Theorem climsqz2 12080

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 2239	. . . . 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 12032	. . . 4 |- ( ( ph /\ x e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x )
9	1	uztrn2 9919	. . . . . . . 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 12068	. . . . . . . . . . . . 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 8879	. . . . . . . . . . 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 11920	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) = ( ( G ` k ) - A ) )
18	13, 10, 11, 16, 14	letrd 8440	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> A <_ ( F ` k ) )
19	13, 11, 18	abssubge0d 11920	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) = ( ( F ` k ) - A ) )
20	15, 17, 19	3brtr4d 4157	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) <_ ( abs ` ( ( F ` k ) - A ) ) )
21	20	adantlr 481	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) <_ ( abs ` ( ( F ` k ) - A ) ) )
22	10	adantlr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( G ` k ) e. RR )
23	12	ad2antrr 492	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> A e. RR )
24	22, 23	resubcld 8698	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. RR )
25	24	recnd 8344	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. CC )
26	25	abscld 11925	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) e. RR )
27	11	adantlr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( F ` k ) e. RR )
28	27, 23	resubcld 8698	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. RR )
29	28	recnd 8344	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. CC )
30	29	abscld 11925	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) e. RR )
31		rpre 10040	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
32	31	ad2antlr 493	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> x e. RR )
33		lelttr 8404	. . . . . . . . . 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 1278	. . . . . . . . 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 433	. . . . . . . 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 404	. . . . . 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 2617	. . . . 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 2652	. . . 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 2623	. 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 2239	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
44	12	recnd 8344	. . 3 |- ( ph -> A e. CC )
45	10	recnd 8344	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
46	1, 2, 42, 43, 44, 45	clim2c 12028	. 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 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   RRcr 8168   < clt 8350   <_ cle 8351   - cmin 8487   ZZcz 9623   ZZ>=cuz 9900   RR+crp 10033   abscabs 11741   ~~> cli 12022

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-rp 10034  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023

This theorem is referenced by:  expcnvap0  12247  expcnvre  12248  explecnv  12250