Theorem clim2ser2 11620

Description: The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)

Hypotheses

Ref	Expression
clim2ser.1	|- Z = ( ZZ>= ` M )
clim2ser.2	|- ( ph -> N e. Z )
clim2ser.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
clim2ser2.5	|- ( ph -> seq ( N + 1 ) ( + , F ) ~~> A )

Assertion

Ref	Expression
clim2ser2	|- ( ph -> seq M ( + , F ) ~~> ( A + ( seq M ( + , F ) ` N ) ) )

Distinct variable groups:   A, k   k, F   k, M   k, N   ph, k   k, Z

Proof of Theorem clim2ser2

Dummy variables j x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2204	. 2 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
2		clim2ser.2	. . . . 5 |- ( ph -> N e. Z )
3		clim2ser.1	. . . . 5 |- Z = ( ZZ>= ` M )
4	2, 3	eleqtrdi 2297	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		peano2uz 9703	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
6	4, 5	syl 14	. . 3 |- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) )
7		eluzelz 9656	. . 3 |- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( N + 1 ) e. ZZ )
8	6, 7	syl 14	. 2 |- ( ph -> ( N + 1 ) e. ZZ )
9		clim2ser2.5	. 2 |- ( ph -> seq ( N + 1 ) ( + , F ) ~~> A )
10		eluzel2 9652	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
11	4, 10	syl 14	. . . 4 |- ( ph -> M e. ZZ )
12		clim2ser.4	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
13	3, 11, 12	serf 10626	. . 3 |- ( ph -> seq M ( + , F ) : Z --> CC )
14	13, 2	ffvelcdmd 5715	. 2 |- ( ph -> ( seq M ( + , F ) ` N ) e. CC )
15		seqex 10592	. . 3 |- seq M ( + , F ) e. _V
16	15	a1i 9	. 2 |- ( ph -> seq M ( + , F ) e. _V )
17	6, 3	eleqtrrdi 2298	. . . . . 6 |- ( ph -> ( N + 1 ) e. Z )
18	3	uztrn2 9665	. . . . . 6 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
19	17, 18	sylan 283	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
20	19, 12	syldan 282	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> ( F ` k ) e. CC )
21	1, 8, 20	serf 10626	. . 3 |- ( ph -> seq ( N + 1 ) ( + , F ) : ( ZZ>= ` ( N + 1 ) ) --> CC )
22	21	ffvelcdmda 5714	. 2 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq ( N + 1 ) ( + , F ) ` j ) e. CC )
23	14	adantr 276	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` N ) e. CC )
24		addcl 8049	. . . . 5 |- ( ( k e. CC /\ x e. CC ) -> ( k + x ) e. CC )
25	24	adantl 277	. . . 4 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k + x ) e. CC )
26		addass 8054	. . . . 5 |- ( ( k e. CC /\ x e. CC /\ y e. CC ) -> ( ( k + x ) + y ) = ( k + ( x + y ) ) )
27	26	adantl 277	. . . 4 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ ( k e. CC /\ x e. CC /\ y e. CC ) ) -> ( ( k + x ) + y ) = ( k + ( x + y ) ) )
28		simpr 110	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> j e. ( ZZ>= ` ( N + 1 ) ) )
29	4	adantr 276	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> N e. ( ZZ>= ` M ) )
30	3	eleq2i 2271	. . . . . 6 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
31	30, 12	sylan2br 288	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
32	31	adantlr 477	. . . 4 |- ( ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
33	25, 27, 28, 29, 32	seq3split 10631	. . 3 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) = ( ( seq M ( + , F ) ` N ) + ( seq ( N + 1 ) ( + , F ) ` j ) ) )
34	23, 22, 33	comraddd 8228	. 2 |- ( ( ph /\ j e. ( ZZ>= ` ( N + 1 ) ) ) -> ( seq M ( + , F ) ` j ) = ( ( seq ( N + 1 ) ( + , F ) ` j ) + ( seq M ( + , F ) ` N ) ) )
35	1, 8, 9, 14, 16, 22, 34	climaddc1 11611	1 |- ( ph -> seq M ( + , F ) ~~> ( A + ( seq M ( + , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1372   e. wcel 2175   _Vcvv 2771   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   CCcc 7922   1c1 7925   + caddc 7927   ZZcz 9371   ZZ>=cuz 9647   seqcseq 10590   ~~> cli 11560

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-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-rp 9775  df-fz 10130  df-seqfrec 10591  df-exp 10682  df-cj 11124  df-re 11125  df-im 11126  df-rsqrt 11280  df-abs 11281  df-clim 11561

This theorem is referenced by:  iserex  11621