Theorem cn1lem 11083

Description: A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Hypotheses

Ref	Expression
cn1lem.1	|- F : CC --> CC
cn1lem.2	|- ( ( z e. CC /\ A e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) )

Assertion

Ref	Expression
cn1lem	|- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )

Distinct variable groups:   x, y, z   y, A, z   y, F

Allowed substitution hints:   A( x)   F( x, z)

Proof of Theorem cn1lem

Step	Hyp	Ref	Expression
1		simpr 109	. 2 |- ( ( A e. CC /\ x e. RR+ ) -> x e. RR+ )
2		simpr 109	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> z e. CC )
3		simpll 518	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> A e. CC )
4		cn1lem.2	. . . . 5 |- ( ( z e. CC /\ A e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) )
5	2, 3, 4	syl2anc 408	. . . 4 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) )
6		cn1lem.1	. . . . . . . . 9 |- F : CC --> CC
7	6	ffvelrni 5554	. . . . . . . 8 |- ( z e. CC -> ( F ` z ) e. CC )
8	2, 7	syl 14	. . . . . . 7 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( F ` z ) e. CC )
9	6	ffvelrni 5554	. . . . . . . 8 |- ( A e. CC -> ( F ` A ) e. CC )
10	3, 9	syl 14	. . . . . . 7 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( F ` A ) e. CC )
11	8, 10	subcld 8073	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( F ` z ) - ( F ` A ) ) e. CC )
12	11	abscld 10953	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) e. RR )
13	2, 3	subcld 8073	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( z - A ) e. CC )
14	13	abscld 10953	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( z - A ) ) e. RR )
15		rpre 9448	. . . . . 6 |- ( x e. RR+ -> x e. RR )
16	15	ad2antlr 480	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> x e. RR )
17		lelttr 7852	. . . . 5 |- ( ( ( abs ` ( ( F ` z ) - ( F ` A ) ) ) e. RR /\ ( abs ` ( z - A ) ) e. RR /\ x e. RR ) -> ( ( ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) /\ ( abs ` ( z - A ) ) < x ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
18	12, 14, 16, 17	syl3anc 1216	. . . 4 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( ( abs ` ( ( F ` z ) - ( F ` A ) ) ) <_ ( abs ` ( z - A ) ) /\ ( abs ` ( z - A ) ) < x ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
19	5, 18	mpand 425	. . 3 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
20	19	ralrimiva 2505	. 2 |- ( ( A e. CC /\ x e. RR+ ) -> A. z e. CC ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
21		breq2 3933	. . . . 5 |- ( y = x -> ( ( abs ` ( z - A ) ) < y <-> ( abs ` ( z - A ) ) < x ) )
22	21	imbi1d 230	. . . 4 |- ( y = x -> ( ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) <-> ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) ) )
23	22	ralbidv 2437	. . 3 |- ( y = x -> ( A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) <-> A. z e. CC ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) ) )
24	23	rspcev 2789	. 2 |- ( ( x e. RR+ /\ A. z e. CC ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
25	1, 20, 24	syl2anc 408	1 |- ( ( A e. CC /\ x e. RR+ ) -> E. y e. RR+ A. z e. CC ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   < clt 7800   <_ cle 7801   - cmin 7933   RR+crp 9441   abscabs 10769

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

This theorem is referenced by:  abscn2  11084  cjcn2  11085  recn2  11086  imcn2  11087