Metamath Proof Explorer

Theorem dvferm2

Description: One-sided version of dvferm . A point U which is the local maximum of its left neighborhood has derivative at least zero. (Contributed by Mario Carneiro, 24-Feb-2015) (Proof shortened by Mario Carneiro, 28-Dec-2016)

Ref Expression
Hypotheses dvferm.a 
|- ( ph -> F : X --> RR )
dvferm.b 
|- ( ph -> X C_ RR )
dvferm.u 
|- ( ph -> U e. ( A (,) B ) )
dvferm.s 
|- ( ph -> ( A (,) B ) C_ X )
dvferm.d 
|- ( ph -> U e. dom ( RR _D F ) )
dvferm2.r 
|- ( ph -> A. y e. ( A (,) U ) ( F ` y ) <_ ( F ` U ) )
Assertion dvferm2 
|- ( ph -> 0 <_ ( ( RR _D F ) ` U ) )

Proof

Step Hyp Ref Expression
1 dvferm.a 
 |-  ( ph -> F : X --> RR )
2 dvferm.b 
 |-  ( ph -> X C_ RR )
3 dvferm.u 
 |-  ( ph -> U e. ( A (,) B ) )
4 dvferm.s 
 |-  ( ph -> ( A (,) B ) C_ X )
5 dvferm.d 
 |-  ( ph -> U e. dom ( RR _D F ) )
6 dvferm2.r 
 |-  ( ph -> A. y e. ( A (,) U ) ( F ` y ) <_ ( F ` U ) )
7 fveq2 
 |-  ( x = z -> ( F ` x ) = ( F ` z ) )
8 7 oveq1d 
 |-  ( x = z -> ( ( F ` x ) - ( F ` U ) ) = ( ( F ` z ) - ( F ` U ) ) )
9 oveq1 
 |-  ( x = z -> ( x - U ) = ( z - U ) )
10 8 9 oveq12d 
 |-  ( x = z -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) = ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) )
11 eqid 
 |-  ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) = ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) )
12 ovex 
 |-  ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) e. _V
13 10 11 12 fvmpt 
 |-  ( z e. ( X \ { U } ) -> ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) = ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) )
14 13 fvoveq1d 
 |-  ( z e. ( X \ { U } ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) = ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) )
15 id 
 |-  ( y = -u ( ( RR _D F ) ` U ) -> y = -u ( ( RR _D F ) ` U ) )
16 14 15 breqan12rd 
 |-  ( ( y = -u ( ( RR _D F ) ` U ) /\ z e. ( X \ { U } ) ) -> ( ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y <-> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
17 16 imbi2d 
 |-  ( ( y = -u ( ( RR _D F ) ` U ) /\ z e. ( X \ { U } ) ) -> ( ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) <-> ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) )
18 17 ralbidva 
 |-  ( y = -u ( ( RR _D F ) ` U ) -> ( A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) <-> A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) )
19 18 rexbidv 
 |-  ( y = -u ( ( RR _D F ) ` U ) -> ( E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) <-> E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) )
20 dvf 
 |-  ( RR _D F ) : dom ( RR _D F ) --> CC
21 ffun 
 |-  ( ( RR _D F ) : dom ( RR _D F ) --> CC -> Fun ( RR _D F ) )
22 funfvbrb 
 |-  ( Fun ( RR _D F ) -> ( U e. dom ( RR _D F ) <-> U ( RR _D F ) ( ( RR _D F ) ` U ) ) )
23 20 21 22 mp2b 
 |-  ( U e. dom ( RR _D F ) <-> U ( RR _D F ) ( ( RR _D F ) ` U ) )
24 5 23 sylib 
 |-  ( ph -> U ( RR _D F ) ( ( RR _D F ) ` U ) )
25 eqid 
 |-  ( ( TopOpen ` CCfld ) |`t RR ) = ( ( TopOpen ` CCfld ) |`t RR )
26 eqid 
 |-  ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
27 ax-resscn 
 |-  RR C_ CC
28 27 a1i 
 |-  ( ph -> RR C_ CC )
29 fss 
 |-  ( ( F : X --> RR /\ RR C_ CC ) -> F : X --> CC )
30 1 27 29 sylancl 
 |-  ( ph -> F : X --> CC )
31 25 26 11 28 30 2 eldv 
 |-  ( ph -> ( U ( RR _D F ) ( ( RR _D F ) ` U ) <-> ( U e. ( ( int ` ( ( TopOpen ` CCfld ) |`t RR ) ) ` X ) /\ ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) limCC U ) ) ) )
32 24 31 mpbid 
 |-  ( ph -> ( U e. ( ( int ` ( ( TopOpen ` CCfld ) |`t RR ) ) ` X ) /\ ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) limCC U ) ) )
33 32 simprd 
 |-  ( ph -> ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) limCC U ) )
34 33 adantr 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) limCC U ) )
35 2 27 sstrdi 
 |-  ( ph -> X C_ CC )
36 4 3 sseldd 
 |-  ( ph -> U e. X )
37 30 35 36 dvlem 
 |-  ( ( ph /\ x e. ( X \ { U } ) ) -> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) e. CC )
38 37 fmpttd 
 |-  ( ph -> ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) : ( X \ { U } ) --> CC )
39 38 adantr 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) : ( X \ { U } ) --> CC )
40 35 adantr 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> X C_ CC )
41 40 ssdifssd 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( X \ { U } ) C_ CC )
42 35 36 sseldd 
 |-  ( ph -> U e. CC )
43 42 adantr 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> U e. CC )
44 39 41 43 ellimc3 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( ( ( RR _D F ) ` U ) e. ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) limCC U ) <-> ( ( ( RR _D F ) ` U ) e. CC /\ A. y e. RR+ E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) ) ) )
45 34 44 mpbid 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( ( ( RR _D F ) ` U ) e. CC /\ A. y e. RR+ E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) ) )
46 45 simprd 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> A. y e. RR+ E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( x e. ( X \ { U } ) |-> ( ( ( F ` x ) - ( F ` U ) ) / ( x - U ) ) ) ` z ) - ( ( RR _D F ) ` U ) ) ) < y ) )
47 dvfre 
 |-  ( ( F : X --> RR /\ X C_ RR ) -> ( RR _D F ) : dom ( RR _D F ) --> RR )
48 1 2 47 syl2anc 
 |-  ( ph -> ( RR _D F ) : dom ( RR _D F ) --> RR )
49 48 5 ffvelrnd 
 |-  ( ph -> ( ( RR _D F ) ` U ) e. RR )
50 49 adantr 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> ( ( RR _D F ) ` U ) e. RR )
51 50 renegcld 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR )
52 49 lt0neg1d 
 |-  ( ph -> ( ( ( RR _D F ) ` U ) < 0 <-> 0 < -u ( ( RR _D F ) ` U ) ) )
53 52 biimpa 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> 0 < -u ( ( RR _D F ) ` U ) )
54 51 53 elrpd 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -u ( ( RR _D F ) ` U ) e. RR+ )
55 19 46 54 rspcdva 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
56 1 ad3antrrr 
 |-  ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> F : X --> RR )
57 2 ad3antrrr 
 |-  ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> X C_ RR )
58 3 ad3antrrr 
 |-  ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> U e. ( A (,) B ) )
59 4 ad3antrrr 
 |-  ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> ( A (,) B ) C_ X )
60 5 ad3antrrr 
 |-  ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> U e. dom ( RR _D F ) )
61 6 ad3antrrr 
 |-  ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> A. y e. ( A (,) U ) ( F ` y ) <_ ( F ` U ) )
62 simpllr 
 |-  ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> ( ( RR _D F ) ` U ) < 0 )
63 simplr 
 |-  ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> u e. RR+ )
64 simpr 
 |-  ( ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) ) -> A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
65 eqid 
 |-  ( ( if ( A <_ ( U - u ) , ( U - u ) , A ) + U ) / 2 ) = ( ( if ( A <_ ( U - u ) , ( U - u ) , A ) + U ) / 2 )
66 56 57 58 59 60 61 62 63 64 65 dvferm2lem 
 |-  -. ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) /\ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
67 66 imnani 
 |-  ( ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) /\ u e. RR+ ) -> -. A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
68 67 nrexdv 
 |-  ( ( ph /\ ( ( RR _D F ) ` U ) < 0 ) -> -. E. u e. RR+ A. z e. ( X \ { U } ) ( ( z =/= U /\ ( abs ` ( z - U ) ) < u ) -> ( abs ` ( ( ( ( F ` z ) - ( F ` U ) ) / ( z - U ) ) - ( ( RR _D F ) ` U ) ) ) < -u ( ( RR _D F ) ` U ) ) )
69 55 68 pm2.65da 
 |-  ( ph -> -. ( ( RR _D F ) ` U ) < 0 )
70 0re 
 |-  0 e. RR
71 lenlt 
 |-  ( ( 0 e. RR /\ ( ( RR _D F ) ` U ) e. RR ) -> ( 0 <_ ( ( RR _D F ) ` U ) <-> -. ( ( RR _D F ) ` U ) < 0 ) )
72 70 49 71 sylancr 
 |-  ( ph -> ( 0 <_ ( ( RR _D F ) ` U ) <-> -. ( ( RR _D F ) ` U ) < 0 ) )
73 69 72 mpbird 
 |-  ( ph -> 0 <_ ( ( RR _D F ) ` U ) )