Theorem gsummptnn0fz 18382

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 1843	. . . . 5 |- F/ x ( S < k -> C = .0. )
3		nfv 1843	. . . . . 6 |- F/ k S < x
4		nfcsb1v 3549	. . . . . . 7 |- F/_ k [_ x / k ]_ C
5	4	nfeq1 2778	. . . . . 6 |- F/ k [_ x / k ]_ C = .0.
6	3, 5	nfim 1825	. . . . 5 |- F/ k ( S < x -> [_ x / k ]_ C = .0. )
7		breq2 4657	. . . . . 6 |- ( k = x -> ( S < k <-> S < x ) )
8		csbeq1a 3542	. . . . . . 7 |- ( k = x -> C = [_ x / k ]_ C )
9	8	eqeq1d 2624	. . . . . 6 |- ( k = x -> ( C = .0. <-> [_ x / k ]_ C = .0. ) )
10	7, 9	imbi12d 334	. . . . 5 |- ( k = x -> ( ( S < k -> C = .0. ) <-> ( S < x -> [_ x / k ]_ C = .0. ) ) )
11	2, 6, 10	cbvral 3167	. . . 4 |- ( A. k e. NN0 ( S < k -> C = .0. ) <-> A. x e. NN0 ( S < x -> [_ x / k ]_ C = .0. ) )
12	1, 11	sylib 208	. . 3 |- ( ph -> A. x e. NN0 ( S < x -> [_ x / k ]_ C = .0. ) )
13		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. NN0 ) -> x e. NN0 )
14		gsummptnn0fz.f	. . . . . . . . . . . . 13 |- ( ph -> A. k e. NN0 C e. B )
15	14	anim2i 593	. . . . . . . . . . . 12 |- ( ( x e. NN0 /\ ph ) -> ( x e. NN0 /\ A. k e. NN0 C e. B ) )
16	15	ancoms 469	. . . . . . . . . . 11 |- ( ( ph /\ x e. NN0 ) -> ( x e. NN0 /\ A. k e. NN0 C e. B ) )
17		rspcsbela 4006	. . . . . . . . . . 11 |- ( ( x e. NN0 /\ A. k e. NN0 C e. B ) -> [_ x / k ]_ C e. B )
18	16, 17	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x e. NN0 ) -> [_ x / k ]_ C e. B )
19	13, 18	jca 554	. . . . . . . . 9 |- ( ( ph /\ x e. NN0 ) -> ( x e. NN0 /\ [_ x / k ]_ C e. B ) )
20	19	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ x e. NN0 ) /\ [_ x / k ]_ C = .0. ) -> ( x e. NN0 /\ [_ x / k ]_ C e. B ) )
21		eqid 2622	. . . . . . . . 9 |- ( k e. NN0 |-> C ) = ( k e. NN0 |-> C )
22	21	fvmpts 6285	. . . . . . . 8 |- ( ( x e. NN0 /\ [_ x / k ]_ C e. B ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
23	20, 22	syl 17	. . . . . . 7 |- ( ( ( ph /\ x e. NN0 ) /\ [_ x / k ]_ C = .0. ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
24		simpr 477	. . . . . . 7 |- ( ( ( ph /\ x e. NN0 ) /\ [_ x / k ]_ C = .0. ) -> [_ x / k ]_ C = .0. )
25	23, 24	eqtrd 2656	. . . . . 6 |- ( ( ( ph /\ x e. NN0 ) /\ [_ x / k ]_ C = .0. ) -> ( ( k e. NN0 |-> C ) ` x ) = .0. )
26	25	ex 450	. . . . 5 |- ( ( ph /\ x e. NN0 ) -> ( [_ x / k ]_ C = .0. -> ( ( k e. NN0 |-> C ) ` x ) = .0. ) )
27	26	imim2d 57	. . . 4 |- ( ( ph /\ x e. NN0 ) -> ( ( S < x -> [_ x / k ]_ C = .0. ) -> ( S < x -> ( ( k e. NN0 |-> C ) ` x ) = .0. ) ) )
28	27	ralimdva 2962	. . 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 6381	. . . . 5 |- ( A. k e. NN0 C e. B <-> ( k e. NN0 |-> C ) : NN0 --> B )
34	14, 33	sylib 208	. . . 4 |- ( ph -> ( k e. NN0 |-> C ) : NN0 --> B )
35		fvex 6201	. . . . . . 7 |- ( Base ` G ) e. _V
36	30, 35	eqeltri 2697	. . . . . 6 |- B e. _V
37		nn0ex 11298	. . . . . 6 |- NN0 e. _V
38	36, 37	pm3.2i 471	. . . . 5 |- ( B e. _V /\ NN0 e. _V )
39		elmapg 7870	. . . . 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 13	. . . 4 |- ( ph -> ( ( k e. NN0 |-> C ) e. ( B ^m NN0 ) <-> ( k e. NN0 |-> C ) : NN0 --> B ) )
41	34, 40	mpbird 247	. . 3 |- ( ph -> ( k e. NN0 |-> C ) e. ( B ^m NN0 ) )
42		gsummptnn0fz.s	. . 3 |- ( ph -> S e. NN0 )
43		fz0ssnn0 12435	. . . . 5 |- ( 0 ... S ) C_ NN0
44		resmpt 5449	. . . . 5 |- ( ( 0 ... S ) C_ NN0 -> ( ( k e. NN0 |-> C ) |` ( 0 ... S ) ) = ( k e. ( 0 ... S ) |-> C ) )
45	43, 44	ax-mp 5	. . . 4 |- ( ( k e. NN0 |-> C ) |` ( 0 ... S ) ) = ( k e. ( 0 ... S ) |-> C )
46	45	eqcomi 2631	. . 3 |- ( k e. ( 0 ... S ) |-> C ) = ( ( k e. NN0 |-> C ) |` ( 0 ... S ) )
47	30, 31, 32, 41, 42, 46	fsfnn0gsumfsffz 18379	. 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 ) ) ) )
48	29, 47	mpd 15	1 |- ( ph -> ( G gsumg ( k e. NN0 |-> C ) ) = ( G gsumg ( k e. ( 0 ... S ) |-> C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   _Vcvv 3200   [_csb 3533   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   < clt 10074   NN0cn0 11292   ...cfz 12326   Basecbs 15857   0gc0g 16100   gsum_g cgsu 16101  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  gsummptnn0fzv  18383  gsummoncoe1  19674  pmatcollpwfi  20587