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Metamath Proof Explorer

Theorem dvne0f1

Description: A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015)

Ref Expression
Hypotheses dvne0.a 
|- ( ph -> A e. RR )
dvne0.b 
|- ( ph -> B e. RR )
dvne0.f 
|- ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
dvne0.d 
|- ( ph -> dom ( RR _D F ) = ( A (,) B ) )
dvne0.z 
|- ( ph -> -. 0 e. ran ( RR _D F ) )
Assertion dvne0f1 
|- ( ph -> F : ( A [,] B ) -1-1-> RR )

Proof

Step Hyp Ref Expression
1 dvne0.a 
 |-  ( ph -> A e. RR )
2 dvne0.b 
 |-  ( ph -> B e. RR )
3 dvne0.f 
 |-  ( ph -> F e. ( ( A [,] B ) -cn-> RR ) )
4 dvne0.d 
 |-  ( ph -> dom ( RR _D F ) = ( A (,) B ) )
5 dvne0.z 
 |-  ( ph -> -. 0 e. ran ( RR _D F ) )
6 1 2 3 4 5 dvne0 
 |-  ( ph -> ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) )
7 isof1o 
 |-  ( F Isom < , < ( ( A [,] B ) , ran F ) -> F : ( A [,] B ) -1-1-onto-> ran F )
8 isof1o 
 |-  ( F Isom < , `' < ( ( A [,] B ) , ran F ) -> F : ( A [,] B ) -1-1-onto-> ran F )
9 7 8 jaoi 
 |-  ( ( F Isom < , < ( ( A [,] B ) , ran F ) \/ F Isom < , `' < ( ( A [,] B ) , ran F ) ) -> F : ( A [,] B ) -1-1-onto-> ran F )
10 f1of1 
 |-  ( F : ( A [,] B ) -1-1-onto-> ran F -> F : ( A [,] B ) -1-1-> ran F )
11 6 9 10 3syl 
 |-  ( ph -> F : ( A [,] B ) -1-1-> ran F )
12 cncff 
 |-  ( F e. ( ( A [,] B ) -cn-> RR ) -> F : ( A [,] B ) --> RR )
13 frn 
 |-  ( F : ( A [,] B ) --> RR -> ran F C_ RR )
14 3 12 13 3syl 
 |-  ( ph -> ran F C_ RR )
15 f1ss 
 |-  ( ( F : ( A [,] B ) -1-1-> ran F /\ ran F C_ RR ) -> F : ( A [,] B ) -1-1-> RR )
16 11 14 15 syl2anc 
 |-  ( ph -> F : ( A [,] B ) -1-1-> RR )