Theorem ser3le 10291

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 2689	. . . . . 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 8143	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( G ` k ) - ( F ` k ) ) e. RR )
6		fveq2 5421	. . . . . . . 8 |- ( x = k -> ( G ` x ) = ( G ` k ) )
7		fveq2 5421	. . . . . . . 8 |- ( x = k -> ( F ` x ) = ( F ` k ) )
8	6, 7	oveq12d 5792	. . . . . . 7 |- ( x = k -> ( ( G ` x ) - ( F ` x ) ) = ( ( G ` k ) - ( F ` k ) ) )
9		eqid 2139	. . . . . . 7 |- ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) = ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) )
10	8, 9	fvmptg 5497	. . . . . 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 410	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
12	11, 5	eqeltrd 2216	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) e. RR )
13		elfzuz 9802	. . . . . . 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 284	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) <_ ( G ` k ) )
16	3, 4	subge0d 8297	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( 0 <_ ( ( G ` k ) - ( F ` k ) ) <-> ( F ` k ) <_ ( G ` k ) ) )
17	13, 16	sylan2 284	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ( 0 <_ ( ( G ` k ) - ( F ` k ) ) <-> ( F ` k ) <_ ( G ` k ) ) )
18	15, 17	mpbird 166	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ ( ( G ` k ) - ( F ` k ) ) )
19	13, 11	sylan2 284	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
20	18, 19	breqtrrd 3956	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) )
21	1, 12, 20	ser3ge0 10290	. . 3 |- ( ph -> 0 <_ ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) )
22	3	recnd 7794	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
23	4	recnd 7794	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
24	1, 22, 23, 11	ser3sub 10279	. . 3 |- ( ph -> ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) = ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
25	21, 24	breqtrd 3954	. 2 |- ( ph -> 0 <_ ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
26		eqid 2139	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
27		eluzel2 9331	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
28	1, 27	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
29	26, 28, 3	serfre 10248	. . . 4 |- ( ph -> seq M ( + , G ) : ( ZZ>= ` M ) --> RR )
30	29, 1	ffvelrnd 5556	. . 3 |- ( ph -> ( seq M ( + , G ) ` N ) e. RR )
31	26, 28, 4	serfre 10248	. . . 4 |- ( ph -> seq M ( + , F ) : ( ZZ>= ` M ) --> RR )
32	31, 1	ffvelrnd 5556	. . 3 |- ( ph -> ( seq M ( + , F ) ` N ) e. RR )
33	30, 32	subge0d 8297	. 2 |- ( ph -> ( 0 <_ ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) <-> ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) ) )
34	25, 33	mpbid 146	1 |- ( ph -> ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2686   class class class wbr 3929   |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   RRcr 7619   0cc0 7620   + caddc 7623   <_ cle 7801   - cmin 7933   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790   seqcseq 10218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-fz 9791  df-fzo 9920  df-seqfrec 10219

This theorem is referenced by:  iserle  11111  cvgcmpub  11245