Theorem ser3le 10629

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 2766	. . . . . 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 8407	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( G ` k ) - ( F ` k ) ) e. RR )
6		fveq2 5558	. . . . . . . 8 |- ( x = k -> ( G ` x ) = ( G ` k ) )
7		fveq2 5558	. . . . . . . 8 |- ( x = k -> ( F ` x ) = ( F ` k ) )
8	6, 7	oveq12d 5940	. . . . . . 7 |- ( x = k -> ( ( G ` x ) - ( F ` x ) ) = ( ( G ` k ) - ( F ` k ) ) )
9		eqid 2196	. . . . . . 7 |- ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) = ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) )
10	8, 9	fvmptg 5637	. . . . . 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 2273	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) e. RR )
13		elfzuz 10096	. . . . . . 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 8562	. . . . . . 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 4061	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) )
21	1, 12, 20	ser3ge0 10628	. . 3 |- ( ph -> 0 <_ ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) )
22	3	recnd 8055	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
23	4	recnd 8055	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
24	1, 22, 23, 11	ser3sub 10615	. . 3 |- ( ph -> ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) = ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
25	21, 24	breqtrd 4059	. 2 |- ( ph -> 0 <_ ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
26		eqid 2196	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
27		eluzel2 9606	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
28	1, 27	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
29	26, 28, 3	serfre 10576	. . . 4 |- ( ph -> seq M ( + , G ) : ( ZZ>= ` M ) --> RR )
30	29, 1	ffvelcdmd 5698	. . 3 |- ( ph -> ( seq M ( + , G ) ` N ) e. RR )
31	26, 28, 4	serfre 10576	. . . 4 |- ( ph -> seq M ( + , F ) : ( ZZ>= ` M ) --> RR )
32	31, 1	ffvelcdmd 5698	. . 3 |- ( ph -> ( seq M ( + , F ) ` N ) e. RR )
33	30, 32	subge0d 8562	. 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 2167   _Vcvv 2763   class class class wbr 4033   |-> cmpt 4094   ` cfv 5258  (class class class)co 5922   RRcr 7878   0cc0 7879   + caddc 7882   <_ cle 8062   - cmin 8197   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   seqcseq 10539

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218  df-seqfrec 10540

This theorem is referenced by:  iserle  11507  cvgcmpub  11641