MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvle Structured version   Unicode version

Theorem dvle 21323
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 20313 . . . . 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 2435 . . . . 5  |-  ( x  e.  ( M [,] N )  |->  A )  =  ( x  e.  ( M [,] N
)  |->  A )
65fmpt 5854 . . . 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 2501 . . . 4  |-  ( x  =  Y  ->  ( A  e.  RR  <->  R  e.  RR ) )
109rspcv 3060 . . 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 20313 . . . . . 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 2435 . . . . . 6  |-  ( x  e.  ( M [,] N )  |->  C )  =  ( x  e.  ( M [,] N
)  |->  C )
1615fmpt 5854 . . . . 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 2501 . . . . 5  |-  ( x  =  Y  ->  ( C  e.  RR  <->  S  e.  RR ) )
2019rspcv 3060 . . . 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 2501 . . . . 5  |-  ( x  =  X  ->  ( C  e.  RR  <->  Q  e.  RR ) )
2524rspcv 3060 . . . 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 9766 . 2  |-  ( ph  ->  ( S  -  Q
)  e.  RR )
28 dvle.p . . . . 5  |-  ( x  =  X  ->  A  =  P )
2928eleq1d 2501 . . . 4  |-  ( x  =  X  ->  ( A  e.  RR  <->  P  e.  RR ) )
3029rspcv 3060 . . 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 9402 . . . . 5  |-  ( ph  ->  R  e.  CC )
3326recnd 9402 . . . . . 6  |-  ( ph  ->  Q  e.  CC )
3421recnd 9402 . . . . . 6  |-  ( ph  ->  S  e.  CC )
3533, 34subcld 9709 . . . . 5  |-  ( ph  ->  ( Q  -  S
)  e.  CC )
3632, 35addcomd 9561 . . . 4  |-  ( ph  ->  ( R  +  ( Q  -  S ) )  =  ( ( Q  -  S )  +  R ) )
3732, 34, 33subsub2d 9738 . . . 4  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( R  +  ( Q  -  S ) ) )
3833, 34, 32subsubd 9737 . . . 4  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  =  ( ( Q  -  S )  +  R ) )
3936, 37, 383eqtr4d 2477 . . 3  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  =  ( Q  -  ( S  -  R ) ) )
4021, 11resubcld 9766 . . . 4  |-  ( ph  ->  ( S  -  R
)  e.  RR )
41 dvle.m . . . . . 6  |-  ( ph  ->  M  e.  RR )
42 dvle.n . . . . . 6  |-  ( ph  ->  N  e.  RR )
43 eqid 2435 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4443subcn 20286 . . . . . . 7  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
45 ax-resscn 9329 . . . . . . 7  |-  RR  C_  CC
46 resubcl 9663 . . . . . . 7  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  -  A
)  e.  RR )
4743, 44, 12, 2, 45, 46cncfmpt2ss 20335 . . . . . 6  |-  ( ph  ->  ( x  e.  ( M [,] N ) 
|->  ( C  -  A
) )  e.  ( ( M [,] N
) -cn-> RR ) )
48 ioossicc 11371 . . . . . . . . . . . . . . . . 17  |-  ( M (,) N )  C_  ( M [,] N )
4948sseli 3342 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( M (,) N )  ->  x  e.  ( M [,] N
) )
5017r19.21bi 2806 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  RR )
5149, 50sylan2 471 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  RR )
52 eqid 2435 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  C )  =  ( x  e.  ( M (,) N
)  |->  C )
5351, 52fmptd 5857 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  C ) : ( M (,) N ) --> RR )
54 ioossre 11347 . . . . . . . . . . . . . 14  |-  ( M (,) N )  C_  RR
55 dvfre 21269 . . . . . . . . . . . . . 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 657 . . . . . . . . . . . . 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 9431 . . . . . . . . . . . . . . . . . . 19  |-  Rel  <_
6160brrelex2i 4869 . . . . . . . . . . . . . . . . . 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 5325 . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  C ) )  =  ( M (,) N ) )
6757, 66feq12d 5538 . . . . . . . . . . . . 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 2435 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  D )  =  ( x  e.  ( M (,) N
)  |->  D )
7069fmpt 5854 . . . . . . . . . . . 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 2806 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  D  e.  RR )
737r19.21bi 2806 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  RR )
7449, 73sylan2 471 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  RR )
75 eqid 2435 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( M (,) N )  |->  A )  =  ( x  e.  ( M (,) N
)  |->  A )
7674, 75fmptd 5857 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  A ) : ( M (,) N ) --> RR )
77 dvfre 21269 . . . . . . . . . . . . . 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 657 . . . . . . . . . . . . 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 4868 . . . . . . . . . . . . . . . . . 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 5325 . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  ( ph  ->  dom  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  ( M (,) N ) )
8779, 86feq12d 5538 . . . . . . . . . . . . 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 2435 . . . . . . . . . . . . 13  |-  ( x  e.  ( M (,) N )  |->  B )  =  ( x  e.  ( M (,) N
)  |->  B )
9089fmpt 5854 . . . . . . . . . . . 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 2806 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  B  e.  RR )
9372, 92resubcld 9766 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  RR )
9472, 92subge0d 9919 . . . . . . . . . 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 11382 . . . . . . . . 9  |-  ( ( D  -  B )  e.  ( 0 [,) +oo )  <->  ( ( D  -  B )  e.  RR  /\  0  <_ 
( D  -  B
) ) )
9793, 95, 96sylanbrc 659 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  ( D  -  B )  e.  ( 0 [,) +oo )
)
98 eqid 2435 . . . . . . . 8  |-  ( x  e.  ( M (,) N )  |->  ( D  -  B ) )  =  ( x  e.  ( M (,) N
)  |->  ( D  -  B ) )
9997, 98fmptd 5857 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( M (,) N ) 
|->  ( D  -  B
) ) : ( M (,) N ) --> ( 0 [,) +oo ) )
10045a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  C_  CC )
101 iccssre 11367 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M [,] N
)  C_  RR )
10241, 42, 101syl2anc 656 . . . . . . . . . 10  |-  ( ph  ->  ( M [,] N
)  C_  RR )
10350, 73resubcld 9766 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  RR )
104103recnd 9402 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  ( C  -  A )  e.  CC )
10543tgioo2 20224 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
106 iccntr 20242 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
10741, 42, 106syl2anc 656 . . . . . . . . . 10  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( M [,] N ) )  =  ( M (,) N
) )
108100, 102, 104, 105, 43, 107dvmptntr 21289 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( RR  _D  ( x  e.  ( M (,) N )  |->  ( C  -  A ) ) ) )
109 reelprrecn 9364 . . . . . . . . . . 11  |-  RR  e.  { RR ,  CC }
110109a1i 11 . . . . . . . . . 10  |-  ( ph  ->  RR  e.  { RR ,  CC } )
11150recnd 9402 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  C  e.  CC )
11249, 111sylan2 471 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  C  e.  CC )
11373recnd 9402 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( M [,] N ) )  ->  A  e.  CC )
11449, 113sylan2 471 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( M (,) N ) )  ->  A  e.  CC )
115110, 112, 62, 57, 114, 82, 79dvmptsub 21285 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M (,) N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
116108, 115eqtrd 2467 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  ( M [,] N )  |->  ( C  -  A ) ) )  =  ( x  e.  ( M (,) N )  |->  ( D  -  B ) ) )
117116feq1d 5536 . . . . . . 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 21322 . . . . 5  |-  ( ph  ->  ( ( x  e.  ( M [,] N
)  |->  ( C  -  A ) ) `  X )  <_  (
( x  e.  ( M [,] N ) 
|->  ( C  -  A
) ) `  Y
) )
12123, 28oveq12d 6100 . . . . . . 7  |-  ( x  =  X  ->  ( C  -  A )  =  ( Q  -  P ) )
122 eqid 2435 . . . . . . 7  |-  ( x  e.  ( M [,] N )  |->  ( C  -  A ) )  =  ( x  e.  ( M [,] N
)  |->  ( C  -  A ) )
123 ovex 6107 . . . . . . 7  |-  ( C  -  A )  e. 
_V
124121, 122, 123fvmpt3i 5768 . . . . . 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 6100 . . . . . . 7  |-  ( x  =  Y  ->  ( C  -  A )  =  ( S  -  R ) )
127126, 122, 123fvmpt3i 5768 . . . . . 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 4311 . . . 4  |-  ( ph  ->  ( Q  -  P
)  <_  ( S  -  R ) )
13026, 31, 40, 129subled 9932 . . 3  |-  ( ph  ->  ( Q  -  ( S  -  R )
)  <_  P )
13139, 130eqbrtrd 4302 . 2  |-  ( ph  ->  ( R  -  ( S  -  Q )
)  <_  P )
13211, 27, 31, 131subled 9932 1  |-  ( ph  ->  ( R  -  P
)  <_  ( S  -  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1757   A.wral 2707   _Vcvv 2964    C_ wss 3318   {cpr 3869   class class class wbr 4282    e. cmpt 4340   dom cdm 4829   ran crn 4830   -->wf 5404   ` cfv 5408  (class class class)co 6082   CCcc 9270   RRcr 9271   0cc0 9272    + caddc 9275   +oocpnf 9405    <_ cle 9409    - cmin 9585   (,)cioo 11290   [,)cico 11292   [,]cicc 11293   TopOpenctopn 14345   topGenctg 14361  ℂfldccnfld 17664   intcnt 18465   -cn->ccncf 20296    _D cdv 21182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-inf2 7837  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349  ax-pre-sup 9350  ax-addf 9351  ax-mulf 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-iin 4164  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-se 4669  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-isom 5417  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6311  df-om 6468  df-1st 6568  df-2nd 6569  df-supp 6682  df-recs 6820  df-rdg 6854  df-1o 6910  df-2o 6911  df-oadd 6914  df-er 7091  df-map 7206  df-pm 7207  df-ixp 7254  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-fsupp 7611  df-fi 7651  df-sup 7681  df-oi 7714  df-card 8099  df-cda 8327  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-div 9984  df-nn 10313  df-2 10370  df-3 10371  df-4 10372  df-5 10373  df-6 10374  df-7 10375  df-8 10376  df-9 10377  df-10 10378  df-n0 10570  df-z 10637  df-dec 10746  df-uz 10852  df-q 10944  df-rp 10982  df-xneg 11079  df-xadd 11080  df-xmul 11081  df-ioo 11294  df-ico 11296  df-icc 11297  df-fz 11427  df-fzo 11535  df-seq 11793  df-exp 11852  df-hash 12090  df-cj 12574  df-re 12575  df-im 12576  df-sqr 12710  df-abs 12711  df-struct 14161  df-ndx 14162  df-slot 14163  df-base 14164  df-sets 14165  df-ress 14166  df-plusg 14236  df-mulr 14237  df-starv 14238  df-sca 14239  df-vsca 14240  df-ip 14241  df-tset 14242  df-ple 14243  df-ds 14245  df-unif 14246  df-hom 14247  df-cco 14248  df-rest 14346  df-topn 14347  df-0g 14365  df-gsum 14366  df-topgen 14367  df-pt 14368  df-prds 14371  df-xrs 14425  df-qtop 14430  df-imas 14431  df-xps 14433  df-mre 14509  df-mrc 14510  df-acs 14512  df-mnd 15400  df-submnd 15450  df-mulg 15530  df-cntz 15817  df-cmn 16261  df-psmet 17655  df-xmet 17656  df-met 17657  df-bl 17658  df-mopn 17659  df-fbas 17660  df-fg 17661  df-cnfld 17665  df-top 18347  df-bases 18349  df-topon 18350  df-topsp 18351  df-cld 18467  df-ntr 18468  df-cls 18469  df-nei 18546  df-lp 18584  df-perf 18585  df-cn 18675  df-cnp 18676  df-haus 18763  df-cmp 18834  df-tx 18979  df-hmeo 19172  df-fil 19263  df-fm 19355  df-flim 19356  df-flf 19357  df-xms 19739  df-ms 19740  df-tms 19741  df-cncf 20298  df-limc 21185  df-dv 21186
This theorem is referenced by:  dvfsumle  21337  dvfsumlem2  21343  loglesqr  22083
  Copyright terms: Public domain W3C validator