Theorem climsqz 11958

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 276	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> M e. ZZ )
4		simpr 110	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> x e. RR+ )
5		eqidd 2232	. . . . 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 11911	. . . 4 |- ( ( ph /\ x e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x )
9	1	uztrn2 9818	. . . . . . . 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 11947	. . . . . . . . . . . . 13 |- ( ph -> A e. RR )
13	12	adantr 276	. . . . . . . . . . . 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 8784	. . . . . . . . . . 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 11800	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) = ( A - ( G ` k ) ) )
18	10, 11, 13, 14, 16	letrd 8345	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( F ` k ) <_ A )
19	10, 13, 18	abssuble0d 11800	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) = ( A - ( F ` k ) ) )
20	15, 17, 19	3brtr4d 4125	. . . . . . . . . 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	11	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 8602	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. RR )
25	24	recnd 8250	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( G ` k ) - A ) e. CC )
26	25	abscld 11804	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( G ` k ) - A ) ) e. RR )
27	10	adantlr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( F ` k ) e. RR )
28	27, 23	resubcld 8602	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. RR )
29	28	recnd 8250	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( ( F ` k ) - A ) e. CC )
30	29	abscld 11804	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) e. RR )
31		rpre 9939	. . . . . . . . . . 11 |- ( x e. RR+ -> x e. RR )
32	31	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ k e. Z ) -> x e. RR )
33		lelttr 8310	. . . . . . . . . 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 1274	. . . . . . . . 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 2600	. . . . 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 2635	. . . 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 2606	. 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 2232	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( G ` k ) )
44	12	recnd 8250	. . 3 |- ( ph -> A e. CC )
45	11	recnd 8250	. . 3 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
46	1, 2, 42, 43, 44, 45	clim2c 11907	. 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 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8074   < clt 8256   <_ cle 8257   - cmin 8392   ZZcz 9523   ZZ>=cuz 9799   RR+crp 9932   abscabs 11620   ~~> cli 11901

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-rp 9933  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902

This theorem is referenced by: (None)