Theorem elcncf1ii 13207

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 13206	. 2 |- ( T. -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )
8	7	mptru 1352	1 |- ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) )

Syntax hints:   -> wi 4   /\ wa 103   T. wtru 1344   e. wcel 2136   C_ wss 3116   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   CCcc 7751   < clt 7933   - cmin 8069   RR+crp 9589   abscabs 10939   -cn->ccncf 13197

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-map 6616  df-cncf 13198

This theorem is referenced by: (None)