Theorem elcncf1ii 15445

Description: Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)

Hypotheses

Ref	Expression
elcncf1i.1	|- F : A --> B
elcncf1i.2	|- ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ )
elcncf1i.3	|- ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )

Assertion

Ref	Expression
elcncf1ii	|- ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) )

Distinct variable groups:   x, w, y, A   w, B, x, y   w, F, x, y   w, Z

Allowed substitution hints:   Z( x, y)

Proof of Theorem elcncf1ii

Step	Hyp	Ref	Expression
1		elcncf1i.1	. . . 4 |- F : A --> B
2	1	a1i 9	. . 3 |- ( T. -> F : A --> B )
3		elcncf1i.2	. . . 4 |- ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ )
4	3	a1i 9	. . 3 |- ( T. -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
5		elcncf1i.3	. . . 4 |- ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
6	5	a1i 9	. . 3 |- ( T. -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
7	2, 4, 6	elcncf1di 15444	. 2 |- ( T. -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )
8	7	mptru 1407	1 |- ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   T. wtru 1399   e. wcel 2203   C_ wss 3211   class class class wbr 4109   -->wf 5348   ` cfv 5352  (class class class)co 6050   CCcc 8125   < clt 8308   - cmin 8444   RR+crp 9986   abscabs 11682   -cn->ccncf 15435

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-map 6884  df-cncf 15436

This theorem is referenced by: (None)