Theorem ser3le 10608

Description: Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)

Hypotheses

Ref	Expression
ser3ge0.1	|- ( ph -> N e. ( ZZ>= ` M ) )
ser3ge0.2	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
ser3le.3	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR )
serle.4	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) <_ ( G ` k ) )

Assertion

Ref	Expression
ser3le	|- ( ph -> ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) )

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

Proof of Theorem ser3le

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ser3ge0.1	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
2		vex 2763	. . . . . 6 |- k e. _V
3		ser3le.3	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR )
4		ser3ge0.2	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )
5	3, 4	resubcld 8400	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( G ` k ) - ( F ` k ) ) e. RR )
6		fveq2 5554	. . . . . . . 8 |- ( x = k -> ( G ` x ) = ( G ` k ) )
7		fveq2 5554	. . . . . . . 8 |- ( x = k -> ( F ` x ) = ( F ` k ) )
8	6, 7	oveq12d 5936	. . . . . . 7 |- ( x = k -> ( ( G ` x ) - ( F ` x ) ) = ( ( G ` k ) - ( F ` k ) ) )
9		eqid 2193	. . . . . . 7 |- ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) = ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) )
10	8, 9	fvmptg 5633	. . . . . 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 414	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
12	11, 5	eqeltrd 2270	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) e. RR )
13		elfzuz 10087	. . . . . . 7 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
14		serle.4	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) <_ ( G ` k ) )
15	13, 14	sylan2 286	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) <_ ( G ` k ) )
16	3, 4	subge0d 8554	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( 0 <_ ( ( G ` k ) - ( F ` k ) ) <-> ( F ` k ) <_ ( G ` k ) ) )
17	13, 16	sylan2 286	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ( 0 <_ ( ( G ` k ) - ( F ` k ) ) <-> ( F ` k ) <_ ( G ` k ) ) )
18	15, 17	mpbird 167	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ ( ( G ` k ) - ( F ` k ) ) )
19	13, 11	sylan2 286	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
20	18, 19	breqtrrd 4057	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) )
21	1, 12, 20	ser3ge0 10607	. . 3 |- ( ph -> 0 <_ ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) )
22	3	recnd 8048	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
23	4	recnd 8048	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
24	1, 22, 23, 11	ser3sub 10594	. . 3 |- ( ph -> ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) = ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
25	21, 24	breqtrd 4055	. 2 |- ( ph -> 0 <_ ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
26		eqid 2193	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
27		eluzel2 9597	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
28	1, 27	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
29	26, 28, 3	serfre 10555	. . . 4 |- ( ph -> seq M ( + , G ) : ( ZZ>= ` M ) --> RR )
30	29, 1	ffvelcdmd 5694	. . 3 |- ( ph -> ( seq M ( + , G ) ` N ) e. RR )
31	26, 28, 4	serfre 10555	. . . 4 |- ( ph -> seq M ( + , F ) : ( ZZ>= ` M ) --> RR )
32	31, 1	ffvelcdmd 5694	. . 3 |- ( ph -> ( seq M ( + , F ) ` N ) e. RR )
33	30, 32	subge0d 8554	. 2 |- ( ph -> ( 0 <_ ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) <-> ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) ) )
34	25, 33	mpbid 147	1 |- ( ph -> ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4029   |-> cmpt 4090   ` cfv 5254  (class class class)co 5918   RRcr 7871   0cc0 7872   + caddc 7875   <_ cle 8055   - cmin 8190   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074   seqcseq 10518

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519

This theorem is referenced by:  iserle  11485  cvgcmpub  11619