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Theorem dvle 22577
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 21566 . . . . 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 2454 . . . . 5  |-  ( x  e.  ( M [,] N )  |->  A )  =  ( x  e.  ( M [,] N
)  |->  A )
65fmpt 6028 . . . 4  |-  ( A. x  e.  ( M [,] N ) A  e.  RR  <->  ( x  e.  ( M [,] N
)  |->  A ) : ( M [,] N
) --> RR )
74, 6sylibr 212 . . 3  |-  ( ph  ->  A. x  e.  ( M [,] N ) A  e.  RR )
8 dvle.r . . . . 5  |-  ( x  =  Y  ->  A  =  R )
98eleq1d 2523 . . . 4  |-  ( x  =  Y  ->  ( A  e.  RR  <->  R  e.  RR ) )
109rspcv 3203 . . 3  |-  ( Y  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) A  e.  RR  ->  R  e.  RR ) )
111, 7, 10sylc 60 . 2  |-  ( ph  ->  R  e.  RR )
12 dvle.c . . . . . 6  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  C )  e.  ( ( M [,] N
) -cn-> RR ) )
13 cncff 21566 . . . . . 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 2454 . . . . . 6  |-  ( x  e.  ( M [,] N )  |->  C )  =  ( x  e.  ( M [,] N
)  |->  C )
1615fmpt 6028 . . . . 5  |-  ( A. x  e.  ( M [,] N ) C  e.  RR  <->  ( x  e.  ( M [,] N
)  |->  C ) : ( M [,] N
) --> RR )
1714, 16sylibr 212 . . . 4  |-  ( ph  ->  A. x  e.  ( M [,] N ) C  e.  RR )
18 dvle.s . . . . . 6  |-  ( x  =  Y  ->  C  =  S )
1918eleq1d 2523 . . . . 5  |-  ( x  =  Y  ->  ( C  e.  RR  <->  S  e.  RR ) )
2019rspcv 3203 . . . 4  |-  ( Y  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) C  e.  RR  ->  S  e.  RR ) )
211, 17, 20sylc 60 . . 3  |-  ( ph  ->  S  e.  RR )
22 dvle.x . . . 4  |-  ( ph  ->  X  e.  ( M [,] N ) )
23 dvle.q . . . . . 6  |-  ( x  =  X  ->  C  =  Q )
2423eleq1d 2523 . . . . 5  |-  ( x  =  X  ->  ( C  e.  RR  <->  Q  e.  RR ) )
2524rspcv 3203 . . . 4  |-  ( X  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) C  e.  RR  ->  Q  e.  RR ) )
2622, 17, 25sylc 60 . . 3  |-  ( ph  ->  Q  e.  RR )
2721, 26resubcld 9983 . 2  |-  ( ph  ->  ( S  -  Q
)  e.  RR )
28 dvle.p . . . . 5  |-  ( x  =  X  ->  A  =  P )
2928eleq1d 2523 . . . 4  |-  ( x  =  X  ->  ( A  e.  RR  <->  P  e.  RR ) )
3029rspcv 3203 . . 3  |-  ( X  e.  ( M [,] N )  ->  ( A. x  e.  ( M [,] N ) A  e.  RR  ->  P  e.  RR ) )
3122, 7, 30sylc 60 . 2  |-  ( ph  ->  P  e.  RR )
3211recnd 9611 . . . . 5  |-  ( ph  ->  R  e.  CC )
3326recnd 9611 . . . . . 6  |-  ( ph  ->  Q  e.  CC )
3421recnd 9611 . . . . . 6  |-  ( ph  ->  S  e.  CC )
3533, 34subcld 9922 . . . . 5  |-  ( ph  ->  ( Q  -  S
)  e.  CC )
3632, 35addcomd 9771 . . . 4  |-  ( ph  ->  ( R  +  ( Q  -  S ) )  =  ( ( Q  -  S )  +  R ) )
3732, 34, 33subsub2d 9951 . . . 4  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( R  +  ( Q  -  S ) ) )
3833, 34, 32subsubd 9950 . . . 4  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  =  ( ( Q  -  S )  +  R ) )
3936, 37, 383eqtr4d 2505 . . 3  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( Q  -  ( S  -  R ) ) )
4021, 11resubcld 9983 . . . 4  |-  ( ph  ->  ( S  -  R
)  e.  RR )
41 dvle.m . . . . . 6  |-  ( ph  ->  M  e.  RR )
42 dvle.n . . . . . 6  |-  ( ph  ->  N  e.  RR )
43 eqid 2454 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4443subcn 21539 . . . . . . 7  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
45 ax-resscn 9538 . . . . . . 7  |-  RR  C_  CC
46 resubcl 9874 . . . . . . 7  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  -  A
)  e.  RR )
4743, 44, 12, 2, 45, 46cncfmpt2ss 21588 . . . . . 6  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  ( C  -  A
) )  e.  ( ( M [,] N
) -cn-> RR ) )
48 ioossicc 11613 . . . . . . . . . . . . . . . . 17  |-  ( M (,) N )  C_  ( M [,] N )
4948sseli 3485 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( M (,) N )  ->  x  e.  ( M [,] N
) )
5017r19.21bi 2823 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  RR )
5149, 50sylan2 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  RR )
52 eqid 2454 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  C )  =  ( x  e.  ( M (,) N
)  |->  C )
5351, 52fmptd 6031 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  C ) : ( M (,) N ) --> RR )
54 ioossre 11589 . . . . . . . . . . . . . 14  |-  ( M (,) N )  C_  RR
55 dvfre 22523 . . . . . . . . . . . . . 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 660 . . . . . . . . . . . . 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 5194 . . . . . . . . . . . . . . 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 9640 . . . . . . . . . . . . . . . . . . 19  |-  Rel  <_
6160brrelex2i 5030 . . . . . . . . . . . . . . . . . 18  |-  ( B  <_  D  ->  D  e.  _V )
6259, 61syl 16 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  D  e.  _V )
6362ralrimiva 2868 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  ( M (,) N ) D  e.  _V )
64 dmmptg 5487 . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  C ) )  =  ( M (,) N ) )
6757, 66feq12d 5702 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  D ) : ( M (,) N ) --> RR )
69 eqid 2454 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  D )  =  ( x  e.  ( M (,) N
)  |->  D )
7069fmpt 6028 . . . . . . . . . . . 12  |-  ( A. x  e.  ( M (,) N ) D  e.  RR  <->  ( x  e.  ( M (,) N
)  |->  D ) : ( M (,) N
) --> RR )
7168, 70sylibr 212 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( M (,) N ) D  e.  RR )
7271r19.21bi 2823 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  D  e.  RR )
737r19.21bi 2823 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  RR )
7449, 73sylan2 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  RR )
75 eqid 2454 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  A )  =  ( x  e.  ( M (,) N
)  |->  A )
7674, 75fmptd 6031 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  A ) : ( M (,) N ) --> RR )
77 dvfre 22523 . . . . . . . . . . . . . 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 660 . . . . . . . . . . . . 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 5194 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  dom  ( x  e.  ( M (,) N )  |->  B ) )
8160brrelexi 5029 . . . . . . . . . . . . . . . . . 18  |-  ( B  <_  D  ->  B  e.  _V )
8259, 81syl 16 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  e.  _V )
8382ralrimiva 2868 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. x  e.  ( M (,) N ) B  e.  _V )
84 dmmptg 5487 . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  ( M (,) N ) )
8779, 86feq12d 5702 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  B ) : ( M (,) N ) --> RR )
89 eqid 2454 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  B )  =  ( x  e.  ( M (,) N
)  |->  B )
9089fmpt 6028 . . . . . . . . . . . 12  |-  ( A. x  e.  ( M (,) N ) B  e.  RR  <->  ( x  e.  ( M (,) N
)  |->  B ) : ( M (,) N
) --> RR )
9188, 90sylibr 212 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( M (,) N ) B  e.  RR )
9291r19.21bi 2823 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  e.  RR )
9372, 92resubcld 9983 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  RR )
9472, 92subge0d 10138 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( 0  <_  ( D  -  B )  <->  B  <_  D ) )
9559, 94mpbird 232 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  0  <_  ( D  -  B ) )
96 elrege0 11630 . . . . . . . . 9  |-  ( ( D  -  B )  e.  ( 0 [,) +oo )  <->  ( ( D  -  B )  e.  RR  /\  0  <_ 
( D  -  B
) ) )
9793, 95, 96sylanbrc 662 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  ( 0 [,) +oo )
)
98 eqid 2454 . . . . . . . 8  |-  ( x  e.  ( M (,) N )  |->  ( D  -  B ) )  =  ( x  e.  ( M (,) N
)  |->  ( D  -  B ) )
9997, 98fmptd 6031 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  ( D  -  B
) ) : ( M (,) N ) --> ( 0 [,) +oo ) )
10045a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
101 iccssre 11609 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M [,] N
)  C_  RR )
10241, 42, 101syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  ( M [,] N
)  C_  RR )
10350, 73resubcld 9983 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  RR )
104103recnd 9611 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  CC )
10543tgioo2 21477 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
106 iccntr 21495 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
10741, 42, 106syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
108100, 102, 104, 105, 43, 107dvmptntr 22543 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( RR  _D  ( x  e.  ( M (,) N )  |->  ( C  -  A ) ) ) )
109 reelprrecn 9573 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
110109a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
11150recnd 9611 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  CC )
11249, 111sylan2 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  CC )
11373recnd 9611 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  CC )
11449, 113sylan2 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  CC )
115110, 112, 62, 57, 114, 82, 79dvmptsub 22539 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
116108, 115eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
117116feq1d 5699 . . . . . . 7  |-  ( ph  ->  ( ( RR  _D  ( x  e.  ( M [,] N )  |->  ( C  -  A ) ) ) : ( M (,) N ) --> ( 0 [,) +oo ) 
<->  ( x  e.  ( M (,) N ) 
|->  ( D  -  B
) ) : ( M (,) N ) --> ( 0 [,) +oo ) ) )
11899, 117mpbird 232 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) ) : ( M (,) N ) --> ( 0 [,) +oo ) )
119 dvle.l . . . . . 6  |-  ( ph  ->  X  <_  Y )
12041, 42, 47, 118, 22, 1, 119dvge0 22576 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  X )  <_  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  Y
) )
12123, 28oveq12d 6288 . . . . . . 7  |-  ( x  =  X  ->  ( C  -  A )  =  ( Q  -  P ) )
122 eqid 2454 . . . . . . 7  |-  ( x  e.  ( M [,] N )  |->  ( C  -  A ) )  =  ( x  e.  ( M [,] N
)  |->  ( C  -  A ) )
123 ovex 6298 . . . . . . 7  |-  ( C  -  A )  e. 
_V
124121, 122, 123fvmpt3i 5935 . . . . . 6  |-  ( X  e.  ( M [,] N )  ->  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  X
)  =  ( Q  -  P ) )
12522, 124syl 16 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  X )  =  ( Q  -  P ) )
12618, 8oveq12d 6288 . . . . . . 7  |-  ( x  =  Y  ->  ( C  -  A )  =  ( S  -  R ) )
127126, 122, 123fvmpt3i 5935 . . . . . 6  |-  ( Y  e.  ( M [,] N )  ->  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  Y
)  =  ( S  -  R ) )
1281, 127syl 16 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  Y )  =  ( S  -  R ) )
129120, 125, 1283brtr3d 4468 . . . 4  |-  ( ph  ->  ( Q  -  P
)  <_  ( S  -  R ) )
13026, 31, 40, 129subled 10151 . . 3  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  <_  P )
13139, 130eqbrtrd 4459 . 2  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  <_  P )
13211, 27, 31, 131subled 10151 1  |-  ( ph  ->  ( R  -  P
)  <_  ( S  -  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    C_ wss 3461   {cpr 4018   class class class wbr 4439    |-> cmpt 4497   dom cdm 4988   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484   +oocpnf 9614    <_ cle 9618    - cmin 9796   (,)cioo 11532   [,)cico 11534   [,]cicc 11535   TopOpenctopn 14914   topGenctg 14930  ℂfldccnfld 18618   intcnt 19688   -cn->ccncf 21549    _D cdv 22436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-haus 19986  df-cmp 20057  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-cncf 21551  df-limc 22439  df-dv 22440
This theorem is referenced by:  dvfsumle  22591  dvfsumlem2  22597  loglesqrt  23303
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