Theorem elcncf1ii 15303

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 15302	. 2 |- ( T. -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )
8	7	mptru 1406	1 |- ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   T. wtru 1398   e. wcel 2202   C_ wss 3200   class class class wbr 4088   -->wf 5322   ` cfv 5326  (class class class)co 6017   CCcc 8029   < clt 8213   - cmin 8349   RR+crp 9887   abscabs 11557   -cn->ccncf 15293

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-map 6818  df-cncf 15294

This theorem is referenced by: (None)