Theorem elcncf1ii 15681

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 15680	. 2 |- ( T. -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )
8	7	mptru 1411	1 |- ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   T. wtru 1403   e. wcel 2209   C_ wss 3220   class class class wbr 4130   -->wf 5373   ` cfv 5377  (class class class)co 6085   CCcc 8177   < clt 8360   - cmin 8497   RR+crp 10054   abscabs 11763   -cn->ccncf 15671

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-map 6924  df-cncf 15672

This theorem is used by: (None)