Theorem iserile 10180

Description: Comparison of the limits of two infinite series. (Contributed by Jim Kingdon, 22-Aug-2021.)

Hypotheses

Ref	Expression
clim2ser.1	|- Z = ( ZZ>= ` M )
iserile.2	|- ( ph -> M e. ZZ )
iserile.4	|- ( ph -> seq M ( + , F , CC ) ~~> A )
iserile.5	|- ( ph -> seq M ( + , G , CC ) ~~> B )
iserile.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
iserile.7	|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
iserile.8	|- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) )

Assertion

Ref	Expression
iserile	|- ( ph -> A <_ B )

Distinct variable groups:   A, k   B, k   k, F   k, M   k, G   ph, k   k, Z

Proof of Theorem iserile

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

Step	Hyp	Ref	Expression
1		clim2ser.1	. 2 |- Z = ( ZZ>= ` M )
2		iserile.2	. 2 |- ( ph -> M e. ZZ )
3		iserile.4	. 2 |- ( ph -> seq M ( + , F , CC ) ~~> A )
4		iserile.5	. 2 |- ( ph -> seq M ( + , G , CC ) ~~> B )
5		cnex 7097	. . . . . . 7 |- CC e. _V
6	5	a1i 9	. . . . . 6 |- ( ph -> CC e. _V )
7		ax-resscn 7068	. . . . . . 7 |- RR C_ CC
8	7	a1i 9	. . . . . 6 |- ( ph -> RR C_ CC )
9	1	eleq2i 2145	. . . . . . 7 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
10		iserile.6	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
11	9, 10	sylan2br 282	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
12		readdcl 7099	. . . . . . 7 |- ( ( k e. RR /\ x e. RR ) -> ( k + x ) e. RR )
13	12	adantl 271	. . . . . 6 |- ( ( ph /\ ( k e. RR /\ x e. RR ) ) -> ( k + x ) e. RR )
14		addcl 7098	. . . . . . 7 |- ( ( k e. CC /\ x e. CC ) -> ( k + x ) e. CC )
15	14	adantl 271	. . . . . 6 |- ( ( ph /\ ( k e. CC /\ x e. CC ) ) -> ( k + x ) e. CC )
16	2, 6, 8, 11, 13, 15	iseqss 9446	. . . . 5 |- ( ph -> seq M ( + , F , RR ) = seq M ( + , F , CC ) )
17	16	adantr 270	. . . 4 |- ( ( ph /\ j e. Z ) -> seq M ( + , F , RR ) = seq M ( + , F , CC ) )
18	17	fveq1d 5200	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F , RR ) ` j ) = ( seq M ( + , F , CC ) ` j ) )
19	1, 2, 10	iserfre 9454	. . . 4 |- ( ph -> seq M ( + , F , RR ) : Z --> RR )
20	19	ffvelrnda 5323	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F , RR ) ` j ) e. RR )
21	18, 20	eqeltrrd 2156	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F , CC ) ` j ) e. RR )
22		iserile.7	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
23	9, 22	sylan2br 282	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR )
24	2, 6, 8, 23, 13, 15	iseqss 9446	. . . . 5 |- ( ph -> seq M ( + , G , RR ) = seq M ( + , G , CC ) )
25	24	adantr 270	. . . 4 |- ( ( ph /\ j e. Z ) -> seq M ( + , G , RR ) = seq M ( + , G , CC ) )
26	25	fveq1d 5200	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G , RR ) ` j ) = ( seq M ( + , G , CC ) ` j ) )
27	1, 2, 22	iserfre 9454	. . . 4 |- ( ph -> seq M ( + , G , RR ) : Z --> RR )
28	27	ffvelrnda 5323	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G , RR ) ` j ) e. RR )
29	26, 28	eqeltrrd 2156	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , G , CC ) ` j ) e. RR )
30		simpr 108	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. Z )
31	30, 1	syl6eleq 2171	. . 3 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
32	11	adantlr 460	. . 3 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
33	23	adantlr 460	. . 3 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR )
34		simpll 495	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ph )
35	9	biimpri 131	. . . . 5 |- ( k e. ( ZZ>= ` M ) -> k e. Z )
36	35	adantl 271	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> k e. Z )
37		iserile.8	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) )
38	34, 36, 37	syl2anc 403	. . 3 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) <_ ( G ` k ) )
39	31, 32, 33, 38	serile 9474	. 2 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , F , CC ) ` j ) <_ ( seq M ( + , G , CC ) ` j ) )
40	1, 2, 3, 4, 21, 29, 39	climle 10172	1 |- ( ph -> A <_ B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   C_ wss 2973   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   CCcc 6979   RRcr 6980   + caddc 6984   <_ cle 7154   ZZcz 8351   ZZ>=cuz 8619   seqcseq 9431   ~~> cli 10117

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-fz 9030  df-fzo 9153  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-clim 10118

This theorem is referenced by:  iserige0  10181