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Theorem dvle 19893
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 18925 . . . . 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 2438 . . . . 5  |-  ( x  e.  ( M [,] N )  |->  A )  =  ( x  e.  ( M [,] N
)  |->  A )
65fmpt 5892 . . . 4  |-  ( A. x  e.  ( M [,] N ) A  e.  RR  <->  ( x  e.  ( M [,] N
)  |->  A ) : ( M [,] N
) --> RR )
74, 6sylibr 205 . . 3  |-  ( ph  ->  A. x  e.  ( M [,] N ) A  e.  RR )
8 dvle.r . . . . 5  |-  ( x  =  Y  ->  A  =  R )
98eleq1d 2504 . . . 4  |-  ( x  =  Y  ->  ( A  e.  RR  <->  R  e.  RR ) )
109rspcv 3050 . . 3  |-  ( Y  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) A  e.  RR  ->  R  e.  RR ) )
111, 7, 10sylc 59 . 2  |-  ( ph  ->  R  e.  RR )
12 dvle.c . . . . . 6  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  C )  e.  ( ( M [,] N
) -cn-> RR ) )
13 cncff 18925 . . . . . 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 2438 . . . . . 6  |-  ( x  e.  ( M [,] N )  |->  C )  =  ( x  e.  ( M [,] N
)  |->  C )
1615fmpt 5892 . . . . 5  |-  ( A. x  e.  ( M [,] N ) C  e.  RR  <->  ( x  e.  ( M [,] N
)  |->  C ) : ( M [,] N
) --> RR )
1714, 16sylibr 205 . . . 4  |-  ( ph  ->  A. x  e.  ( M [,] N ) C  e.  RR )
18 dvle.s . . . . . 6  |-  ( x  =  Y  ->  C  =  S )
1918eleq1d 2504 . . . . 5  |-  ( x  =  Y  ->  ( C  e.  RR  <->  S  e.  RR ) )
2019rspcv 3050 . . . 4  |-  ( Y  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) C  e.  RR  ->  S  e.  RR ) )
211, 17, 20sylc 59 . . 3  |-  ( ph  ->  S  e.  RR )
22 dvle.x . . . 4  |-  ( ph  ->  X  e.  ( M [,] N ) )
23 dvle.q . . . . . 6  |-  ( x  =  X  ->  C  =  Q )
2423eleq1d 2504 . . . . 5  |-  ( x  =  X  ->  ( C  e.  RR  <->  Q  e.  RR ) )
2524rspcv 3050 . . . 4  |-  ( X  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) C  e.  RR  ->  Q  e.  RR ) )
2622, 17, 25sylc 59 . . 3  |-  ( ph  ->  Q  e.  RR )
2721, 26resubcld 9467 . 2  |-  ( ph  ->  ( S  -  Q
)  e.  RR )
28 dvle.p . . . . 5  |-  ( x  =  X  ->  A  =  P )
2928eleq1d 2504 . . . 4  |-  ( x  =  X  ->  ( A  e.  RR  <->  P  e.  RR ) )
3029rspcv 3050 . . 3  |-  ( X  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) A  e.  RR  ->  P  e.  RR ) )
3122, 7, 30sylc 59 . 2  |-  ( ph  ->  P  e.  RR )
3211recnd 9116 . . . . 5  |-  ( ph  ->  R  e.  CC )
3326recnd 9116 . . . . . 6  |-  ( ph  ->  Q  e.  CC )
3421recnd 9116 . . . . . 6  |-  ( ph  ->  S  e.  CC )
3533, 34subcld 9413 . . . . 5  |-  ( ph  ->  ( Q  -  S
)  e.  CC )
3632, 35addcomd 9270 . . . 4  |-  ( ph  ->  ( R  +  ( Q  -  S ) )  =  ( ( Q  -  S )  +  R ) )
3732, 34, 33subsub2d 9442 . . . 4  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( R  +  ( Q  -  S ) ) )
3833, 34, 32subsubd 9441 . . . 4  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  =  ( ( Q  -  S )  +  R ) )
3936, 37, 383eqtr4d 2480 . . 3  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( Q  -  ( S  -  R ) ) )
4021, 11resubcld 9467 . . . 4  |-  ( ph  ->  ( S  -  R
)  e.  RR )
41 dvle.m . . . . . 6  |-  ( ph  ->  M  e.  RR )
42 dvle.n . . . . . 6  |-  ( ph  ->  N  e.  RR )
43 eqid 2438 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4443subcn 18898 . . . . . . 7  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
45 ax-resscn 9049 . . . . . . 7  |-  RR  C_  CC
46 resubcl 9367 . . . . . . 7  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  -  A
)  e.  RR )
4743, 44, 12, 2, 45, 46cncfmpt2ss 18947 . . . . . 6  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  ( C  -  A
) )  e.  ( ( M [,] N
) -cn-> RR ) )
48 ioossicc 10998 . . . . . . . . . . . . . . . . 17  |-  ( M (,) N )  C_  ( M [,] N )
4948sseli 3346 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( M (,) N )  ->  x  e.  ( M [,] N
) )
5017r19.21bi 2806 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  RR )
5149, 50sylan2 462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  RR )
52 eqid 2438 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  C )  =  ( x  e.  ( M (,) N
)  |->  C )
5351, 52fmptd 5895 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  C ) : ( M (,) N ) --> RR )
54 ioossre 10974 . . . . . . . . . . . . . 14  |-  ( M (,) N )  C_  RR
55 dvfre 19839 . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . 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 5074 . . . . . . . . . . . . . . 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 9144 . . . . . . . . . . . . . . . . . . 19  |-  Rel  <_
6160brrelex2i 4921 . . . . . . . . . . . . . . . . . 18  |-  ( B  <_  D  ->  D  e.  _V )
6259, 61syl 16 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  D  e.  _V )
6362ralrimiva 2791 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  ( M (,) N ) D  e.  _V )
64 dmmptg 5369 . . . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  C ) )  =  ( M (,) N ) )
6757, 66feq12d 5584 . . . . . . . . . . . . 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 203 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  D ) : ( M (,) N ) --> RR )
69 eqid 2438 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  D )  =  ( x  e.  ( M (,) N
)  |->  D )
7069fmpt 5892 . . . . . . . . . . . 12  |-  ( A. x  e.  ( M (,) N ) D  e.  RR  <->  ( x  e.  ( M (,) N
)  |->  D ) : ( M (,) N
) --> RR )
7168, 70sylibr 205 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( M (,) N ) D  e.  RR )
7271r19.21bi 2806 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  D  e.  RR )
737r19.21bi 2806 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  RR )
7449, 73sylan2 462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  RR )
75 eqid 2438 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  A )  =  ( x  e.  ( M (,) N
)  |->  A )
7674, 75fmptd 5895 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  A ) : ( M (,) N ) --> RR )
77 dvfre 19839 . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . 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 5074 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  dom  ( x  e.  ( M (,) N )  |->  B ) )
8160brrelexi 4920 . . . . . . . . . . . . . . . . . 18  |-  ( B  <_  D  ->  B  e.  _V )
8259, 81syl 16 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  e.  _V )
8382ralrimiva 2791 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  ( M (,) N ) B  e.  _V )
84 dmmptg 5369 . . . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  ( M (,) N ) )
8779, 86feq12d 5584 . . . . . . . . . . . . 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 203 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  B ) : ( M (,) N ) --> RR )
89 eqid 2438 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  B )  =  ( x  e.  ( M (,) N
)  |->  B )
9089fmpt 5892 . . . . . . . . . . . 12  |-  ( A. x  e.  ( M (,) N ) B  e.  RR  <->  ( x  e.  ( M (,) N
)  |->  B ) : ( M (,) N
) --> RR )
9188, 90sylibr 205 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( M (,) N ) B  e.  RR )
9291r19.21bi 2806 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  e.  RR )
9372, 92resubcld 9467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  RR )
9472, 92subge0d 9618 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( 0  <_  ( D  -  B )  <->  B  <_  D ) )
9559, 94mpbird 225 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  0  <_  ( D  -  B ) )
96 elrege0 11009 . . . . . . . . 9  |-  ( ( D  -  B )  e.  ( 0 [,) 
+oo )  <->  ( ( D  -  B )  e.  RR  /\  0  <_ 
( D  -  B
) ) )
9793, 95, 96sylanbrc 647 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  ( 0 [,)  +oo )
)
98 eqid 2438 . . . . . . . 8  |-  ( x  e.  ( M (,) N )  |->  ( D  -  B ) )  =  ( x  e.  ( M (,) N
)  |->  ( D  -  B ) )
9997, 98fmptd 5895 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  ( D  -  B
) ) : ( M (,) N ) --> ( 0 [,)  +oo ) )
10045a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
101 iccssre 10994 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M [,] N
)  C_  RR )
10241, 42, 101syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( M [,] N
)  C_  RR )
10350, 73resubcld 9467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  RR )
104103recnd 9116 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  CC )
10543tgioo2 18836 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
106 iccntr 18854 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
10741, 42, 106syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
108100, 102, 104, 105, 43, 107dvmptntr 19859 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( RR  _D  ( x  e.  ( M (,) N )  |->  ( C  -  A ) ) ) )
109 reex 9083 . . . . . . . . . . . 12  |-  RR  e.  _V
110109prid1 3914 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
111110a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
11250recnd 9116 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  CC )
11349, 112sylan2 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  CC )
11473recnd 9116 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  CC )
11549, 114sylan2 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  CC )
116111, 113, 62, 57, 115, 82, 79dvmptsub 19855 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
117108, 116eqtrd 2470 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
118117feq1d 5582 . . . . . . 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 225 . . . . . 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 19892 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  X )  <_  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  Y
) )
12223, 28oveq12d 6101 . . . . . . 7  |-  ( x  =  X  ->  ( C  -  A )  =  ( Q  -  P ) )
123 eqid 2438 . . . . . . 7  |-  ( x  e.  ( M [,] N )  |->  ( C  -  A ) )  =  ( x  e.  ( M [,] N
)  |->  ( C  -  A ) )
124 ovex 6108 . . . . . . 7  |-  ( C  -  A )  e. 
_V
125122, 123, 124fvmpt3i 5811 . . . . . 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 6101 . . . . . . 7  |-  ( x  =  Y  ->  ( C  -  A )  =  ( S  -  R ) )
128127, 123, 124fvmpt3i 5811 . . . . . 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 4243 . . . 4  |-  ( ph  ->  ( Q  -  P
)  <_  ( S  -  R ) )
13126, 31, 40, 130subled 9631 . . 3  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  <_  P )
13239, 131eqbrtrd 4234 . 2  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  <_  P )
13311, 27, 31, 132subled 9631 1  |-  ( ph  ->  ( R  -  P
)  <_  ( S  -  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322   {cpr 3817   class class class wbr 4214    e. cmpt 4268   dom cdm 4880   ran crn 4881   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992    + caddc 8995    +oocpnf 9119    <_ cle 9123    - cmin 9293   (,)cioo 10918   [,)cico 10920   [,]cicc 10921   TopOpenctopn 13651   topGenctg 13667  ℂfldccnfld 16705   intcnt 17083   -cn->ccncf 18908    _D cdv 19752
This theorem is referenced by:  dvfsumle  19907  dvfsumlem2  19913  loglesqr  20644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-cmp 17452  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756
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