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Theorem dyadmbl 21083
Description: Any union of dyadic rational intervals is measurable. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypotheses
Ref Expression
dyadmbl.1  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
dyadmbl.2  |-  G  =  { z  e.  A  |  A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }
dyadmbl.3  |-  ( ph  ->  A  C_  ran  F )
Assertion
Ref Expression
dyadmbl  |-  ( ph  ->  U. ( [,] " A
)  e.  dom  vol )
Distinct variable groups:    x, y    z, w, ph    x, w, y, A, z    z, G   
w, F, x, y, z
Allowed substitution hints:    ph( x, y)    G( x, y, w)

Proof of Theorem dyadmbl
Dummy variables  f 
a  b  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dyadmbl.1 . . 3  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
2 dyadmbl.2 . . 3  |-  G  =  { z  e.  A  |  A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }
3 dyadmbl.3 . . 3  |-  ( ph  ->  A  C_  ran  F )
41, 2, 3dyadmbllem 21082 . 2  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
5 isfinite 7861 . . . 4  |-  ( G  e.  Fin  <->  G  ~<  om )
6 iccf 11391 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
7 ffun 5564 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
8 funiunfv 5968 . . . . . 6  |-  ( Fun 
[,]  ->  U_ n  e.  G  ( [,] `  n )  =  U. ( [,] " G ) )
96, 7, 8mp2b 10 . . . . 5  |-  U_ n  e.  G  ( [,] `  n )  =  U. ( [,] " G )
10 simpr 461 . . . . . 6  |-  ( (
ph  /\  G  e.  Fin )  ->  G  e. 
Fin )
11 ssrab2 3440 . . . . . . . . . . . . . . . 16  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
122, 11eqsstri 3389 . . . . . . . . . . . . . . 15  |-  G  C_  A
1312, 3syl5ss 3370 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  C_  ran  F )
141dyadf 21074 . . . . . . . . . . . . . . . 16  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
15 frn 5568 . . . . . . . . . . . . . . . 16  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
1614, 15ax-mp 5 . . . . . . . . . . . . . . 15  |-  ran  F  C_  (  <_  i^i  ( RR  X.  RR ) )
17 inss2 3574 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
1816, 17sstri 3368 . . . . . . . . . . . . . 14  |-  ran  F  C_  ( RR  X.  RR )
1913, 18syl6ss 3371 . . . . . . . . . . . . 13  |-  ( ph  ->  G  C_  ( RR  X.  RR ) )
2019adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  e.  Fin )  ->  G  C_  ( RR  X.  RR ) )
2120sselda 3359 . . . . . . . . . . 11  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  n  e.  ( RR  X.  RR ) )
22 1st2nd2 6616 . . . . . . . . . . 11  |-  ( n  e.  ( RR  X.  RR )  ->  n  = 
<. ( 1st `  n
) ,  ( 2nd `  n ) >. )
2321, 22syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  n  =  <. ( 1st `  n
) ,  ( 2nd `  n ) >. )
2423fveq2d 5698 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. ) )
25 df-ov 6097 . . . . . . . . 9  |-  ( ( 1st `  n ) [,] ( 2nd `  n
) )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. )
2624, 25syl6eqr 2493 . . . . . . . 8  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( ( 1st `  n
) [,] ( 2nd `  n ) ) )
27 xp1st 6609 . . . . . . . . . 10  |-  ( n  e.  ( RR  X.  RR )  ->  ( 1st `  n )  e.  RR )
2821, 27syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( 1st `  n )  e.  RR )
29 xp2nd 6610 . . . . . . . . . 10  |-  ( n  e.  ( RR  X.  RR )  ->  ( 2nd `  n )  e.  RR )
3021, 29syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( 2nd `  n )  e.  RR )
31 iccmbl 21050 . . . . . . . . 9  |-  ( ( ( 1st `  n
)  e.  RR  /\  ( 2nd `  n )  e.  RR )  -> 
( ( 1st `  n
) [,] ( 2nd `  n ) )  e. 
dom  vol )
3228, 30, 31syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  (
( 1st `  n
) [,] ( 2nd `  n ) )  e. 
dom  vol )
3326, 32eqeltrd 2517 . . . . . . 7  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  e. 
dom  vol )
3433ralrimiva 2802 . . . . . 6  |-  ( (
ph  /\  G  e.  Fin )  ->  A. n  e.  G  ( [,] `  n )  e.  dom  vol )
35 finiunmbl 21028 . . . . . 6  |-  ( ( G  e.  Fin  /\  A. n  e.  G  ( [,] `  n )  e.  dom  vol )  ->  U_ n  e.  G  ( [,] `  n )  e.  dom  vol )
3610, 34, 35syl2anc 661 . . . . 5  |-  ( (
ph  /\  G  e.  Fin )  ->  U_ n  e.  G  ( [,] `  n )  e.  dom  vol )
379, 36syl5eqelr 2528 . . . 4  |-  ( (
ph  /\  G  e.  Fin )  ->  U. ( [,] " G )  e. 
dom  vol )
385, 37sylan2br 476 . . 3  |-  ( (
ph  /\  G  ~<  om )  ->  U. ( [,] " G )  e. 
dom  vol )
39 nnenom 11805 . . . . . . 7  |-  NN  ~~  om
40 ensym 7361 . . . . . . 7  |-  ( G 
~~  om  ->  om  ~~  G )
41 entr 7364 . . . . . . 7  |-  ( ( NN  ~~  om  /\  om 
~~  G )  ->  NN  ~~  G )
4239, 40, 41sylancr 663 . . . . . 6  |-  ( G 
~~  om  ->  NN  ~~  G )
43 bren 7322 . . . . . 6  |-  ( NN 
~~  G  <->  E. f 
f : NN -1-1-onto-> G )
4442, 43sylib 196 . . . . 5  |-  ( G 
~~  om  ->  E. f 
f : NN -1-1-onto-> G )
45 rnco2 5348 . . . . . . . . . 10  |-  ran  ( [,]  o.  f )  =  ( [,] " ran  f )
46 f1ofo 5651 . . . . . . . . . . . . 13  |-  ( f : NN -1-1-onto-> G  ->  f : NN -onto-> G )
4746adantl 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN -onto-> G )
48 forn 5626 . . . . . . . . . . . 12  |-  ( f : NN -onto-> G  ->  ran  f  =  G
)
4947, 48syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  f  =  G )
5049imaeq2d 5172 . . . . . . . . . 10  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ( [,] " ran  f )  =  ( [,] " G
) )
5145, 50syl5eq 2487 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  ( [,]  o.  f )  =  ( [,] " G
) )
5251unieqd 4104 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  =  U. ( [,] " G ) )
53 f1of 5644 . . . . . . . . . 10  |-  ( f : NN -1-1-onto-> G  ->  f : NN
--> G )
5413, 16syl6ss 3371 . . . . . . . . . 10  |-  ( ph  ->  G  C_  (  <_  i^i  ( RR  X.  RR ) ) )
55 fss 5570 . . . . . . . . . 10  |-  ( ( f : NN --> G  /\  G  C_  (  <_  i^i  ( RR  X.  RR ) ) )  -> 
f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
5653, 54, 55syl2anr 478 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
57 fss 5570 . . . . . . . . . . . . . . 15  |-  ( ( f : NN --> G  /\  G  C_  ran  F )  ->  f : NN --> ran  F )
5853, 13, 57syl2anr 478 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> ran  F )
59 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  a  e.  NN )
60 ffvelrn 5844 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  F  /\  a  e.  NN )  ->  ( f `  a )  e.  ran  F )
6158, 59, 60syl2an 477 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  ran  F )
62 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  b  e.  NN )
63 ffvelrn 5844 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  F  /\  b  e.  NN )  ->  ( f `  b )  e.  ran  F )
6458, 62, 63syl2an 477 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  ran  F )
651dyaddisj 21079 . . . . . . . . . . . . 13  |-  ( ( ( f `  a
)  e.  ran  F  /\  ( f `  b
)  e.  ran  F
)  ->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  \/  ( [,] `  (
f `  b )
)  C_  ( [,] `  ( f `  a
) )  \/  (
( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
6661, 64, 65syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  \/  ( [,] `  (
f `  b )
)  C_  ( [,] `  ( f `  a
) )  \/  (
( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
6753adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> G )
68 ffvelrn 5844 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> G  /\  b  e.  NN )  ->  ( f `  b
)  e.  G )
6967, 62, 68syl2an 477 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  G
)
7012, 69sseldi 3357 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  A
)
71 ffvelrn 5844 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> G  /\  a  e.  NN )  ->  ( f `  a
)  e.  G )
7267, 59, 71syl2an 477 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  G
)
73 fveq2 5694 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  a )  ->  ( [,] `  z )  =  ( [,] `  (
f `  a )
) )
7473sseq1d 3386 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  a )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  w ) ) )
75 eqeq1 2449 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  a )  ->  (
z  =  w  <->  ( f `  a )  =  w ) )
7674, 75imbi12d 320 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( f `  a )  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7776ralbidv 2738 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  ( f `  a )  ->  ( A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7877, 2elrab2 3122 . . . . . . . . . . . . . . . . 17  |-  ( ( f `  a )  e.  G  <->  ( (
f `  a )  e.  A  /\  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7978simprbi 464 . . . . . . . . . . . . . . . 16  |-  ( ( f `  a )  e.  G  ->  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) )
8072, 79syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) )
81 fveq2 5694 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  b )  ->  ( [,] `  w )  =  ( [,] `  (
f `  b )
) )
8281sseq2d 3387 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  b )  ->  (
( [,] `  (
f `  a )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  ( f `  b ) ) ) )
83 eqeq2 2452 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  b )  ->  (
( f `  a
)  =  w  <->  ( f `  a )  =  ( f `  b ) ) )
8482, 83imbi12d 320 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( f `  b )  ->  (
( ( [,] `  (
f `  a )
)  C_  ( [,] `  w )  ->  (
f `  a )  =  w )  <->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  ->  ( f `  a )  =  ( f `  b ) ) ) )
8584rspcv 3072 . . . . . . . . . . . . . . 15  |-  ( ( f `  b )  e.  A  ->  ( A. w  e.  A  ( ( [,] `  (
f `  a )
)  C_  ( [,] `  w )  ->  (
f `  a )  =  w )  ->  (
( [,] `  (
f `  a )
)  C_  ( [,] `  ( f `  b
) )  ->  (
f `  a )  =  ( f `  b ) ) ) )
8670, 80, 85sylc 60 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  ->  ( f `  a )  =  ( f `  b ) ) )
87 f1of1 5643 . . . . . . . . . . . . . . . . 17  |-  ( f : NN -1-1-onto-> G  ->  f : NN
-1-1-> G )
8887adantl 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN -1-1-> G )
89 f1fveq 5978 . . . . . . . . . . . . . . . 16  |-  ( ( f : NN -1-1-> G  /\  ( a  e.  NN  /\  b  e.  NN ) )  ->  ( (
f `  a )  =  ( f `  b )  <->  a  =  b ) )
9088, 89sylan 471 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
f `  a )  =  ( f `  b )  <->  a  =  b ) )
91 orc 385 . . . . . . . . . . . . . . 15  |-  ( a  =  b  ->  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
9290, 91syl6bi 228 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
f `  a )  =  ( f `  b )  ->  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) ) )
9386, 92syld 44 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  ->  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) ) )
9412, 72sseldi 3357 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  A
)
95 fveq2 5694 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  b )  ->  ( [,] `  z )  =  ( [,] `  (
f `  b )
) )
9695sseq1d 3386 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  b )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  w ) ) )
97 eqeq1 2449 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  b )  ->  (
z  =  w  <->  ( f `  b )  =  w ) )
9896, 97imbi12d 320 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( f `  b )  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
9998ralbidv 2738 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  ( f `  b )  ->  ( A. w  e.  A  ( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
10099, 2elrab2 3122 . . . . . . . . . . . . . . . . 17  |-  ( ( f `  b )  e.  G  <->  ( (
f `  b )  e.  A  /\  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
101100simprbi 464 . . . . . . . . . . . . . . . 16  |-  ( ( f `  b )  e.  G  ->  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) )
10269, 101syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) )
103 fveq2 5694 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  ( [,] `  w )  =  ( [,] `  (
f `  a )
) )
104103sseq2d 3387 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  a )  ->  (
( [,] `  (
f `  b )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  ( f `  a ) ) ) )
105 eqeq2 2452 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  (
( f `  b
)  =  w  <->  ( f `  b )  =  ( f `  a ) ) )
106 eqcom 2445 . . . . . . . . . . . . . . . . . 18  |-  ( ( f `  b )  =  ( f `  a )  <->  ( f `  a )  =  ( f `  b ) )
107105, 106syl6bb 261 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  a )  ->  (
( f `  b
)  =  w  <->  ( f `  a )  =  ( f `  b ) ) )
108104, 107imbi12d 320 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( f `  a )  ->  (
( ( [,] `  (
f `  b )
)  C_  ( [,] `  w )  ->  (
f `  b )  =  w )  <->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  ->  ( f `  a )  =  ( f `  b ) ) ) )
109108rspcv 3072 . . . . . . . . . . . . . . 15  |-  ( ( f `  a )  e.  A  ->  ( A. w  e.  A  ( ( [,] `  (
f `  b )
)  C_  ( [,] `  w )  ->  (
f `  b )  =  w )  ->  (
( [,] `  (
f `  b )
)  C_  ( [,] `  ( f `  a
) )  ->  (
f `  a )  =  ( f `  b ) ) ) )
11094, 102, 109sylc 60 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  ->  ( f `  a )  =  ( f `  b ) ) )
111110, 92syld 44 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  ->  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) ) )
112 olc 384 . . . . . . . . . . . . . 14  |-  ( ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/)  ->  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) )
113112a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/)  ->  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) ) )
11493, 111, 1133jaod 1282 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
( [,] `  (
f `  a )
)  C_  ( [,] `  ( f `  b
) )  \/  ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  \/  ( ( (,) `  ( f `  a
) )  i^i  ( (,) `  ( f `  b ) ) )  =  (/) )  ->  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) ) )
11566, 114mpd 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( a  =  b  \/  (
( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
116115ralrimivva 2811 . . . . . . . . . 10  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  A. a  e.  NN  A. b  e.  NN  ( a  =  b  \/  ( ( (,) `  ( f `
 a ) )  i^i  ( (,) `  (
f `  b )
) )  =  (/) ) )
117 fveq2 5694 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
f `  a )  =  ( f `  b ) )
118117fveq2d 5698 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( (,) `  ( f `  a ) )  =  ( (,) `  (
f `  b )
) )
119118disjor 4279 . . . . . . . . . 10  |-  (Disj  a  e.  NN  ( (,) `  (
f `  a )
)  <->  A. a  e.  NN  A. b  e.  NN  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
120116, 119sylibr 212 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  -> Disj  a  e.  NN  ( (,) `  (
f `  a )
) )
121 eqid 2443 . . . . . . . . 9  |-  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq 1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
12256, 120, 121uniiccmbl 21073 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  e.  dom  vol )
12352, 122eqeltrrd 2518 . . . . . . 7  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ( [,] " G )  e. 
dom  vol )
124123ex 434 . . . . . 6  |-  ( ph  ->  ( f : NN -1-1-onto-> G  ->  U. ( [,] " G
)  e.  dom  vol ) )
125124exlimdv 1690 . . . . 5  |-  ( ph  ->  ( E. f  f : NN -1-1-onto-> G  ->  U. ( [,] " G )  e. 
dom  vol ) )
12644, 125syl5 32 . . . 4  |-  ( ph  ->  ( G  ~~  om  ->  U. ( [,] " G
)  e.  dom  vol ) )
127126imp 429 . . 3  |-  ( (
ph  /\  G  ~~  om )  ->  U. ( [,] " G )  e. 
dom  vol )
128 reex 9376 . . . . . . . . 9  |-  RR  e.  _V
129128, 128xpex 6511 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
130129inex2 4437 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
131130, 16ssexi 4440 . . . . . 6  |-  ran  F  e.  _V
132 ssdomg 7358 . . . . . 6  |-  ( ran 
F  e.  _V  ->  ( G  C_  ran  F  ->  G  ~<_  ran  F )
)
133131, 13, 132mpsyl 63 . . . . 5  |-  ( ph  ->  G  ~<_  ran  F )
134 omelon 7855 . . . . . . . 8  |-  om  e.  On
135 znnen 13498 . . . . . . . . . . . 12  |-  ZZ  ~~  NN
136135, 39entri 7366 . . . . . . . . . . 11  |-  ZZ  ~~  om
137 nn0ennn 11804 . . . . . . . . . . . 12  |-  NN0  ~~  NN
138137, 39entri 7366 . . . . . . . . . . 11  |-  NN0  ~~  om
139 xpen 7477 . . . . . . . . . . 11  |-  ( ( ZZ  ~~  om  /\  NN0  ~~  om )  ->  ( ZZ  X.  NN0 )  ~~  ( om  X.  om )
)
140136, 138, 139mp2an 672 . . . . . . . . . 10  |-  ( ZZ 
X.  NN0 )  ~~  ( om  X.  om )
141 xpomen 8185 . . . . . . . . . 10  |-  ( om 
X.  om )  ~~  om
142140, 141entri 7366 . . . . . . . . 9  |-  ( ZZ 
X.  NN0 )  ~~  om
143142ensymi 7362 . . . . . . . 8  |-  om  ~~  ( ZZ  X.  NN0 )
144 isnumi 8119 . . . . . . . 8  |-  ( ( om  e.  On  /\  om 
~~  ( ZZ  X.  NN0 ) )  ->  ( ZZ  X.  NN0 )  e. 
dom  card )
145134, 143, 144mp2an 672 . . . . . . 7  |-  ( ZZ 
X.  NN0 )  e.  dom  card
146 ffn 5562 . . . . . . . . 9  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
14714, 146ax-mp 5 . . . . . . . 8  |-  F  Fn  ( ZZ  X.  NN0 )
148 dffn4 5629 . . . . . . . 8  |-  ( F  Fn  ( ZZ  X.  NN0 )  <->  F : ( ZZ 
X.  NN0 ) -onto-> ran  F
)
149147, 148mpbi 208 . . . . . . 7  |-  F :
( ZZ  X.  NN0 ) -onto-> ran  F
150 fodomnum 8230 . . . . . . 7  |-  ( ( ZZ  X.  NN0 )  e.  dom  card  ->  ( F : ( ZZ  X.  NN0 ) -onto-> ran  F  ->  ran  F  ~<_  ( ZZ  X.  NN0 ) ) )
151145, 149, 150mp2 9 . . . . . 6  |-  ran  F  ~<_  ( ZZ  X.  NN0 )
152 domentr 7371 . . . . . 6  |-  ( ( ran  F  ~<_  ( ZZ 
X.  NN0 )  /\  ( ZZ  X.  NN0 )  ~~  om )  ->  ran  F  ~<_  om )
153151, 142, 152mp2an 672 . . . . 5  |-  ran  F  ~<_  om
154 domtr 7365 . . . . 5  |-  ( ( G  ~<_  ran  F  /\  ran  F  ~<_  om )  ->  G  ~<_  om )
155133, 153, 154sylancl 662 . . . 4  |-  ( ph  ->  G  ~<_  om )
156 brdom2 7342 . . . 4  |-  ( G  ~<_  om  <->  ( G  ~<  om  \/  G  ~~  om ) )
157155, 156sylib 196 . . 3  |-  ( ph  ->  ( G  ~<  om  \/  G  ~~  om ) )
15838, 127, 157mpjaodan 784 . 2  |-  ( ph  ->  U. ( [,] " G
)  e.  dom  vol )
1594, 158eqeltrd 2517 1  |-  ( ph  ->  U. ( [,] " A
)  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2718   {crab 2722   _Vcvv 2975    i^i cin 3330    C_ wss 3331   (/)c0 3640   ~Pcpw 3863   <.cop 3886   U.cuni 4094   U_ciun 4174  Disj wdisj 4265   class class class wbr 4295   Oncon0 4722    X. cxp 4841   dom cdm 4843   ran crn 4844   "cima 4846    o. ccom 4847   Fun wfun 5415    Fn wfn 5416   -->wf 5417   -1-1->wf1 5418   -onto->wfo 5419   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6094    e. cmpt2 6096   omcom 6479   1stc1st 6578   2ndc2nd 6579    ~~ cen 7310    ~<_ cdom 7311    ~< csdm 7312   Fincfn 7313   cardccrd 8108   RRcr 9284   1c1 9286    + caddc 9288   RR*cxr 9420    <_ cle 9422    - cmin 9598    / cdiv 9996   NNcn 10325   2c2 10374   NN0cn0 10582   ZZcz 10649   (,)cioo 11303   [,]cicc 11306    seqcseq 11809   ^cexp 11868   abscabs 12726   volcvol 20950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-disj 4266  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-omul 6928  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fi 7664  df-sup 7694  df-oi 7727  df-card 8112  df-acn 8115  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-n0 10583  df-z 10650  df-uz 10865  df-q 10957  df-rp 10995  df-xneg 11092  df-xadd 11093  df-xmul 11094  df-ioo 11307  df-ico 11309  df-icc 11310  df-fz 11441  df-fzo 11552  df-fl 11645  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-rlim 12970  df-sum 13167  df-rest 14364  df-topgen 14385  df-psmet 17812  df-xmet 17813  df-met 17814  df-bl 17815  df-mopn 17816  df-top 18506  df-bases 18508  df-topon 18509  df-cmp 18993  df-ovol 20951  df-vol 20952
This theorem is referenced by:  opnmbllem  21084
  Copyright terms: Public domain W3C validator