	mpd 15	1 |- ( ph -> ( G gsumg ( k e. NN0 |-> C ) ) = ( G gsumg ( k e. ( 0 ... S ) |-> C ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   F/wnf 1617   e. wcel 1819   A.wral 2807   _Vcvv 3109   [_csb 3430   C_ wss 3471   class class class wbr 4456   |-> cmpt 4515   |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296   ^m cmap 7438   0cc0 9509   < clt 9645   NN0cn0 10816   ...cfz 11697   Basecbs 14644   0gc0g 14857   gsum_g cgsu 14858  CMndccmn 16925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-0g 14859  df-gsum 14860  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-cntz 16482  df-cmn 16927

This theorem is referenced by:  gsummptnn0fzv  17142  gsummoncoe1  18473  pmatcollpwfi  19410