Theorem cn1lem 11314

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 5648	. . . . . . . 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 5648	. . . . . . . 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 8263	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( ( F ` z ) - ( F ` A ) ) e. CC )
12	11	abscld 11182	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) e. RR )
13	2, 3	subcld 8263	. . . . . 6 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( z - A ) e. CC )
14	13	abscld 11182	. . . . 5 |- ( ( ( A e. CC /\ x e. RR+ ) /\ z e. CC ) -> ( abs ` ( z - A ) ) e. RR )
15		rpre 9655	. . . . . 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 8041	. . . . 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 1238	. . . 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 2550	. 2 |- ( ( A e. CC /\ x e. RR+ ) -> A. z e. CC ( ( abs ` ( z - A ) ) < x -> ( abs ` ( ( F ` z ) - ( F ` A ) ) ) < x ) )
21		breq2 4006	. . . . 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 2477	. . 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 2841	. 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 2148   A.wral 2455   E.wrex 2456   class class class wbr 4002   -->wf 5210   ` cfv 5214  (class class class)co 5871   CCcc 7805   RRcr 7806   < clt 7987   <_ cle 7988   - cmin 8123   RR+crp 9648   abscabs 10998

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7898  ax-resscn 7899  ax-1cn 7900  ax-1re 7901  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-mulrcl 7906  ax-addcom 7907  ax-mulcom 7908  ax-addass 7909  ax-mulass 7910  ax-distr 7911  ax-i2m1 7912  ax-0lt1 7913  ax-1rid 7914  ax-0id 7915  ax-rnegex 7916  ax-precex 7917  ax-cnre 7918  ax-pre-ltirr 7919  ax-pre-ltwlin 7920  ax-pre-lttrn 7921  ax-pre-apti 7922  ax-pre-ltadd 7923  ax-pre-mulgt0 7924  ax-pre-mulext 7925  ax-arch 7926  ax-caucvg 7927

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-1st 6137  df-2nd 6138  df-recs 6302  df-frec 6388  df-pnf 7989  df-mnf 7990  df-xr 7991  df-ltxr 7992  df-le 7993  df-sub 8125  df-neg 8126  df-reap 8527  df-ap 8534  df-div 8625  df-inn 8915  df-2 8973  df-3 8974  df-4 8975  df-n0 9172  df-z 9249  df-uz 9524  df-rp 9649  df-seqfrec 10440  df-exp 10514  df-cj 10843  df-re 10844  df-im 10845  df-rsqrt 10999  df-abs 11000

This theorem is referenced by:  abscn2  11315  cjcn2  11316  recn2  11317  imcn2  11318