Theorem serile 9474

Description: Comparison of partial sums of two infinite series of reals. (Contributed by Jim Kingdon, 22-Aug-2021.)

Hypotheses

Ref	Expression
serige0.1	|- ( ph -> N e. ( ZZ>= ` M ) )
serige0.2	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
serile.3	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR )
serile.4	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) <_ ( G ` k ) )

Assertion

Ref	Expression
serile	|- ( ph -> ( seq M ( + , F , CC ) ` N ) <_ ( seq M ( + , G , CC ) ` N ) )

Distinct variable groups:   k, F   k, G   k, M   k, N   ph, k

Proof of Theorem serile

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		serige0.1	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
2		vex 2604	. . . . . 6 |- k e. _V
3		serile.3	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR )
4		serige0.2	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
5	3, 4	resubcld 7485	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( G ` k ) - ( F ` k ) ) e. RR )
6		fveq2 5198	. . . . . . . 8 |- ( x = k -> ( G ` x ) = ( G ` k ) )
7		fveq2 5198	. . . . . . . 8 |- ( x = k -> ( F ` x ) = ( F ` k ) )
8	6, 7	oveq12d 5550	. . . . . . 7 |- ( x = k -> ( ( G ` x ) - ( F ` x ) ) = ( ( G ` k ) - ( F ` k ) ) )
9		eqid 2081	. . . . . . 7 |- ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) = ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) )
10	8, 9	fvmptg 5269	. . . . . 6 |- ( ( k e. _V /\ ( ( G ` k ) - ( F ` k ) ) e. RR ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
11	2, 5, 10	sylancr 405	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
12	11, 5	eqeltrd 2155	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) e. RR )
13		serile.4	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) <_ ( G ` k ) )
14	3, 4	subge0d 7635	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( 0 <_ ( ( G ` k ) - ( F ` k ) ) <-> ( F ` k ) <_ ( G ` k ) ) )
15	13, 14	mpbird 165	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> 0 <_ ( ( G ` k ) - ( F ` k ) ) )
16	15, 11	breqtrrd 3811	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> 0 <_ ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) )
17	1, 12, 16	serige0 9473	. . 3 |- ( ph -> 0 <_ ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) , CC ) ` N ) )
18	3	recnd 7147	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
19	4	recnd 7147	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
20	1, 18, 19, 11	isersub 9464	. . 3 |- ( ph -> ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) , CC ) ` N ) = ( ( seq M ( + , G , CC ) ` N ) - ( seq M ( + , F , CC ) ` N ) ) )
21	17, 20	breqtrd 3809	. 2 |- ( ph -> 0 <_ ( ( seq M ( + , G , CC ) ` N ) - ( seq M ( + , F , CC ) ` N ) ) )
22		eluzel2 8624	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
23	1, 22	syl 14	. . . . . 6 |- ( ph -> M e. ZZ )
24		cnex 7097	. . . . . . 7 |- CC e. _V
25	24	a1i 9	. . . . . 6 |- ( ph -> CC e. _V )
26		ax-resscn 7068	. . . . . . 7 |- RR C_ CC
27	26	a1i 9	. . . . . 6 |- ( ph -> RR C_ CC )
28		readdcl 7099	. . . . . . 7 |- ( ( k e. RR /\ x e. RR ) -> ( k + x ) e. RR )
29	28	adantl 271	. . . . . 6 |- ( ( ph /\ ( k e. RR /\ x e. RR ) ) -> ( k + x ) e. RR )
30		addcl 7098	. . . . . . 7 |- ( ( k e. CC /\ x e. CC ) -> ( k + x ) e. CC )
31	30	adantl 271	. . . . . 6 |- ( ( ph /\ ( k e. CC /\ x e. CC ) ) -> ( k + x ) e. CC )
32	23, 25, 27, 3, 29, 31	iseqss 9446	. . . . 5 |- ( ph -> seq M ( + , G , RR ) = seq M ( + , G , CC ) )
33	32	fveq1d 5200	. . . 4 |- ( ph -> ( seq M ( + , G , RR ) ` N ) = ( seq M ( + , G , CC ) ` N ) )
34		reex 7107	. . . . . 6 |- RR e. _V
35	34	a1i 9	. . . . 5 |- ( ph -> RR e. _V )
36	1, 35, 3, 29	iseqcl 9443	. . . 4 |- ( ph -> ( seq M ( + , G , RR ) ` N ) e. RR )
37	33, 36	eqeltrrd 2156	. . 3 |- ( ph -> ( seq M ( + , G , CC ) ` N ) e. RR )
38	23, 25, 27, 4, 29, 31	iseqss 9446	. . . . 5 |- ( ph -> seq M ( + , F , RR ) = seq M ( + , F , CC ) )
39	38	fveq1d 5200	. . . 4 |- ( ph -> ( seq M ( + , F , RR ) ` N ) = ( seq M ( + , F , CC ) ` N ) )
40	1, 35, 4, 29	iseqcl 9443	. . . 4 |- ( ph -> ( seq M ( + , F , RR ) ` N ) e. RR )
41	39, 40	eqeltrrd 2156	. . 3 |- ( ph -> ( seq M ( + , F , CC ) ` N ) e. RR )
42	37, 41	subge0d 7635	. 2 |- ( ph -> ( 0 <_ ( ( seq M ( + , G , CC ) ` N ) - ( seq M ( + , F , CC ) ` N ) ) <-> ( seq M ( + , F , CC ) ` N ) <_ ( seq M ( + , G , CC ) ` N ) ) )
43	21, 42	mpbid 145	1 |- ( ph -> ( seq M ( + , F , CC ) ` N ) <_ ( seq M ( + , G , CC ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   C_ wss 2973   class class class wbr 3785   |-> cmpt 3839   ` cfv 4922  (class class class)co 5532   CCcc 6979   RRcr 6980   0cc0 6981   + caddc 6984   <_ cle 7154   - cmin 7279   ZZcz 8351   ZZ>=cuz 8619   seqcseq 9431

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-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  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-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-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-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-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030  df-fzo 9153  df-iseq 9432

This theorem is referenced by:  iserile  10180