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Metamath Proof Explorer

Theorem mulc1cncf

Description: Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007) (Revised by Mario Carneiro, 30-Apr-2014)

Ref Expression
Hypothesis mulc1cncf.1 
|- F = ( x e. CC |-> ( A x. x ) )
Assertion mulc1cncf 
|- ( A e. CC -> F e. ( CC -cn-> CC ) )

Proof

Step Hyp Ref Expression
1 mulc1cncf.1 
 |-  F = ( x e. CC |-> ( A x. x ) )
2 mulcl 
 |-  ( ( A e. CC /\ x e. CC ) -> ( A x. x ) e. CC )
3 2 1 fmptd 
 |-  ( A e. CC -> F : CC --> CC )
4 simprr 
 |-  ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) -> z e. RR+ )
5 simpl 
 |-  ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) -> A e. CC )
6 simprl 
 |-  ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) -> y e. CC )
7 mulcn2 
 |-  ( ( z e. RR+ /\ A e. CC /\ y e. CC ) -> E. t e. RR+ E. w e. RR+ A. v e. CC A. u e. CC ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) )
8 4 5 6 7 syl3anc 
 |-  ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) -> E. t e. RR+ E. w e. RR+ A. v e. CC A. u e. CC ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) )
9 fvoveq1 
 |-  ( v = A -> ( abs ` ( v - A ) ) = ( abs ` ( A - A ) ) )
10 9 breq1d 
 |-  ( v = A -> ( ( abs ` ( v - A ) ) < t <-> ( abs ` ( A - A ) ) < t ) )
11 10 anbi1d 
 |-  ( v = A -> ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) <-> ( ( abs ` ( A - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) ) )
12 oveq1 
 |-  ( v = A -> ( v x. u ) = ( A x. u ) )
13 12 fvoveq1d 
 |-  ( v = A -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) = ( abs ` ( ( A x. u ) - ( A x. y ) ) ) )
14 13 breq1d 
 |-  ( v = A -> ( ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z <-> ( abs ` ( ( A x. u ) - ( A x. y ) ) ) < z ) )
15 11 14 imbi12d 
 |-  ( v = A -> ( ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) <-> ( ( ( abs ` ( A - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( A x. u ) - ( A x. y ) ) ) < z ) ) )
16 15 ralbidv 
 |-  ( v = A -> ( A. u e. CC ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) <-> A. u e. CC ( ( ( abs ` ( A - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( A x. u ) - ( A x. y ) ) ) < z ) ) )
17 16 rspcv 
 |-  ( A e. CC -> ( A. v e. CC A. u e. CC ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) -> A. u e. CC ( ( ( abs ` ( A - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( A x. u ) - ( A x. y ) ) ) < z ) ) )
18 17 ad2antrr 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( t e. RR+ /\ w e. RR+ ) ) -> ( A. v e. CC A. u e. CC ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) -> A. u e. CC ( ( ( abs ` ( A - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( A x. u ) - ( A x. y ) ) ) < z ) ) )
19 subid 
 |-  ( A e. CC -> ( A - A ) = 0 )
20 19 ad2antrr 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( A - A ) = 0 )
21 20 abs00bd 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( abs ` ( A - A ) ) = 0 )
22 simprll 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> t e. RR+ )
23 22 rpgt0d 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> 0 < t )
24 21 23 eqbrtrd 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( abs ` ( A - A ) ) < t )
25 24 biantrurd 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( ( abs ` ( u - y ) ) < w <-> ( ( abs ` ( A - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) ) )
26 simprr 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> u e. CC )
27 oveq2 
 |-  ( x = u -> ( A x. x ) = ( A x. u ) )
28 ovex 
 |-  ( A x. u ) e. _V
29 27 1 28 fvmpt 
 |-  ( u e. CC -> ( F ` u ) = ( A x. u ) )
30 26 29 syl 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( F ` u ) = ( A x. u ) )
31 simplrl 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> y e. CC )
32 oveq2 
 |-  ( x = y -> ( A x. x ) = ( A x. y ) )
33 ovex 
 |-  ( A x. y ) e. _V
34 32 1 33 fvmpt 
 |-  ( y e. CC -> ( F ` y ) = ( A x. y ) )
35 31 34 syl 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( F ` y ) = ( A x. y ) )
36 30 35 oveq12d 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( ( F ` u ) - ( F ` y ) ) = ( ( A x. u ) - ( A x. y ) ) )
37 36 fveq2d 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) = ( abs ` ( ( A x. u ) - ( A x. y ) ) ) )
38 37 breq1d 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z <-> ( abs ` ( ( A x. u ) - ( A x. y ) ) ) < z ) )
39 25 38 imbi12d 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( ( t e. RR+ /\ w e. RR+ ) /\ u e. CC ) ) -> ( ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) <-> ( ( ( abs ` ( A - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( A x. u ) - ( A x. y ) ) ) < z ) ) )
40 39 anassrs 
 |-  ( ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( t e. RR+ /\ w e. RR+ ) ) /\ u e. CC ) -> ( ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) <-> ( ( ( abs ` ( A - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( A x. u ) - ( A x. y ) ) ) < z ) ) )
41 40 ralbidva 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( t e. RR+ /\ w e. RR+ ) ) -> ( A. u e. CC ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) <-> A. u e. CC ( ( ( abs ` ( A - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( A x. u ) - ( A x. y ) ) ) < z ) ) )
42 18 41 sylibrd 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ ( t e. RR+ /\ w e. RR+ ) ) -> ( A. v e. CC A. u e. CC ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) -> A. u e. CC ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) ) )
43 42 anassrs 
 |-  ( ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ t e. RR+ ) /\ w e. RR+ ) -> ( A. v e. CC A. u e. CC ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) -> A. u e. CC ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) ) )
44 43 reximdva 
 |-  ( ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) /\ t e. RR+ ) -> ( E. w e. RR+ A. v e. CC A. u e. CC ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) -> E. w e. RR+ A. u e. CC ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) ) )
45 44 rexlimdva 
 |-  ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) -> ( E. t e. RR+ E. w e. RR+ A. v e. CC A. u e. CC ( ( ( abs ` ( v - A ) ) < t /\ ( abs ` ( u - y ) ) < w ) -> ( abs ` ( ( v x. u ) - ( A x. y ) ) ) < z ) -> E. w e. RR+ A. u e. CC ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) ) )
46 8 45 mpd 
 |-  ( ( A e. CC /\ ( y e. CC /\ z e. RR+ ) ) -> E. w e. RR+ A. u e. CC ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) )
47 46 ralrimivva 
 |-  ( A e. CC -> A. y e. CC A. z e. RR+ E. w e. RR+ A. u e. CC ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) )
48 ssid 
 |-  CC C_ CC
49 elcncf2 
 |-  ( ( CC C_ CC /\ CC C_ CC ) -> ( F e. ( CC -cn-> CC ) <-> ( F : CC --> CC /\ A. y e. CC A. z e. RR+ E. w e. RR+ A. u e. CC ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) ) ) )
50 48 48 49 mp2an 
 |-  ( F e. ( CC -cn-> CC ) <-> ( F : CC --> CC /\ A. y e. CC A. z e. RR+ E. w e. RR+ A. u e. CC ( ( abs ` ( u - y ) ) < w -> ( abs ` ( ( F ` u ) - ( F ` y ) ) ) < z ) ) )
51 3 47 50 sylanbrc 
 |-  ( A e. CC -> F e. ( CC -cn-> CC ) )