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Theorem dvle 19370
Description: If  A
( x ) ,  C ( x ) are differentiable functions and  A `  <_  C `
, then for  x  <_  y,  A ( y )  -  A ( x )  <_  C
( y )  -  C ( x ). (Contributed by Mario Carneiro, 16-May-2016.)
Hypotheses
Ref Expression
dvle.m  |-  ( ph  ->  M  e.  RR )
dvle.n  |-  ( ph  ->  N  e.  RR )
dvle.a  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  A )  e.  ( ( M [,] N
) -cn-> RR ) )
dvle.b  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  A ) )  =  ( x  e.  ( M (,) N )  |->  B ) )
dvle.c  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  C )  e.  ( ( M [,] N
) -cn-> RR ) )
dvle.d  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  C ) )  =  ( x  e.  ( M (,) N )  |->  D ) )
dvle.f  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  <_  D )
dvle.x  |-  ( ph  ->  X  e.  ( M [,] N ) )
dvle.y  |-  ( ph  ->  Y  e.  ( M [,] N ) )
dvle.l  |-  ( ph  ->  X  <_  Y )
dvle.p  |-  ( x  =  X  ->  A  =  P )
dvle.q  |-  ( x  =  X  ->  C  =  Q )
dvle.r  |-  ( x  =  Y  ->  A  =  R )
dvle.s  |-  ( x  =  Y  ->  C  =  S )
Assertion
Ref Expression
dvle  |-  ( ph  ->  ( R  -  P
)  <_  ( S  -  Q ) )
Distinct variable groups:    x, M    x, N    x, P    x, Q    x, R    x, S    x, X    ph, x    x, Y
Allowed substitution hints:    A( x)    B( x)    C( x)    D( x)

Proof of Theorem dvle
StepHypRef Expression
1 dvle.y . . 3  |-  ( ph  ->  Y  e.  ( M [,] N ) )
2 dvle.a . . . . 5  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  A )  e.  ( ( M [,] N
) -cn-> RR ) )
3 cncff 18413 . . . . 5  |-  ( ( x  e.  ( M [,] N )  |->  A )  e.  ( ( M [,] N )
-cn-> RR )  ->  (
x  e.  ( M [,] N )  |->  A ) : ( M [,] N ) --> RR )
42, 3syl 15 . . . 4  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  A ) : ( M [,] N ) --> RR )
5 eqid 2296 . . . . 5  |-  ( x  e.  ( M [,] N )  |->  A )  =  ( x  e.  ( M [,] N
)  |->  A )
65fmpt 5697 . . . 4  |-  ( A. x  e.  ( M [,] N ) A  e.  RR  <->  ( x  e.  ( M [,] N
)  |->  A ) : ( M [,] N
) --> RR )
74, 6sylibr 203 . . 3  |-  ( ph  ->  A. x  e.  ( M [,] N ) A  e.  RR )
8 dvle.r . . . . 5  |-  ( x  =  Y  ->  A  =  R )
98eleq1d 2362 . . . 4  |-  ( x  =  Y  ->  ( A  e.  RR  <->  R  e.  RR ) )
109rspcv 2893 . . 3  |-  ( Y  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) A  e.  RR  ->  R  e.  RR ) )
111, 7, 10sylc 56 . 2  |-  ( ph  ->  R  e.  RR )
12 dvle.c . . . . . 6  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  C )  e.  ( ( M [,] N
) -cn-> RR ) )
13 cncff 18413 . . . . . 6  |-  ( ( x  e.  ( M [,] N )  |->  C )  e.  ( ( M [,] N )
-cn-> RR )  ->  (
x  e.  ( M [,] N )  |->  C ) : ( M [,] N ) --> RR )
1412, 13syl 15 . . . . 5  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  C ) : ( M [,] N ) --> RR )
15 eqid 2296 . . . . . 6  |-  ( x  e.  ( M [,] N )  |->  C )  =  ( x  e.  ( M [,] N
)  |->  C )
1615fmpt 5697 . . . . 5  |-  ( A. x  e.  ( M [,] N ) C  e.  RR  <->  ( x  e.  ( M [,] N
)  |->  C ) : ( M [,] N
) --> RR )
1714, 16sylibr 203 . . . 4  |-  ( ph  ->  A. x  e.  ( M [,] N ) C  e.  RR )
18 dvle.s . . . . . 6  |-  ( x  =  Y  ->  C  =  S )
1918eleq1d 2362 . . . . 5  |-  ( x  =  Y  ->  ( C  e.  RR  <->  S  e.  RR ) )
2019rspcv 2893 . . . 4  |-  ( Y  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) C  e.  RR  ->  S  e.  RR ) )
211, 17, 20sylc 56 . . 3  |-  ( ph  ->  S  e.  RR )
22 dvle.x . . . 4  |-  ( ph  ->  X  e.  ( M [,] N ) )
23 dvle.q . . . . . 6  |-  ( x  =  X  ->  C  =  Q )
2423eleq1d 2362 . . . . 5  |-  ( x  =  X  ->  ( C  e.  RR  <->  Q  e.  RR ) )
2524rspcv 2893 . . . 4  |-  ( X  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) C  e.  RR  ->  Q  e.  RR ) )
2622, 17, 25sylc 56 . . 3  |-  ( ph  ->  Q  e.  RR )
2721, 26resubcld 9227 . 2  |-  ( ph  ->  ( S  -  Q
)  e.  RR )
28 dvle.p . . . . 5  |-  ( x  =  X  ->  A  =  P )
2928eleq1d 2362 . . . 4  |-  ( x  =  X  ->  ( A  e.  RR  <->  P  e.  RR ) )
3029rspcv 2893 . . 3  |-  ( X  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) A  e.  RR  ->  P  e.  RR ) )
3122, 7, 30sylc 56 . 2  |-  ( ph  ->  P  e.  RR )
3211recnd 8877 . . . . 5  |-  ( ph  ->  R  e.  CC )
3326recnd 8877 . . . . . 6  |-  ( ph  ->  Q  e.  CC )
3421recnd 8877 . . . . . 6  |-  ( ph  ->  S  e.  CC )
3533, 34subcld 9173 . . . . 5  |-  ( ph  ->  ( Q  -  S
)  e.  CC )
3632, 35addcomd 9030 . . . 4  |-  ( ph  ->  ( R  +  ( Q  -  S ) )  =  ( ( Q  -  S )  +  R ) )
3732, 34, 33subsub2d 9202 . . . 4  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( R  +  ( Q  -  S ) ) )
3833, 34, 32subsubd 9201 . . . 4  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  =  ( ( Q  -  S )  +  R ) )
3936, 37, 383eqtr4d 2338 . . 3  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( Q  -  ( S  -  R ) ) )
4021, 11resubcld 9227 . . . 4  |-  ( ph  ->  ( S  -  R
)  e.  RR )
41 dvle.m . . . . . 6  |-  ( ph  ->  M  e.  RR )
42 dvle.n . . . . . 6  |-  ( ph  ->  N  e.  RR )
43 eqid 2296 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4443subcn 18386 . . . . . . 7  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
45 ax-resscn 8810 . . . . . . 7  |-  RR  C_  CC
46 resubcl 9127 . . . . . . 7  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  -  A
)  e.  RR )
4743, 44, 12, 2, 45, 46cncfmpt2ss 18435 . . . . . 6  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  ( C  -  A
) )  e.  ( ( M [,] N
) -cn-> RR ) )
48 ioossicc 10751 . . . . . . . . . . . . . . . . 17  |-  ( M (,) N )  C_  ( M [,] N )
4948sseli 3189 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( M (,) N )  ->  x  e.  ( M [,] N
) )
5017r19.21bi 2654 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  RR )
5149, 50sylan2 460 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  RR )
52 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  C )  =  ( x  e.  ( M (,) N
)  |->  C )
5351, 52fmptd 5700 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  C ) : ( M (,) N ) --> RR )
54 ioossre 10728 . . . . . . . . . . . . . 14  |-  ( M (,) N )  C_  RR
55 dvfre 19316 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  ( M (,) N ) 
|->  C ) : ( M (,) N ) --> RR  /\  ( M (,) N )  C_  RR )  ->  ( RR 
_D  ( x  e.  ( M (,) N
)  |->  C ) ) : dom  ( RR 
_D  ( x  e.  ( M (,) N
)  |->  C ) ) --> RR )
5653, 54, 55sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  C ) ) : dom  ( RR  _D  (
x  e.  ( M (,) N )  |->  C ) ) --> RR )
57 dvle.d . . . . . . . . . . . . . 14  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  C ) )  =  ( x  e.  ( M (,) N )  |->  D ) )
5857dmeqd 4897 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  C ) )  =  dom  ( x  e.  ( M (,) N )  |->  D ) )
59 dvle.f . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  <_  D )
60 lerel 8905 . . . . . . . . . . . . . . . . . . 19  |-  Rel  <_
6160brrelex2i 4746 . . . . . . . . . . . . . . . . . 18  |-  ( B  <_  D  ->  D  e.  _V )
6259, 61syl 15 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  D  e.  _V )
6362ralrimiva 2639 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  ( M (,) N ) D  e.  _V )
64 dmmptg 5186 . . . . . . . . . . . . . . . 16  |-  ( A. x  e.  ( M (,) N ) D  e. 
_V  ->  dom  ( x  e.  ( M (,) N
)  |->  D )  =  ( M (,) N
) )
6563, 64syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( x  e.  ( M (,) N
)  |->  D )  =  ( M (,) N
) )
6658, 65eqtrd 2328 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  C ) )  =  ( M (,) N ) )
6757, 66feq12d 5397 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( RR  _D  ( x  e.  ( M (,) N )  |->  C ) ) : dom  ( RR  _D  (
x  e.  ( M (,) N )  |->  C ) ) --> RR  <->  ( x  e.  ( M (,) N
)  |->  D ) : ( M (,) N
) --> RR ) )
6856, 67mpbid 201 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  D ) : ( M (,) N ) --> RR )
69 eqid 2296 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  D )  =  ( x  e.  ( M (,) N
)  |->  D )
7069fmpt 5697 . . . . . . . . . . . 12  |-  ( A. x  e.  ( M (,) N ) D  e.  RR  <->  ( x  e.  ( M (,) N
)  |->  D ) : ( M (,) N
) --> RR )
7168, 70sylibr 203 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( M (,) N ) D  e.  RR )
7271r19.21bi 2654 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  D  e.  RR )
737r19.21bi 2654 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  RR )
7449, 73sylan2 460 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  RR )
75 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  A )  =  ( x  e.  ( M (,) N
)  |->  A )
7674, 75fmptd 5700 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  A ) : ( M (,) N ) --> RR )
77 dvfre 19316 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  ( M (,) N ) 
|->  A ) : ( M (,) N ) --> RR  /\  ( M (,) N )  C_  RR )  ->  ( RR 
_D  ( x  e.  ( M (,) N
)  |->  A ) ) : dom  ( RR 
_D  ( x  e.  ( M (,) N
)  |->  A ) ) --> RR )
7876, 54, 77sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  A ) ) : dom  ( RR  _D  (
x  e.  ( M (,) N )  |->  A ) ) --> RR )
79 dvle.b . . . . . . . . . . . . . 14  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  A ) )  =  ( x  e.  ( M (,) N )  |->  B ) )
8079dmeqd 4897 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  dom  ( x  e.  ( M (,) N )  |->  B ) )
8160brrelexi 4745 . . . . . . . . . . . . . . . . . 18  |-  ( B  <_  D  ->  B  e.  _V )
8259, 81syl 15 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  e.  _V )
8382ralrimiva 2639 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  ( M (,) N ) B  e.  _V )
84 dmmptg 5186 . . . . . . . . . . . . . . . 16  |-  ( A. x  e.  ( M (,) N ) B  e. 
_V  ->  dom  ( x  e.  ( M (,) N
)  |->  B )  =  ( M (,) N
) )
8583, 84syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( x  e.  ( M (,) N
)  |->  B )  =  ( M (,) N
) )
8680, 85eqtrd 2328 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  ( M (,) N ) )
8779, 86feq12d 5397 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) ) : dom  ( RR  _D  (
x  e.  ( M (,) N )  |->  A ) ) --> RR  <->  ( x  e.  ( M (,) N
)  |->  B ) : ( M (,) N
) --> RR ) )
8878, 87mpbid 201 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  B ) : ( M (,) N ) --> RR )
89 eqid 2296 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  B )  =  ( x  e.  ( M (,) N
)  |->  B )
9089fmpt 5697 . . . . . . . . . . . 12  |-  ( A. x  e.  ( M (,) N ) B  e.  RR  <->  ( x  e.  ( M (,) N
)  |->  B ) : ( M (,) N
) --> RR )
9188, 90sylibr 203 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( M (,) N ) B  e.  RR )
9291r19.21bi 2654 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  e.  RR )
9372, 92resubcld 9227 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  RR )
9472, 92subge0d 9378 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( 0  <_  ( D  -  B )  <->  B  <_  D ) )
9559, 94mpbird 223 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  0  <_  ( D  -  B ) )
96 elrege0 10762 . . . . . . . . 9  |-  ( ( D  -  B )  e.  ( 0 [,) 
+oo )  <->  ( ( D  -  B )  e.  RR  /\  0  <_ 
( D  -  B
) ) )
9793, 95, 96sylanbrc 645 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  ( 0 [,)  +oo )
)
98 eqid 2296 . . . . . . . 8  |-  ( x  e.  ( M (,) N )  |->  ( D  -  B ) )  =  ( x  e.  ( M (,) N
)  |->  ( D  -  B ) )
9997, 98fmptd 5700 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  ( D  -  B
) ) : ( M (,) N ) --> ( 0 [,)  +oo ) )
10045a1i 10 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
101 iccssre 10747 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M [,] N
)  C_  RR )
10241, 42, 101syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( M [,] N
)  C_  RR )
10350, 73resubcld 9227 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  RR )
104103recnd 8877 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  CC )
10543tgioo2 18325 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
106 iccntr 18342 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
10741, 42, 106syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
108100, 102, 104, 105, 43, 107dvmptntr 19336 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( RR  _D  ( x  e.  ( M (,) N )  |->  ( C  -  A ) ) ) )
109 reex 8844 . . . . . . . . . . . 12  |-  RR  e.  _V
110109prid1 3747 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
111110a1i 10 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
11250recnd 8877 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  CC )
11349, 112sylan2 460 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  CC )
11473recnd 8877 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  CC )
11549, 114sylan2 460 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  CC )
116111, 113, 62, 57, 115, 82, 79dvmptsub 19332 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
117108, 116eqtrd 2328 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
118117feq1d 5395 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  ( x  e.  ( M [,] N )  |->  ( C  -  A ) ) ) : ( M (,) N ) --> ( 0 [,)  +oo ) 
<->  ( x  e.  ( M (,) N ) 
|->  ( D  -  B
) ) : ( M (,) N ) --> ( 0 [,)  +oo ) ) )
11999, 118mpbird 223 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) ) : ( M (,) N ) --> ( 0 [,)  +oo ) )
120 dvle.l . . . . . 6  |-  ( ph  ->  X  <_  Y )
12141, 42, 47, 119, 22, 1, 120dvge0 19369 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  X )  <_  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  Y
) )
12223, 28oveq12d 5892 . . . . . . 7  |-  ( x  =  X  ->  ( C  -  A )  =  ( Q  -  P ) )
123 eqid 2296 . . . . . . 7  |-  ( x  e.  ( M [,] N )  |->  ( C  -  A ) )  =  ( x  e.  ( M [,] N
)  |->  ( C  -  A ) )
124 ovex 5899 . . . . . . 7  |-  ( C  -  A )  e. 
_V
125122, 123, 124fvmpt3i 5621 . . . . . 6  |-  ( X  e.  ( M [,] N )  ->  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  X
)  =  ( Q  -  P ) )
12622, 125syl 15 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  X )  =  ( Q  -  P ) )
12718, 8oveq12d 5892 . . . . . . 7  |-  ( x  =  Y  ->  ( C  -  A )  =  ( S  -  R ) )
128127, 123, 124fvmpt3i 5621 . . . . . 6  |-  ( Y  e.  ( M [,] N )  ->  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  Y
)  =  ( S  -  R ) )
1291, 128syl 15 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  Y )  =  ( S  -  R ) )
130121, 126, 1293brtr3d 4068 . . . 4  |-  ( ph  ->  ( Q  -  P
)  <_  ( S  -  R ) )
13126, 31, 40, 130subled 9391 . . 3  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  <_  P )
13239, 131eqbrtrd 4059 . 2  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  <_  P )
13311, 27, 31, 132subled 9391 1  |-  ( ph  ->  ( R  -  P
)  <_  ( S  -  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   {cpr 3654   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756    +oocpnf 8880    <_ cle 8884    - cmin 9053   (,)cioo 10672   [,)cico 10674   [,]cicc 10675   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393   intcnt 16770   -cn->ccncf 18396    _D cdv 19229
This theorem is referenced by:  dvfsumle  19384  dvfsumlem2  19390  loglesqr  20114
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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