Theorem climsqz2 11385

Description: Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)

Hypotheses

Ref	Expression
climadd.1	|- Z = ( ZZ>= ` M )
climadd.2	|- ( ph -> M e. ZZ )
climadd.4	|- ( ph -> F ~~> A )
climsqz.5	|- ( ph -> G e. W )
climsqz.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
climsqz.7	|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
climsqz2.8	|- ( ( ph /\ k e. Z ) -> ( G ` k ) <_ ( F ` k ) )
climsqz2.9	|- ( ( ph /\ k e. Z ) -> A <_ ( G ` k ) )

Assertion

Ref	Expression
climsqz2	|- ( ph -> G ~~> A )

Distinct variable groups:   k, F   ph, k   A, k   k, G   k, M   k, Z

Allowed substitution hint:   W( k)

Proof of Theorem climsqz2

Dummy variables x j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climadd.1	. . . . 5 |- Z = ( ZZ>= ` M )
2		climadd.2	. . . . . 6 |- ( ph -> M e. ZZ )
3	2	adantr 276	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> M e. ZZ )
4		simpr 110	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> x e. RR+ )
5		eqidd 2190	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
6		climadd.4	. . . . . 6 |- ( ph -> F ~~> A )
7	6	adantr 276	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> F ~~> A )
8	1, 3, 4, 5, 7	climi2 11337	. . . 4 |- ( ( ph /\ x e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x )
9	1	uztrn2 9581	. . . . . . . 8 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
10		climsqz.7	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
11		climsqz.6	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
12	1, 2, 6, 11	climrecl 11373	. . . . . . . . . . . . 13 |- ( ph -> A e. RR )
13	12	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> A e. RR )
14		climsqz2.8	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( G ` k ) <_ ( F ` k ) )
15	10, 11, 13, 14	lesub1dd 8553	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( ( G ` k ) - A ) <_ ( ( F ` k ) - A ) )
16		climsqz2.9	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> A <_ ( G ` k ) )
17	13, 10, 16	abssubge0d 11226	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) = ( ( G ` k ) - A ) )
18	13, 10, 11, 16, 14	letrd 8116	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> A <_ ( F ` k ) )
19	13, 11, 18	abssubge0d 11226	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) = ( ( F ` k ) - A ) )
20	15, 17, 19	3brtr4d 4053	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) <_ ( abs ` ( ( F ` k ) - A ) ) )
21	20	adantlr 477	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) <_ ( abs ` ( ( F ` k ) - A ) ) )
22	10	adantlr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( G ` k ) e. RR )
23	12	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> A e. RR )
24	22, 23	resubcld 8373	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. RR )
25	24	recnd 8021	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. CC )
26	25	abscld 11231	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) e. RR )
27	11	adantlr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( F ` k ) e. RR )
28	27, 23	resubcld 8373	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. RR )
29	28	recnd 8021	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. CC )
30	29	abscld 11231	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) e. RR )
31		rpre 9696	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
32	31	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> x e. RR )
33		lelttr 8081	. . . . . . . . . 10 |- ( ( ( abs ` ( ( G ` k ) - A ) ) e. RR /\ ( abs ` ( ( F ` k ) - A ) ) e. RR /\ x e. RR ) -> ( ( ( abs ` ( ( G ` k ) - A ) ) <_ ( abs ` ( ( F ` k ) - A ) ) /\ ( abs ` ( ( F ` k ) - A ) ) < x ) -> ( abs ` ( ( G ` k ) - A ) ) < x ) )
34	26, 30, 32, 33	syl3anc 1249	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( ( abs ` ( ( G ` k ) - A ) ) <_ ( abs ` ( ( F ` k ) - A ) ) /\ ( abs ` ( ( F ` k ) - A ) ) < x ) -> ( abs ` ( ( G ` k ) - A ) ) < x ) )
35	21, 34	mpand 429	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( abs ` ( ( F ` k ) - A ) ) < x -> ( abs ` ( ( G ` k ) - A ) ) < x ) )
36	9, 35	sylan2 286	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( ( F ` k ) - A ) ) < x -> ( abs ` ( ( G ` k ) - A ) ) < x ) )
37	36	anassrs 400	. . . . . 6 |- ( ( ( ( ph /\ x e. RR+ ) /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( ( F ` k ) - A ) ) < x -> ( abs ` ( ( G ` k ) - A ) ) < x ) )
38	37	ralimdva 2557	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ j e. Z ) -> ( A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x -> A. k e. ( ZZ>= ` j ) ( abs ` ( ( G ` k ) - A ) ) < x ) )
39	38	reximdva 2592	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( G ` k ) - A ) ) < x ) )
40	8, 39	mpd 13	. . 3 |- ( ( ph /\ x e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( G ` k ) - A ) ) < x )
41	40	ralrimiva 2563	. 2 |- ( ph -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( G ` k ) - A ) ) < x )
42		climsqz.5	. . 3 |- ( ph -> G e. W )
43		eqidd 2190	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
44	12	recnd 8021	. . 3 |- ( ph -> A e. CC )
45	10	recnd 8021	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
46	1, 2, 42, 43, 44, 45	clim2c 11333	. 2 |- ( ph -> ( G ~~> A <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( G ` k ) - A ) ) < x ) )
47	41, 46	mpbird 167	1 |- ( ph -> G ~~> A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468   E.wrex 2469   class class class wbr 4021   ` cfv 5238  (class class class)co 5900   RRcr 7845   < clt 8027   <_ cle 8028   - cmin 8163   ZZcz 9288   ZZ>=cuz 9563   RR+crp 9689   abscabs 11047   ~~> cli 11327

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964  ax-arch 7965  ax-caucvg 7966

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-n0 9212  df-z 9289  df-uz 9564  df-rp 9690  df-seqfrec 10485  df-exp 10560  df-cj 10892  df-re 10893  df-im 10894  df-rsqrt 11048  df-abs 11049  df-clim 11328

This theorem is referenced by:  expcnvap0  11551  expcnvre  11552  explecnv  11554