Theorem ser3le 10800

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 2805	. . . . . 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 8560	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( G ` k ) - ( F ` k ) ) e. RR )
6		fveq2 5639	. . . . . . . 8 |- ( x = k -> ( G ` x ) = ( G ` k ) )
7		fveq2 5639	. . . . . . . 8 |- ( x = k -> ( F ` x ) = ( F ` k ) )
8	6, 7	oveq12d 6036	. . . . . . 7 |- ( x = k -> ( ( G ` x ) - ( F ` x ) ) = ( ( G ` k ) - ( F ` k ) ) )
9		eqid 2231	. . . . . . 7 |- ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) = ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) )
10	8, 9	fvmptg 5722	. . . . . 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 2308	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) e. RR )
13		elfzuz 10256	. . . . . . 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 8715	. . . . . . 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 4116	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> 0 <_ ( ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ` k ) )
21	1, 12, 20	ser3ge0 10799	. . 3 |- ( ph -> 0 <_ ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) )
22	3	recnd 8208	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
23	4	recnd 8208	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
24	1, 22, 23, 11	ser3sub 10786	. . 3 |- ( ph -> ( seq M ( + , ( x e. _V |-> ( ( G ` x ) - ( F ` x ) ) ) ) ` N ) = ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
25	21, 24	breqtrd 4114	. 2 |- ( ph -> 0 <_ ( ( seq M ( + , G ) ` N ) - ( seq M ( + , F ) ` N ) ) )
26		eqid 2231	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
27		eluzel2 9760	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
28	1, 27	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
29	26, 28, 3	serfre 10747	. . . 4 |- ( ph -> seq M ( + , G ) : ( ZZ>= ` M ) --> RR )
30	29, 1	ffvelcdmd 5783	. . 3 |- ( ph -> ( seq M ( + , G ) ` N ) e. RR )
31	26, 28, 4	serfre 10747	. . . 4 |- ( ph -> seq M ( + , F ) : ( ZZ>= ` M ) --> RR )
32	31, 1	ffvelcdmd 5783	. . 3 |- ( ph -> ( seq M ( + , F ) ` N ) e. RR )
33	30, 32	subge0d 8715	. 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 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   |-> cmpt 4150   ` cfv 5326  (class class class)co 6018   RRcr 8031   0cc0 8032   + caddc 8035   <_ cle 8215   - cmin 8350   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243   seqcseq 10710

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-fzo 10378  df-seqfrec 10711

This theorem is referenced by:  iserle  11920  cvgcmpub  12055