Theorem cn1lem 11115

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 519	. . . . 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 409	. . . 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 5562	. . . . . . . 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 5562	. . . . . . . 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 8097	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( F ` z ) - ( F ` A ) ) e. CC )
12	11	abscld 10985	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) e. RR )
13	2, 3	subcld 8097	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( z - A ) e. CC )
14	13	abscld 10985	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( z - A ) ) e. RR )
15		rpre 9477	. . . . . 6 |- ( x e. RR+ -> x e. RR )
16	15	ad2antlr 481	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> x e. RR )
17		lelttr 7876	. . . . 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 1217	. . . 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 426	. . 3 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
20	19	ralrimiva 2508	. 2 |- ( ( A e. CC /\ x e. RR+ ) -> A. z e. CC ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
21		breq2 3941	. . . . 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 2438	. . 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 2793	. 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 409	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 1481   A.wral 2417   E.wrex 2418   class class class wbr 3937   -->wf 5127   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   < clt 7824   <_ cle 7825   - cmin 7957   RR+crp 9470   abscabs 10801

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-rp 9471  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803

This theorem is referenced by:  abscn2  11116  cjcn2  11117  recn2  11118  imcn2  11119