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Theorem dvle 19844
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 18876 . . . . 5  |-  ( ( x  e.  ( M [,] N )  |->  A )  e.  ( ( M [,] N )
-cn-> RR )  ->  (
x  e.  ( M [,] N )  |->  A ) : ( M [,] N ) --> RR )
42, 3syl 16 . . . 4  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  A ) : ( M [,] N ) --> RR )
5 eqid 2404 . . . . 5  |-  ( x  e.  ( M [,] N )  |->  A )  =  ( x  e.  ( M [,] N
)  |->  A )
65fmpt 5849 . . . 4  |-  ( A. x  e.  ( M [,] N ) A  e.  RR  <->  ( x  e.  ( M [,] N
)  |->  A ) : ( M [,] N
) --> RR )
74, 6sylibr 204 . . 3  |-  ( ph  ->  A. x  e.  ( M [,] N ) A  e.  RR )
8 dvle.r . . . . 5  |-  ( x  =  Y  ->  A  =  R )
98eleq1d 2470 . . . 4  |-  ( x  =  Y  ->  ( A  e.  RR  <->  R  e.  RR ) )
109rspcv 3008 . . 3  |-  ( Y  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) A  e.  RR  ->  R  e.  RR ) )
111, 7, 10sylc 58 . 2  |-  ( ph  ->  R  e.  RR )
12 dvle.c . . . . . 6  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  C )  e.  ( ( M [,] N
) -cn-> RR ) )
13 cncff 18876 . . . . . 6  |-  ( ( x  e.  ( M [,] N )  |->  C )  e.  ( ( M [,] N )
-cn-> RR )  ->  (
x  e.  ( M [,] N )  |->  C ) : ( M [,] N ) --> RR )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  C ) : ( M [,] N ) --> RR )
15 eqid 2404 . . . . . 6  |-  ( x  e.  ( M [,] N )  |->  C )  =  ( x  e.  ( M [,] N
)  |->  C )
1615fmpt 5849 . . . . 5  |-  ( A. x  e.  ( M [,] N ) C  e.  RR  <->  ( x  e.  ( M [,] N
)  |->  C ) : ( M [,] N
) --> RR )
1714, 16sylibr 204 . . . 4  |-  ( ph  ->  A. x  e.  ( M [,] N ) C  e.  RR )
18 dvle.s . . . . . 6  |-  ( x  =  Y  ->  C  =  S )
1918eleq1d 2470 . . . . 5  |-  ( x  =  Y  ->  ( C  e.  RR  <->  S  e.  RR ) )
2019rspcv 3008 . . . 4  |-  ( Y  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) C  e.  RR  ->  S  e.  RR ) )
211, 17, 20sylc 58 . . 3  |-  ( ph  ->  S  e.  RR )
22 dvle.x . . . 4  |-  ( ph  ->  X  e.  ( M [,] N ) )
23 dvle.q . . . . . 6  |-  ( x  =  X  ->  C  =  Q )
2423eleq1d 2470 . . . . 5  |-  ( x  =  X  ->  ( C  e.  RR  <->  Q  e.  RR ) )
2524rspcv 3008 . . . 4  |-  ( X  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) C  e.  RR  ->  Q  e.  RR ) )
2622, 17, 25sylc 58 . . 3  |-  ( ph  ->  Q  e.  RR )
2721, 26resubcld 9421 . 2  |-  ( ph  ->  ( S  -  Q
)  e.  RR )
28 dvle.p . . . . 5  |-  ( x  =  X  ->  A  =  P )
2928eleq1d 2470 . . . 4  |-  ( x  =  X  ->  ( A  e.  RR  <->  P  e.  RR ) )
3029rspcv 3008 . . 3  |-  ( X  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) A  e.  RR  ->  P  e.  RR ) )
3122, 7, 30sylc 58 . 2  |-  ( ph  ->  P  e.  RR )
3211recnd 9070 . . . . 5  |-  ( ph  ->  R  e.  CC )
3326recnd 9070 . . . . . 6  |-  ( ph  ->  Q  e.  CC )
3421recnd 9070 . . . . . 6  |-  ( ph  ->  S  e.  CC )
3533, 34subcld 9367 . . . . 5  |-  ( ph  ->  ( Q  -  S
)  e.  CC )
3632, 35addcomd 9224 . . . 4  |-  ( ph  ->  ( R  +  ( Q  -  S ) )  =  ( ( Q  -  S )  +  R ) )
3732, 34, 33subsub2d 9396 . . . 4  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( R  +  ( Q  -  S ) ) )
3833, 34, 32subsubd 9395 . . . 4  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  =  ( ( Q  -  S )  +  R ) )
3936, 37, 383eqtr4d 2446 . . 3  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( Q  -  ( S  -  R ) ) )
4021, 11resubcld 9421 . . . 4  |-  ( ph  ->  ( S  -  R
)  e.  RR )
41 dvle.m . . . . . 6  |-  ( ph  ->  M  e.  RR )
42 dvle.n . . . . . 6  |-  ( ph  ->  N  e.  RR )
43 eqid 2404 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4443subcn 18849 . . . . . . 7  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
45 ax-resscn 9003 . . . . . . 7  |-  RR  C_  CC
46 resubcl 9321 . . . . . . 7  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  -  A
)  e.  RR )
4743, 44, 12, 2, 45, 46cncfmpt2ss 18898 . . . . . 6  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  ( C  -  A
) )  e.  ( ( M [,] N
) -cn-> RR ) )
48 ioossicc 10952 . . . . . . . . . . . . . . . . 17  |-  ( M (,) N )  C_  ( M [,] N )
4948sseli 3304 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( M (,) N )  ->  x  e.  ( M [,] N
) )
5017r19.21bi 2764 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  RR )
5149, 50sylan2 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  RR )
52 eqid 2404 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  C )  =  ( x  e.  ( M (,) N
)  |->  C )
5351, 52fmptd 5852 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  C ) : ( M (,) N ) --> RR )
54 ioossre 10928 . . . . . . . . . . . . . 14  |-  ( M (,) N )  C_  RR
55 dvfre 19790 . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . 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 5031 . . . . . . . . . . . . . . 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 9098 . . . . . . . . . . . . . . . . . . 19  |-  Rel  <_
6160brrelex2i 4878 . . . . . . . . . . . . . . . . . 18  |-  ( B  <_  D  ->  D  e.  _V )
6259, 61syl 16 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  D  e.  _V )
6362ralrimiva 2749 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  ( M (,) N ) D  e.  _V )
64 dmmptg 5326 . . . . . . . . . . . . . . . 16  |-  ( A. x  e.  ( M (,) N ) D  e. 
_V  ->  dom  ( x  e.  ( M (,) N
)  |->  D )  =  ( M (,) N
) )
6563, 64syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( x  e.  ( M (,) N
)  |->  D )  =  ( M (,) N
) )
6658, 65eqtrd 2436 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  C ) )  =  ( M (,) N ) )
6757, 66feq12d 5541 . . . . . . . . . . . . 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 202 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  D ) : ( M (,) N ) --> RR )
69 eqid 2404 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  D )  =  ( x  e.  ( M (,) N
)  |->  D )
7069fmpt 5849 . . . . . . . . . . . 12  |-  ( A. x  e.  ( M (,) N ) D  e.  RR  <->  ( x  e.  ( M (,) N
)  |->  D ) : ( M (,) N
) --> RR )
7168, 70sylibr 204 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( M (,) N ) D  e.  RR )
7271r19.21bi 2764 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  D  e.  RR )
737r19.21bi 2764 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  RR )
7449, 73sylan2 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  RR )
75 eqid 2404 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  A )  =  ( x  e.  ( M (,) N
)  |->  A )
7674, 75fmptd 5852 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  A ) : ( M (,) N ) --> RR )
77 dvfre 19790 . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . 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 5031 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  dom  ( x  e.  ( M (,) N )  |->  B ) )
8160brrelexi 4877 . . . . . . . . . . . . . . . . . 18  |-  ( B  <_  D  ->  B  e.  _V )
8259, 81syl 16 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  e.  _V )
8382ralrimiva 2749 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  ( M (,) N ) B  e.  _V )
84 dmmptg 5326 . . . . . . . . . . . . . . . 16  |-  ( A. x  e.  ( M (,) N ) B  e. 
_V  ->  dom  ( x  e.  ( M (,) N
)  |->  B )  =  ( M (,) N
) )
8583, 84syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( x  e.  ( M (,) N
)  |->  B )  =  ( M (,) N
) )
8680, 85eqtrd 2436 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  ( M (,) N ) )
8779, 86feq12d 5541 . . . . . . . . . . . . 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 202 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  B ) : ( M (,) N ) --> RR )
89 eqid 2404 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  B )  =  ( x  e.  ( M (,) N
)  |->  B )
9089fmpt 5849 . . . . . . . . . . . 12  |-  ( A. x  e.  ( M (,) N ) B  e.  RR  <->  ( x  e.  ( M (,) N
)  |->  B ) : ( M (,) N
) --> RR )
9188, 90sylibr 204 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( M (,) N ) B  e.  RR )
9291r19.21bi 2764 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  e.  RR )
9372, 92resubcld 9421 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  RR )
9472, 92subge0d 9572 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( 0  <_  ( D  -  B )  <->  B  <_  D ) )
9559, 94mpbird 224 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  0  <_  ( D  -  B ) )
96 elrege0 10963 . . . . . . . . 9  |-  ( ( D  -  B )  e.  ( 0 [,) 
+oo )  <->  ( ( D  -  B )  e.  RR  /\  0  <_ 
( D  -  B
) ) )
9793, 95, 96sylanbrc 646 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  ( 0 [,)  +oo )
)
98 eqid 2404 . . . . . . . 8  |-  ( x  e.  ( M (,) N )  |->  ( D  -  B ) )  =  ( x  e.  ( M (,) N
)  |->  ( D  -  B ) )
9997, 98fmptd 5852 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  ( D  -  B
) ) : ( M (,) N ) --> ( 0 [,)  +oo ) )
10045a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
101 iccssre 10948 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M [,] N
)  C_  RR )
10241, 42, 101syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( M [,] N
)  C_  RR )
10350, 73resubcld 9421 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  RR )
104103recnd 9070 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  CC )
10543tgioo2 18787 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
106 iccntr 18805 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
10741, 42, 106syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
108100, 102, 104, 105, 43, 107dvmptntr 19810 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( RR  _D  ( x  e.  ( M (,) N )  |->  ( C  -  A ) ) ) )
109 reex 9037 . . . . . . . . . . . 12  |-  RR  e.  _V
110109prid1 3872 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
111110a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
11250recnd 9070 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  CC )
11349, 112sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  CC )
11473recnd 9070 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  CC )
11549, 114sylan2 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  CC )
116111, 113, 62, 57, 115, 82, 79dvmptsub 19806 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
117108, 116eqtrd 2436 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
118117feq1d 5539 . . . . . . 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 224 . . . . . 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 19843 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  X )  <_  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  Y
) )
12223, 28oveq12d 6058 . . . . . . 7  |-  ( x  =  X  ->  ( C  -  A )  =  ( Q  -  P ) )
123 eqid 2404 . . . . . . 7  |-  ( x  e.  ( M [,] N )  |->  ( C  -  A ) )  =  ( x  e.  ( M [,] N
)  |->  ( C  -  A ) )
124 ovex 6065 . . . . . . 7  |-  ( C  -  A )  e. 
_V
125122, 123, 124fvmpt3i 5768 . . . . . 6  |-  ( X  e.  ( M [,] N )  ->  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  X
)  =  ( Q  -  P ) )
12622, 125syl 16 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  X )  =  ( Q  -  P ) )
12718, 8oveq12d 6058 . . . . . . 7  |-  ( x  =  Y  ->  ( C  -  A )  =  ( S  -  R ) )
128127, 123, 124fvmpt3i 5768 . . . . . 6  |-  ( Y  e.  ( M [,] N )  ->  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  Y
)  =  ( S  -  R ) )
1291, 128syl 16 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  Y )  =  ( S  -  R ) )
130121, 126, 1293brtr3d 4201 . . . 4  |-  ( ph  ->  ( Q  -  P
)  <_  ( S  -  R ) )
13126, 31, 40, 130subled 9585 . . 3  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  <_  P )
13239, 131eqbrtrd 4192 . 2  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  <_  P )
13311, 27, 31, 132subled 9585 1  |-  ( ph  ->  ( R  -  P
)  <_  ( S  -  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   {cpr 3775   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    +oocpnf 9073    <_ cle 9077    - cmin 9247   (,)cioo 10872   [,)cico 10874   [,]cicc 10875   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658   intcnt 17036   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  dvfsumle  19858  dvfsumlem2  19864  loglesqr  20595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
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