Theorem climsqz2 11344

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 2178	. . . . 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 11296	. . . 4 |- ( ( ph /\ x e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x )
9	1	uztrn2 9545	. . . . . . . 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 11332	. . . . . . . . . . . . 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 8518	. . . . . . . . . . 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 11185	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) = ( ( G ` k ) - A ) )
18	13, 10, 11, 16, 14	letrd 8081	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> A <_ ( F ` k ) )
19	13, 11, 18	abssubge0d 11185	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) = ( ( F ` k ) - A ) )
20	15, 17, 19	3brtr4d 4036	. . . . . . . . . 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 8338	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. RR )
25	24	recnd 7986	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. CC )
26	25	abscld 11190	. . . . . . . . . 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 8338	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. RR )
29	28	recnd 7986	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. CC )
30	29	abscld 11190	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) e. RR )
31		rpre 9660	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
32	31	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> x e. RR )
33		lelttr 8046	. . . . . . . . . 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 1238	. . . . . . . . 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 2544	. . . . 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 2579	. . . 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 2550	. 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 2178	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
44	12	recnd 7986	. . 3 |- ( ph -> A e. CC )
45	10	recnd 7986	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
46	1, 2, 42, 43, 44, 45	clim2c 11292	. 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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   RRcr 7810   < clt 7992   <_ cle 7993   - cmin 8128   ZZcz 9253   ZZ>=cuz 9528   RR+crp 9653   abscabs 11006   ~~> cli 11286

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-rp 9654  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-clim 11287

This theorem is referenced by:  expcnvap0  11510  expcnvre  11511  explecnv  11513