Theorem gsummptnn0fz 30811

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 1673	. . . . 5 |- F/ x ( S < k -> C = .0. )
3		nfv 1673	. . . . . 6 |- F/ k S < x
4		nfcsb1v 3309	. . . . . . 7 |- F/_ k [_ x / k ]_ C
5	4	nfeq1 2593	. . . . . 6 |- F/ k [_ x / k ]_ C = .0.
6	3, 5	nfim 1853	. . . . 5 |- F/ k ( S < x -> [_ x / k ]_ C = .0. )
7		breq2 4301	. . . . . 6 |- ( k = x -> ( S < k <-> S < x ) )
8		csbeq1a 3302	. . . . . . 7 |- ( k = x -> C = [_ x / k ]_ C )
9	8	eqeq1d 2451	. . . . . 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 2948	. . . 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 3710	. . . . . . . . . . 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 2443	. . . . . . . . 9 |- ( k e. NN0 |-> C ) = ( k e. NN0 |-> C )
22	21	fvmpts 5781	. . . . . . . 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 2475	. . . . . 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 2799	. . 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 5869	. . . . 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 5706	. . . . . . 7 |- ( Base ` G ) e. _V
36	30, 35	eqeltri 2513	. . . . . 6 |- B e. _V
37		nn0ex 10590	. . . . . 6 |- NN0 e. _V
38	36, 37	pm3.2i 455	. . . . 5 |- ( B e. _V /\ NN0 e. _V )
39		elmapg 7232	. . . . 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 11486	. . . . . 6 |- ( k e. ( 0 ... S ) -> k e. NN0 )
44	43	ssriv 3365	. . . . 5 |- ( 0 ... S ) C_ NN0
45		resmpt 5161	. . . . 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 2447	. . 3 |- ( k e. ( 0 ... S ) |-> C ) = ( ( k e. NN0 |-> C ) |` ( 0 ... S ) )
48	30, 31, 32, 41, 42, 47	fsfnn0gsumfsffz 30808	. 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 1369   F/wnf 1589   e. wcel 1756   A.wral 2720   _Vcvv 2977   [_csb 3293   C_ wss 3333   class class class wbr 4297   e. cmpt 4355   |` cres 4847   -->wf 5419   ` cfv 5423  (class class class)co 6096   ^m cmap 7219   0cc0 9287   < clt 9423   NN0cn0 10584   ...cfz 11442   Basecbs 14179   0gc0g 14383   gsum_g cgsu 14384  CMndccmn 16282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-0g 14385  df-gsum 14386  df-mnd 15420  df-cntz 15840  df-cmn 16284

This theorem is referenced by:  gsummptnn0fzv  30812  gsummoncoe1  30848