Theorem climsqz 11104

Description: Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-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 )
climsqz.8	|- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) )
climsqz.9	|- ( ( ph /\ k e. Z ) -> ( G ` k ) <_ A )

Assertion

Ref	Expression
climsqz	|- ( 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 climsqz

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 274	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> M e. ZZ )
4		simpr 109	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> x e. RR+ )
5		eqidd 2140	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
6		climadd.4	. . . . . 6 |- ( ph -> F ~~> A )
7	6	adantr 274	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> F ~~> A )
8	1, 3, 4, 5, 7	climi2 11057	. . . 4 |- ( ( ph /\ x e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x )
9	1	uztrn2 9343	. . . . . . . 8 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
10		climsqz.6	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR )
11		climsqz.7	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
12	1, 2, 6, 10	climrecl 11093	. . . . . . . . . . . . 13 |- ( ph -> A e. RR )
13	12	adantr 274	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> A e. RR )
14		climsqz.8	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ ( G ` k ) )
15	10, 11, 13, 14	lesub2dd 8324	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( A - ( G ` k ) ) <_ ( A - ( F ` k ) ) )
16		climsqz.9	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( G ` k ) <_ A )
17	11, 13, 16	abssuble0d 10949	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) = ( A - ( G ` k ) ) )
18	10, 11, 13, 14, 16	letrd 7886	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ A )
19	10, 13, 18	abssuble0d 10949	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) = ( A - ( F ` k ) ) )
20	15, 17, 19	3brtr4d 3960	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) <_ ( abs ` ( ( F ` k ) - A ) ) )
21	20	adantlr 468	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) <_ ( abs ` ( ( F ` k ) - A ) ) )
22	11	adantlr 468	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( G ` k ) e. RR )
23	12	ad2antrr 479	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> A e. RR )
24	22, 23	resubcld 8143	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. RR )
25	24	recnd 7794	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. CC )
26	25	abscld 10953	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) e. RR )
27	10	adantlr 468	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( F ` k ) e. RR )
28	27, 23	resubcld 8143	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. RR )
29	28	recnd 7794	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. CC )
30	29	abscld 10953	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) e. RR )
31		rpre 9448	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
32	31	ad2antlr 480	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> x e. RR )
33		lelttr 7852	. . . . . . . . . 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 1216	. . . . . . . . 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 425	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( abs ` ( ( F ` k ) - A ) ) < x -> ( abs ` ( ( G ` k ) - A ) ) < x ) )
36	9, 35	sylan2 284	. . . . . . 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 397	. . . . . 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 2499	. . . . 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 2534	. . . 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 2505	. 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 2140	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
44	12	recnd 7794	. . 3 |- ( ph -> A e. CC )
45	11	recnd 7794	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
46	1, 2, 42, 43, 44, 45	clim2c 11053	. 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 166	1 |- ( ph -> G ~~> A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   RRcr 7619   < clt 7800   <_ cle 7801   - cmin 7933   ZZcz 9054   ZZ>=cuz 9326   RR+crp 9441   abscabs 10769   ~~> cli 11047

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-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  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-rmo 2424  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-if 3475  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-po 4218  df-iso 4219  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-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-rp 9442  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048

This theorem is referenced by: (None)