Theorem cn1lem 11457

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 110	. 2 |- ( ( A e. CC /\ x e. RR+ ) -> x e. RR+ )
2		simpr 110	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> z e. CC )
3		simpll 527	. . . . 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 411	. . . 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	ffvelcdmi 5692	. . . . . . . 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	ffvelcdmi 5692	. . . . . . . 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 8330	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( F ` z ) - ( F ` A ) ) e. CC )
12	11	abscld 11325	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) e. RR )
13	2, 3	subcld 8330	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( z - A ) e. CC )
14	13	abscld 11325	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( z - A ) ) e. RR )
15		rpre 9726	. . . . . 6 |- ( x e. RR+ -> x e. RR )
16	15	ad2antlr 489	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> x e. RR )
17		lelttr 8108	. . . . 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 1249	. . . 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 429	. . 3 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
20	19	ralrimiva 2567	. 2 |- ( ( A e. CC /\ x e. RR+ ) -> A. z e. CC ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
21		breq2 4033	. . . . 5 |- ( y = x -> ( ( abs ` ( z - A ) ) < y <-> ( abs ` ( z - A ) ) < x ) )
22	21	imbi1d 231	. . . 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 2494	. . 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 2864	. 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 411	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 104   e. wcel 2164   A.wral 2472   E.wrex 2473   class class class wbr 4029   -->wf 5250   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   < clt 8054   <_ cle 8055   - cmin 8190   RR+crp 9719   abscabs 11141

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143

This theorem is referenced by:  abscn2  11458  cjcn2  11459  recn2  11460  imcn2  11461