Theorem sge0p1 40631

Description: The addition of the next term in a finite sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0p1.1	|- ( ph -> N e. ( ZZ>= ` M ) )
sge0p1.2	|- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. ( 0 [,] +oo ) )
sge0p1.3	|- ( k = ( N + 1 ) -> A = B )

Assertion

Ref	Expression
sge0p1	|- ( ph -> (Σ^ ` ( k e. ( M ... ( N + 1 ) ) |-> A ) ) = ( (Σ^ ` ( k e. ( M ... N ) |-> A ) ) +e B ) )

Distinct variable groups:   B, k   k, M   k, N   ph, k

Allowed substitution hint:   A( k)

Proof of Theorem sge0p1

Step	Hyp	Ref	Expression
1		sge0p1.1	. . . . 5 |- ( ph -> N e. ( ZZ>= ` M ) )
2		fzsuc 12388	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
3	1, 2	syl 17	. . . 4 |- ( ph -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
4	3	mpteq1d 4738	. . 3 |- ( ph -> ( k e. ( M ... ( N + 1 ) ) |-> A ) = ( k e. ( ( M ... N ) u. { ( N + 1 ) } ) |-> A ) )
5	4	fveq2d 6195	. 2 |- ( ph -> (Σ^ ` ( k e. ( M ... ( N + 1 ) ) |-> A ) ) = (Σ^ ` ( k e. ( ( M ... N ) u. { ( N + 1 ) } ) |-> A ) ) )
6		nfv 1843	. . 3 |- F/ k ph
7		ovex 6678	. . . 4 |- ( M ... N ) e. _V
8	7	a1i 11	. . 3 |- ( ph -> ( M ... N ) e. _V )
9		snex 4908	. . . 4 |- { ( N + 1 ) } e. _V
10	9	a1i 11	. . 3 |- ( ph -> { ( N + 1 ) } e. _V )
11		fzp1disj 12399	. . . 4 |- ( ( M ... N ) i^i { ( N + 1 ) } ) = (/)
12	11	a1i 11	. . 3 |- ( ph -> ( ( M ... N ) i^i { ( N + 1 ) } ) = (/) )
13		0xr 10086	. . . . 5 |- 0 e. RR*
14	13	a1i 11	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 e. RR* )
15		pnfxr 10092	. . . . 5 |- +oo e. RR*
16	15	a1i 11	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> +oo e. RR* )
17		iccssxr 12256	. . . . 5 |- ( 0 [,] +oo ) C_ RR*
18		simpl 473	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ph )
19		fzelp1 12393	. . . . . . 7 |- ( k e. ( M ... N ) -> k e. ( M ... ( N + 1 ) ) )
20	19	adantl 482	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> k e. ( M ... ( N + 1 ) ) )
21		sge0p1.2	. . . . . 6 |- ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. ( 0 [,] +oo ) )
22	18, 20, 21	syl2anc 693	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. ( 0 [,] +oo ) )
23	17, 22	sseldi 3601	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. RR* )
24		iccgelb 12230	. . . . 5 |- ( ( 0 e. RR* /\ +oo e. RR* /\ A e. ( 0 [,] +oo ) ) -> 0 <_ A )
25	14, 16, 22, 24	syl3anc 1326	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ A )
26		iccleub 12229	. . . . 5 |- ( ( 0 e. RR* /\ +oo e. RR* /\ A e. ( 0 [,] +oo ) ) -> A <_ +oo )
27	14, 16, 22, 26	syl3anc 1326	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> A <_ +oo )
28	14, 16, 23, 25, 27	eliccxrd 39753	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. ( 0 [,] +oo ) )
29		simpl 473	. . . 4 |- ( ( ph /\ k e. { ( N + 1 ) } ) -> ph )
30		elsni 4194	. . . . . 6 |- ( k e. { ( N + 1 ) } -> k = ( N + 1 ) )
31	30	adantl 482	. . . . 5 |- ( ( ph /\ k e. { ( N + 1 ) } ) -> k = ( N + 1 ) )
32		simpr 477	. . . . . 6 |- ( ( ph /\ k = ( N + 1 ) ) -> k = ( N + 1 ) )
33		peano2uz 11741	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
34		eluzfz2 12349	. . . . . . . 8 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( M ... ( N + 1 ) ) )
35	1, 33, 34	3syl 18	. . . . . . 7 |- ( ph -> ( N + 1 ) e. ( M ... ( N + 1 ) ) )
36	35	adantr 481	. . . . . 6 |- ( ( ph /\ k = ( N + 1 ) ) -> ( N + 1 ) e. ( M ... ( N + 1 ) ) )
37	32, 36	eqeltrd 2701	. . . . 5 |- ( ( ph /\ k = ( N + 1 ) ) -> k e. ( M ... ( N + 1 ) ) )
38	29, 31, 37	syl2anc 693	. . . 4 |- ( ( ph /\ k e. { ( N + 1 ) } ) -> k e. ( M ... ( N + 1 ) ) )
39	29, 38, 21	syl2anc 693	. . 3 |- ( ( ph /\ k e. { ( N + 1 ) } ) -> A e. ( 0 [,] +oo ) )
40	6, 8, 10, 12, 28, 39	sge0splitmpt 40628	. 2 |- ( ph -> (Σ^ ` ( k e. ( ( M ... N ) u. { ( N + 1 ) } ) |-> A ) ) = ( (Σ^ ` ( k e. ( M ... N ) |-> A ) ) +e (Σ^ ` ( k e. { ( N + 1 ) } |-> A ) ) ) )
41		ovex 6678	. . . . 5 |- ( N + 1 ) e. _V
42	41	a1i 11	. . . 4 |- ( ph -> ( N + 1 ) e. _V )
43		id 22	. . . . 5 |- ( ph -> ph )
44		eleq1 2689	. . . . . . . . 9 |- ( k = ( N + 1 ) -> ( k e. ( M ... ( N + 1 ) ) <-> ( N + 1 ) e. ( M ... ( N + 1 ) ) ) )
45	44	anbi2d 740	. . . . . . . 8 |- ( k = ( N + 1 ) -> ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) <-> ( ph /\ ( N + 1 ) e. ( M ... ( N + 1 ) ) ) ) )
46		sge0p1.3	. . . . . . . . 9 |- ( k = ( N + 1 ) -> A = B )
47	46	eleq1d 2686	. . . . . . . 8 |- ( k = ( N + 1 ) -> ( A e. ( 0 [,] +oo ) <-> B e. ( 0 [,] +oo ) ) )
48	45, 47	imbi12d 334	. . . . . . 7 |- ( k = ( N + 1 ) -> ( ( ( ph /\ k e. ( M ... ( N + 1 ) ) ) -> A e. ( 0 [,] +oo ) ) <-> ( ( ph /\ ( N + 1 ) e. ( M ... ( N + 1 ) ) ) -> B e. ( 0 [,] +oo ) ) ) )
49	48, 21	vtoclg 3266	. . . . . 6 |- ( ( N + 1 ) e. _V -> ( ( ph /\ ( N + 1 ) e. ( M ... ( N + 1 ) ) ) -> B e. ( 0 [,] +oo ) ) )
50	41, 49	ax-mp 5	. . . . 5 |- ( ( ph /\ ( N + 1 ) e. ( M ... ( N + 1 ) ) ) -> B e. ( 0 [,] +oo ) )
51	43, 35, 50	syl2anc 693	. . . 4 |- ( ph -> B e. ( 0 [,] +oo ) )
52	42, 51, 46	sge0snmpt 40600	. . 3 |- ( ph -> (Σ^ ` ( k e. { ( N + 1 ) } |-> A ) ) = B )
53	52	oveq2d 6666	. 2 |- ( ph -> ( (Σ^ ` ( k e. ( M ... N ) |-> A ) ) +e (Σ^ ` ( k e. { ( N + 1 ) } |-> A ) ) ) = ( (Σ^ ` ( k e. ( M ... N ) |-> A ) ) +e B ) )
54	5, 40, 53	3eqtrd 2660	1 |- ( ph -> (Σ^ ` ( k e. ( M ... ( N + 1 ) ) |-> A ) ) = ( (Σ^ ` ( k e. ( M ... N ) |-> A ) ) +e B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   ZZ>=cuz 11687   +ecxad 11944   [,]cicc 12178   ...cfz 12326  Σ^{^}csumge0 40579

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-inf2 8538  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  ax-pre-sup 10014

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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-sumge0 40580

This theorem is referenced by:  caratheodorylem1  40740