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Theorem dyadmbl 21882
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 21881 . 2  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
5 isfinite 8072 . . . 4  |-  ( G  e.  Fin  <->  G  ~<  om )
6 iccf 11632 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
7 ffun 5723 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
8 funiunfv 6145 . . . . . 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 3570 . . . . . . . . . . . . . . . 16  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
122, 11eqsstri 3519 . . . . . . . . . . . . . . 15  |-  G  C_  A
1312, 3syl5ss 3500 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  C_  ran  F )
141dyadf 21873 . . . . . . . . . . . . . . . 16  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
15 frn 5727 . . . . . . . . . . . . . . . 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 3704 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
1816, 17sstri 3498 . . . . . . . . . . . . . 14  |-  ran  F  C_  ( RR  X.  RR )
1913, 18syl6ss 3501 . . . . . . . . . . . . 13  |-  ( ph  ->  G  C_  ( RR  X.  RR ) )
2019adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  e.  Fin )  ->  G  C_  ( RR  X.  RR ) )
2120sselda 3489 . . . . . . . . . . 11  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  n  e.  ( RR  X.  RR ) )
22 1st2nd2 6822 . . . . . . . . . . 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 5860 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. ) )
25 df-ov 6284 . . . . . . . . 9  |-  ( ( 1st `  n ) [,] ( 2nd `  n
) )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. )
2624, 25syl6eqr 2502 . . . . . . . 8  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( ( 1st `  n
) [,] ( 2nd `  n ) ) )
27 xp1st 6815 . . . . . . . . . 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 6816 . . . . . . . . . 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 21849 . . . . . . . . 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 2531 . . . . . . 7  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  e. 
dom  vol )
3433ralrimiva 2857 . . . . . 6  |-  ( (
ph  /\  G  e.  Fin )  ->  A. n  e.  G  ( [,] `  n )  e.  dom  vol )
35 finiunmbl 21827 . . . . . 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 2536 . . . 4  |-  ( (
ph  /\  G  e.  Fin )  ->  U. ( [,] " G )  e. 
dom  vol )
385, 37sylan2br 476 . . 3  |-  ( (
ph  /\  G  ~<  om )  ->  U. ( [,] " G )  e. 
dom  vol )
39 nnenom 12069 . . . . . . 7  |-  NN  ~~  om
40 ensym 7566 . . . . . . 7  |-  ( G 
~~  om  ->  om  ~~  G )
41 entr 7569 . . . . . . 7  |-  ( ( NN  ~~  om  /\  om 
~~  G )  ->  NN  ~~  G )
4239, 40, 41sylancr 663 . . . . . 6  |-  ( G 
~~  om  ->  NN  ~~  G )
43 bren 7527 . . . . . 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 5504 . . . . . . . . . 10  |-  ran  ( [,]  o.  f )  =  ( [,] " ran  f )
46 f1ofo 5813 . . . . . . . . . . . . 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 5788 . . . . . . . . . . . 12  |-  ( f : NN -onto-> G  ->  ran  f  =  G
)
4947, 48syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  f  =  G )
5049imaeq2d 5327 . . . . . . . . . 10  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ( [,] " ran  f )  =  ( [,] " G
) )
5145, 50syl5eq 2496 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  ( [,]  o.  f )  =  ( [,] " G
) )
5251unieqd 4244 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  =  U. ( [,] " G ) )
53 f1of 5806 . . . . . . . . . 10  |-  ( f : NN -1-1-onto-> G  ->  f : NN
--> G )
5413, 16syl6ss 3501 . . . . . . . . . 10  |-  ( ph  ->  G  C_  (  <_  i^i  ( RR  X.  RR ) ) )
55 fss 5729 . . . . . . . . . 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 5729 . . . . . . . . . . . . . . 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 6014 . . . . . . . . . . . . . 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 6014 . . . . . . . . . . . . . 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 21878 . . . . . . . . . . . . 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 6014 . . . . . . . . . . . . . . . . 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 3487 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  A
)
71 ffvelrn 6014 . . . . . . . . . . . . . . . . 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 5856 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  a )  ->  ( [,] `  z )  =  ( [,] `  (
f `  a )
) )
7473sseq1d 3516 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  a )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  w ) ) )
75 eqeq1 2447 . . . . . . . . . . . . . . . . . . . 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 2882 . . . . . . . . . . . . . . . . . 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 3245 . . . . . . . . . . . . . . . . 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 5856 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  b )  ->  ( [,] `  w )  =  ( [,] `  (
f `  b )
) )
8281sseq2d 3517 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  b )  ->  (
( [,] `  (
f `  a )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  ( f `  b ) ) ) )
83 eqeq2 2458 . . . . . . . . . . . . . . . . 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 3192 . . . . . . . . . . . . . . 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 5805 . . . . . . . . . . . . . . . . 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 6155 . . . . . . . . . . . . . . . 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 3487 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  A
)
95 fveq2 5856 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  b )  ->  ( [,] `  z )  =  ( [,] `  (
f `  b )
) )
9695sseq1d 3516 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  b )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  w ) ) )
97 eqeq1 2447 . . . . . . . . . . . . . . . . . . . 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 2882 . . . . . . . . . . . . . . . . . 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 3245 . . . . . . . . . . . . . . . . 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 5856 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  ( [,] `  w )  =  ( [,] `  (
f `  a )
) )
104103sseq2d 3517 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  a )  ->  (
( [,] `  (
f `  b )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  ( f `  a ) ) ) )
105 eqeq2 2458 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  (
( f `  b
)  =  w  <->  ( f `  b )  =  ( f `  a ) ) )
106 eqcom 2452 . . . . . . . . . . . . . . . . . 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 3192 . . . . . . . . . . . . . . 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 1293 . . . . . . . . . . . 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 2864 . . . . . . . . . 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 5856 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
f `  a )  =  ( f `  b ) )
118117fveq2d 5860 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( (,) `  ( f `  a ) )  =  ( (,) `  (
f `  b )
) )
119118disjor 4421 . . . . . . . . . 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 21872 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  e.  dom  vol )
12352, 122eqeltrrd 2532 . . . . . . 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 1711 . . . . 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 9586 . . . . . . . . 9  |-  RR  e.  _V
129128, 128xpex 6589 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
130129inex2 4579 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
131130, 16ssexi 4582 . . . . . 6  |-  ran  F  e.  _V
132 ssdomg 7563 . . . . . 6  |-  ( ran 
F  e.  _V  ->  ( G  C_  ran  F  ->  G  ~<_  ran  F )
)
133131, 13, 132mpsyl 63 . . . . 5  |-  ( ph  ->  G  ~<_  ran  F )
134 omelon 8066 . . . . . . . 8  |-  om  e.  On
135 znnen 13823 . . . . . . . . . . . 12  |-  ZZ  ~~  NN
136135, 39entri 7571 . . . . . . . . . . 11  |-  ZZ  ~~  om
137 nn0ennn 12068 . . . . . . . . . . . 12  |-  NN0  ~~  NN
138137, 39entri 7571 . . . . . . . . . . 11  |-  NN0  ~~  om
139 xpen 7682 . . . . . . . . . . 11  |-  ( ( ZZ  ~~  om  /\  NN0  ~~  om )  ->  ( ZZ  X.  NN0 )  ~~  ( om  X.  om )
)
140136, 138, 139mp2an 672 . . . . . . . . . 10  |-  ( ZZ 
X.  NN0 )  ~~  ( om  X.  om )
141 xpomen 8396 . . . . . . . . . 10  |-  ( om 
X.  om )  ~~  om
142140, 141entri 7571 . . . . . . . . 9  |-  ( ZZ 
X.  NN0 )  ~~  om
143142ensymi 7567 . . . . . . . 8  |-  om  ~~  ( ZZ  X.  NN0 )
144 isnumi 8330 . . . . . . . 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 5721 . . . . . . . . 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 5791 . . . . . . . 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 8441 . . . . . . 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 7576 . . . . . 6  |-  ( ( ran  F  ~<_  ( ZZ 
X.  NN0 )  /\  ( ZZ  X.  NN0 )  ~~  om )  ->  ran  F  ~<_  om )
153151, 142, 152mp2an 672 . . . . 5  |-  ran  F  ~<_  om
154 domtr 7570 . . . . 5  |-  ( ( G  ~<_  ran  F  /\  ran  F  ~<_  om )  ->  G  ~<_  om )
155133, 153, 154sylancl 662 . . . 4  |-  ( ph  ->  G  ~<_  om )
156 brdom2 7547 . . . 4  |-  ( G  ~<_  om  <->  ( G  ~<  om  \/  G  ~~  om ) )
157155, 156sylib 196 . . 3  |-  ( ph  ->  ( G  ~<  om  \/  G  ~~  om ) )
15838, 127, 157mpjaodan 786 . 2  |-  ( ph  ->  U. ( [,] " G
)  e.  dom  vol )
1594, 158eqeltrd 2531 1  |-  ( ph  ->  U. ( [,] " A
)  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 973    = wceq 1383   E.wex 1599    e. wcel 1804   A.wral 2793   {crab 2797   _Vcvv 3095    i^i cin 3460    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   <.cop 4020   U.cuni 4234   U_ciun 4315  Disj wdisj 4407   class class class wbr 4437   Oncon0 4868    X. cxp 4987   dom cdm 4989   ran crn 4990   "cima 4992    o. ccom 4993   Fun wfun 5572    Fn wfn 5573   -->wf 5574   -1-1->wf1 5575   -onto->wfo 5576   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   omcom 6685   1stc1st 6783   2ndc2nd 6784    ~~ cen 7515    ~<_ cdom 7516    ~< csdm 7517   Fincfn 7518   cardccrd 8319   RRcr 9494   1c1 9496    + caddc 9498   RR*cxr 9630    <_ cle 9632    - cmin 9810    / cdiv 10212   NNcn 10542   2c2 10591   NN0cn0 10801   ZZcz 10870   (,)cioo 11538   [,]cicc 11541    seqcseq 12086   ^cexp 12145   abscabs 13046   volcvol 21748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-n0 10802  df-z 10871  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-rlim 13291  df-sum 13488  df-rest 14697  df-topgen 14718  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-top 19272  df-bases 19274  df-topon 19275  df-cmp 19760  df-ovol 21749  df-vol 21750
This theorem is referenced by:  opnmbllem  21883
  Copyright terms: Public domain W3C validator