Theorem serle 12856

Description: Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
serge0.1	|- ( ph -> N e. ( ZZ>= ` M ) )
serge0.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )
serle.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. RR )
serle.4	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) <_ ( G ` k ) )

Assertion

Ref	Expression
serle	|- ( 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 serle

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		serge0.1	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
2		vex 3203	. . . . . 6 |- k e. _V
3		fveq2 6191	. . . . . . . 8 |- ( x = k -> ( G ` x ) = ( G ` k ) )
4		fveq2 6191	. . . . . . . 8 |- ( x = k -> ( F ` x ) = ( F ` k ) )
5	3, 4	oveq12d 6668	. . . . . . 7 |- ( x = k -> ( ( G ` x ) - ( F ` x ) ) = ( ( G ` k ) - ( F ` k ) ) )
6		eqid 2622	. . . . . . 7 |- ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) = ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) )
7		ovex 6678	. . . . . . 7 |- ( ( G ` k ) - ( F ` k ) ) e. _V
8	5, 6, 7	fvmpt 6282	. . . . . 6 |- ( k e. _V -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
9	2, 8	ax-mp 5	. . . . 5 |- ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) )
10		serle.3	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. RR )
11		serge0.2	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. RR )
12	10, 11	resubcld 10458	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( G ` k ) - ( F ` k ) ) e. RR )
13	9, 12	syl5eqel 2705	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) e. RR )
14		serle.4	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) <_ ( G ` k ) )
15	10, 11	subge0d 10617	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> ( 0 <_ ( ( G ` k ) - ( F ` k ) ) <-> ( F ` k ) <_ ( G ` k ) ) )
16	14, 15	mpbird 247	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ ( ( G ` k ) - ( F ` k ) ) )
17	16, 9	syl6breqr 4695	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) )
18	1, 13, 17	serge0 12855	. . 3 |- ( ph -> 0 <_ ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) )
19	10	recnd 10068	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. CC )
20	11	recnd 10068	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
21	9	a1i 11	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) = ( ( G ` k ) - ( F ` k ) ) )
22	1, 19, 20, 21	sersub 12844	. . 3 |- ( ph -> ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) = ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
23	18, 22	breqtrd 4679	. 2 |- ( ph -> 0 <_ ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
24		readdcl 10019	. . . . 5 |- ( ( k e. RR /\ x e. RR ) -> ( k + x ) e. RR )
25	24	adantl 482	. . . 4 |- ( ( ph /\ ( k e. RR /\ x e. RR ) ) -> ( k + x ) e. RR )
26	1, 10, 25	seqcl 12821	. . 3 |- ( ph -> ( seq M ( + , G ) ` N ) e. RR )
27	1, 11, 25	seqcl 12821	. . 3 |- ( ph -> ( seq M ( + , F ) ` N ) e. RR )
28	26, 27	subge0d 10617	. 2 |- ( ph -> ( 0 <_ ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) <-> ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) ) )
29	23, 28	mpbid 222	1 |- ( ph -> ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   <_ cle 10075   - cmin 10266   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802

This theorem is referenced by:  iserle  14390  cvgcmpub  14549  ioombl1lem4  23329  stirlinglem10  40300