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Theorem dyadmbl 19494
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 19493 . 2  |-  ( ph  ->  U. ( [,] " A
)  =  U. ( [,] " G ) )
5 isfinite 7609 . . . 4  |-  ( G  e.  Fin  <->  G  ~<  om )
6 iccf 11005 . . . . . 6  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
7 ffun 5595 . . . . . 6  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
8 funiunfv 5997 . . . . . 6  |-  ( Fun 
[,]  ->  U_ n  e.  G  ( [,] `  n )  =  U. ( [,] " G ) )
96, 7, 8mp2b 10 . . . . 5  |-  U_ n  e.  G  ( [,] `  n )  =  U. ( [,] " G )
10 simpr 449 . . . . . 6  |-  ( (
ph  /\  G  e.  Fin )  ->  G  e. 
Fin )
11 ssrab2 3430 . . . . . . . . . . . . . . . 16  |-  { z  e.  A  |  A. w  e.  A  (
( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w ) }  C_  A
122, 11eqsstri 3380 . . . . . . . . . . . . . . 15  |-  G  C_  A
1312, 3syl5ss 3361 . . . . . . . . . . . . . 14  |-  ( ph  ->  G  C_  ran  F )
141dyadf 19485 . . . . . . . . . . . . . . . 16  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
15 frn 5599 . . . . . . . . . . . . . . . 16  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
1614, 15ax-mp 8 . . . . . . . . . . . . . . 15  |-  ran  F  C_  (  <_  i^i  ( RR  X.  RR ) )
17 inss2 3564 . . . . . . . . . . . . . . 15  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
1816, 17sstri 3359 . . . . . . . . . . . . . 14  |-  ran  F  C_  ( RR  X.  RR )
1913, 18syl6ss 3362 . . . . . . . . . . . . 13  |-  ( ph  ->  G  C_  ( RR  X.  RR ) )
2019adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  e.  Fin )  ->  G  C_  ( RR  X.  RR ) )
2120sselda 3350 . . . . . . . . . . 11  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  n  e.  ( RR  X.  RR ) )
22 1st2nd2 6388 . . . . . . . . . . 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 5734 . . . . . . . . 9  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. ) )
25 df-ov 6086 . . . . . . . . 9  |-  ( ( 1st `  n ) [,] ( 2nd `  n
) )  =  ( [,] `  <. ( 1st `  n ) ,  ( 2nd `  n
) >. )
2624, 25syl6eqr 2488 . . . . . . . 8  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  =  ( ( 1st `  n
) [,] ( 2nd `  n ) ) )
27 xp1st 6378 . . . . . . . . . 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 6379 . . . . . . . . . 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 19462 . . . . . . . . 9  |-  ( ( ( 1st `  n
)  e.  RR  /\  ( 2nd `  n )  e.  RR )  -> 
( ( 1st `  n
) [,] ( 2nd `  n ) )  e. 
dom  vol )
3228, 30, 31syl2anc 644 . . . . . . . 8  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  (
( 1st `  n
) [,] ( 2nd `  n ) )  e. 
dom  vol )
3326, 32eqeltrd 2512 . . . . . . 7  |-  ( ( ( ph  /\  G  e.  Fin )  /\  n  e.  G )  ->  ( [,] `  n )  e. 
dom  vol )
3433ralrimiva 2791 . . . . . 6  |-  ( (
ph  /\  G  e.  Fin )  ->  A. n  e.  G  ( [,] `  n )  e.  dom  vol )
35 finiunmbl 19440 . . . . . 6  |-  ( ( G  e.  Fin  /\  A. n  e.  G  ( [,] `  n )  e.  dom  vol )  ->  U_ n  e.  G  ( [,] `  n )  e.  dom  vol )
3610, 34, 35syl2anc 644 . . . . 5  |-  ( (
ph  /\  G  e.  Fin )  ->  U_ n  e.  G  ( [,] `  n )  e.  dom  vol )
379, 36syl5eqelr 2523 . . . 4  |-  ( (
ph  /\  G  e.  Fin )  ->  U. ( [,] " G )  e. 
dom  vol )
385, 37sylan2br 464 . . 3  |-  ( (
ph  /\  G  ~<  om )  ->  U. ( [,] " G )  e. 
dom  vol )
39 nnenom 11321 . . . . . . 7  |-  NN  ~~  om
40 ensym 7158 . . . . . . 7  |-  ( G 
~~  om  ->  om  ~~  G )
41 entr 7161 . . . . . . 7  |-  ( ( NN  ~~  om  /\  om 
~~  G )  ->  NN  ~~  G )
4239, 40, 41sylancr 646 . . . . . 6  |-  ( G 
~~  om  ->  NN  ~~  G )
43 bren 7119 . . . . . 6  |-  ( NN 
~~  G  <->  E. f 
f : NN -1-1-onto-> G )
4442, 43sylib 190 . . . . 5  |-  ( G 
~~  om  ->  E. f 
f : NN -1-1-onto-> G )
45 rnco2 5379 . . . . . . . . . 10  |-  ran  ( [,]  o.  f )  =  ( [,] " ran  f )
46 f1ofo 5683 . . . . . . . . . . . . 13  |-  ( f : NN -1-1-onto-> G  ->  f : NN -onto-> G )
4746adantl 454 . . . . . . . . . . . 12  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN -onto-> G )
48 forn 5658 . . . . . . . . . . . 12  |-  ( f : NN -onto-> G  ->  ran  f  =  G
)
4947, 48syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  f  =  G )
5049imaeq2d 5205 . . . . . . . . . 10  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ( [,] " ran  f )  =  ( [,] " G
) )
5145, 50syl5eq 2482 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  ran  ( [,]  o.  f )  =  ( [,] " G
) )
5251unieqd 4028 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  =  U. ( [,] " G ) )
53 f1of 5676 . . . . . . . . . 10  |-  ( f : NN -1-1-onto-> G  ->  f : NN
--> G )
5413, 16syl6ss 3362 . . . . . . . . . 10  |-  ( ph  ->  G  C_  (  <_  i^i  ( RR  X.  RR ) ) )
55 fss 5601 . . . . . . . . . 10  |-  ( ( f : NN --> G  /\  G  C_  (  <_  i^i  ( RR  X.  RR ) ) )  -> 
f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
5653, 54, 55syl2anr 466 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
57 fss 5601 . . . . . . . . . . . . . . 15  |-  ( ( f : NN --> G  /\  G  C_  ran  F )  ->  f : NN --> ran  F )
5853, 13, 57syl2anr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> ran  F )
59 simpl 445 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  a  e.  NN )
60 ffvelrn 5870 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  F  /\  a  e.  NN )  ->  ( f `  a )  e.  ran  F )
6158, 59, 60syl2an 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  ran  F )
62 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( a  e.  NN  /\  b  e.  NN )  ->  b  e.  NN )
63 ffvelrn 5870 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  F  /\  b  e.  NN )  ->  ( f `  b )  e.  ran  F )
6458, 62, 63syl2an 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  ran  F )
651dyaddisj 19490 . . . . . . . . . . . . 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 644 . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN --> G )
68 ffvelrn 5870 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> G  /\  b  e.  NN )  ->  ( f `  b
)  e.  G )
6967, 62, 68syl2an 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  G
)
7012, 69sseldi 3348 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  b )  e.  A
)
71 ffvelrn 5870 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> G  /\  a  e.  NN )  ->  ( f `  a
)  e.  G )
7267, 59, 71syl2an 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  G
)
73 fveq2 5730 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  a )  ->  ( [,] `  z )  =  ( [,] `  (
f `  a )
) )
7473sseq1d 3377 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  a )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  w ) ) )
75 eqeq1 2444 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  a )  ->  (
z  =  w  <->  ( f `  a )  =  w ) )
7674, 75imbi12d 313 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( f `  a )  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7776ralbidv 2727 . . . . . . . . . . . . . . . . . 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 3096 . . . . . . . . . . . . . . . . 17  |-  ( ( f `  a )  e.  G  <->  ( (
f `  a )  e.  A  /\  A. w  e.  A  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  w )  ->  ( f `  a )  =  w ) ) )
7978simprbi 452 . . . . . . . . . . . . . . . 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 5730 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  b )  ->  ( [,] `  w )  =  ( [,] `  (
f `  b )
) )
8281sseq2d 3378 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  b )  ->  (
( [,] `  (
f `  a )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  a
) )  C_  ( [,] `  ( f `  b ) ) ) )
83 eqeq2 2447 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  b )  ->  (
( f `  a
)  =  w  <->  ( f `  a )  =  ( f `  b ) ) )
8482, 83imbi12d 313 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( f `  b )  ->  (
( ( [,] `  (
f `  a )
)  C_  ( [,] `  w )  ->  (
f `  a )  =  w )  <->  ( ( [,] `  ( f `  a ) )  C_  ( [,] `  ( f `
 b ) )  ->  ( f `  a )  =  ( f `  b ) ) ) )
8584rspcv 3050 . . . . . . . . . . . . . . 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 59 . . . . . . . . . . . . . 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 5675 . . . . . . . . . . . . . . . . 17  |-  ( f : NN -1-1-onto-> G  ->  f : NN
-1-1-> G )
8887adantl 454 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  f : NN -1-1-> G )
89 f1fveq 6010 . . . . . . . . . . . . . . . 16  |-  ( ( f : NN -1-1-> G  /\  ( a  e.  NN  /\  b  e.  NN ) )  ->  ( (
f `  a )  =  ( f `  b )  <->  a  =  b ) )
9088, 89sylan 459 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( (
f `  a )  =  ( f `  b )  <->  a  =  b ) )
91 orc 376 . . . . . . . . . . . . . . 15  |-  ( a  =  b  ->  (
a  =  b  \/  ( ( (,) `  (
f `  a )
)  i^i  ( (,) `  ( f `  b
) ) )  =  (/) ) )
9290, 91syl6bi 221 . . . . . . . . . . . . . 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 43 . . . . . . . . . . . . 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 3348 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f : NN -1-1-onto-> G )  /\  (
a  e.  NN  /\  b  e.  NN )
)  ->  ( f `  a )  e.  A
)
95 fveq2 5730 . . . . . . . . . . . . . . . . . . . . 21  |-  ( z  =  ( f `  b )  ->  ( [,] `  z )  =  ( [,] `  (
f `  b )
) )
9695sseq1d 3377 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  b )  ->  (
( [,] `  z
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  w ) ) )
97 eqeq1 2444 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  =  ( f `  b )  ->  (
z  =  w  <->  ( f `  b )  =  w ) )
9896, 97imbi12d 313 . . . . . . . . . . . . . . . . . . 19  |-  ( z  =  ( f `  b )  ->  (
( ( [,] `  z
)  C_  ( [,] `  w )  ->  z  =  w )  <->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
9998ralbidv 2727 . . . . . . . . . . . . . . . . . 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 3096 . . . . . . . . . . . . . . . . 17  |-  ( ( f `  b )  e.  G  <->  ( (
f `  b )  e.  A  /\  A. w  e.  A  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  w )  ->  ( f `  b )  =  w ) ) )
101100simprbi 452 . . . . . . . . . . . . . . . 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 5730 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  ( [,] `  w )  =  ( [,] `  (
f `  a )
) )
104103sseq2d 3378 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  a )  ->  (
( [,] `  (
f `  b )
)  C_  ( [,] `  w )  <->  ( [,] `  ( f `  b
) )  C_  ( [,] `  ( f `  a ) ) ) )
105 eqeq2 2447 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ( f `  a )  ->  (
( f `  b
)  =  w  <->  ( f `  b )  =  ( f `  a ) ) )
106 eqcom 2440 . . . . . . . . . . . . . . . . . 18  |-  ( ( f `  b )  =  ( f `  a )  <->  ( f `  a )  =  ( f `  b ) )
107105, 106syl6bb 254 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( f `  a )  ->  (
( f `  b
)  =  w  <->  ( f `  a )  =  ( f `  b ) ) )
108104, 107imbi12d 313 . . . . . . . . . . . . . . . 16  |-  ( w  =  ( f `  a )  ->  (
( ( [,] `  (
f `  b )
)  C_  ( [,] `  w )  ->  (
f `  b )  =  w )  <->  ( ( [,] `  ( f `  b ) )  C_  ( [,] `  ( f `
 a ) )  ->  ( f `  a )  =  ( f `  b ) ) ) )
109108rspcv 3050 . . . . . . . . . . . . . . 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 59 . . . . . . . . . . . . . 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 43 . . . . . . . . . . . . 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 375 . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . 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 2800 . . . . . . . . . 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 5730 . . . . . . . . . . . 12  |-  ( a  =  b  ->  (
f `  a )  =  ( f `  b ) )
118117fveq2d 5734 . . . . . . . . . . 11  |-  ( a  =  b  ->  ( (,) `  ( f `  a ) )  =  ( (,) `  (
f `  b )
) )
119118disjor 4198 . . . . . . . . . 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 205 . . . . . . . . 9  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  -> Disj  a  e.  NN ( (,) `  (
f `  a )
) )
121 eqid 2438 . . . . . . . . 9  |-  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)  =  seq  1
(  +  ,  ( ( abs  o.  -  )  o.  f )
)
12256, 120, 121uniiccmbl 19484 . . . . . . . 8  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ran  ( [,]  o.  f )  e.  dom  vol )
12352, 122eqeltrrd 2513 . . . . . . 7  |-  ( (
ph  /\  f : NN
-1-1-onto-> G )  ->  U. ( [,] " G )  e. 
dom  vol )
124123ex 425 . . . . . 6  |-  ( ph  ->  ( f : NN -1-1-onto-> G  ->  U. ( [,] " G
)  e.  dom  vol ) )
125124exlimdv 1647 . . . . 5  |-  ( ph  ->  ( E. f  f : NN -1-1-onto-> G  ->  U. ( [,] " G )  e. 
dom  vol ) )
12644, 125syl5 31 . . . 4  |-  ( ph  ->  ( G  ~~  om  ->  U. ( [,] " G
)  e.  dom  vol ) )
127126imp 420 . . 3  |-  ( (
ph  /\  G  ~~  om )  ->  U. ( [,] " G )  e. 
dom  vol )
128 reex 9083 . . . . . . . . 9  |-  RR  e.  _V
129128, 128xpex 4992 . . . . . . . 8  |-  ( RR 
X.  RR )  e. 
_V
130129inex2 4347 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  e.  _V
131130, 16ssexi 4350 . . . . . 6  |-  ran  F  e.  _V
132 ssdomg 7155 . . . . . 6  |-  ( ran 
F  e.  _V  ->  ( G  C_  ran  F  ->  G  ~<_  ran  F )
)
133131, 13, 132mpsyl 62 . . . . 5  |-  ( ph  ->  G  ~<_  ran  F )
134 omelon 7603 . . . . . . . 8  |-  om  e.  On
135 znnen 12814 . . . . . . . . . . . 12  |-  ZZ  ~~  NN
136135, 39entri 7163 . . . . . . . . . . 11  |-  ZZ  ~~  om
137 nn0ennn 11320 . . . . . . . . . . . 12  |-  NN0  ~~  NN
138137, 39entri 7163 . . . . . . . . . . 11  |-  NN0  ~~  om
139 xpen 7272 . . . . . . . . . . 11  |-  ( ( ZZ  ~~  om  /\  NN0  ~~  om )  ->  ( ZZ  X.  NN0 )  ~~  ( om  X.  om )
)
140136, 138, 139mp2an 655 . . . . . . . . . 10  |-  ( ZZ 
X.  NN0 )  ~~  ( om  X.  om )
141 xpomen 7899 . . . . . . . . . 10  |-  ( om 
X.  om )  ~~  om
142140, 141entri 7163 . . . . . . . . 9  |-  ( ZZ 
X.  NN0 )  ~~  om
143142ensymi 7159 . . . . . . . 8  |-  om  ~~  ( ZZ  X.  NN0 )
144 isnumi 7835 . . . . . . . 8  |-  ( ( om  e.  On  /\  om 
~~  ( ZZ  X.  NN0 ) )  ->  ( ZZ  X.  NN0 )  e. 
dom  card )
145134, 143, 144mp2an 655 . . . . . . 7  |-  ( ZZ 
X.  NN0 )  e.  dom  card
146 ffn 5593 . . . . . . . . 9  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
14714, 146ax-mp 8 . . . . . . . 8  |-  F  Fn  ( ZZ  X.  NN0 )
148 dffn4 5661 . . . . . . . 8  |-  ( F  Fn  ( ZZ  X.  NN0 )  <->  F : ( ZZ 
X.  NN0 ) -onto-> ran  F
)
149147, 148mpbi 201 . . . . . . 7  |-  F :
( ZZ  X.  NN0 ) -onto-> ran  F
150 fodomnum 7940 . . . . . . 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 7168 . . . . . 6  |-  ( ( ran  F  ~<_  ( ZZ 
X.  NN0 )  /\  ( ZZ  X.  NN0 )  ~~  om )  ->  ran  F  ~<_  om )
153151, 142, 152mp2an 655 . . . . 5  |-  ran  F  ~<_  om
154 domtr 7162 . . . . 5  |-  ( ( G  ~<_  ran  F  /\  ran  F  ~<_  om )  ->  G  ~<_  om )
155133, 153, 154sylancl 645 . . . 4  |-  ( ph  ->  G  ~<_  om )
156 brdom2 7139 . . . 4  |-  ( G  ~<_  om  <->  ( G  ~<  om  \/  G  ~~  om ) )
157155, 156sylib 190 . . 3  |-  ( ph  ->  ( G  ~<  om  \/  G  ~~  om ) )
15838, 127, 157mpjaodan 763 . 2  |-  ( ph  ->  U. ( [,] " G
)  e.  dom  vol )
1594, 158eqeltrd 2512 1  |-  ( ph  ->  U. ( [,] " A
)  e.  dom  vol )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 936   E.wex 1551    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   <.cop 3819   U.cuni 4017   U_ciun 4095  Disj wdisj 4184   class class class wbr 4214   Oncon0 4583   omcom 4847    X. cxp 4878   dom cdm 4880   ran crn 4881   "cima 4883    o. ccom 4884   Fun wfun 5450    Fn wfn 5451   -->wf 5452   -1-1->wf1 5453   -onto->wfo 5454   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350    ~~ cen 7108    ~<_ cdom 7109    ~< csdm 7110   Fincfn 7111   cardccrd 7824   RRcr 8991   1c1 8993    + caddc 8995   RR*cxr 9121    <_ cle 9123    - cmin 9293    / cdiv 9679   NNcn 10002   2c2 10051   NN0cn0 10223   ZZcz 10284   (,)cioo 10918   [,]cicc 10921    seq cseq 11325   ^cexp 11384   abscabs 12041   volcvol 19362
This theorem is referenced by:  opnmbllem  19495
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
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-disj 4185  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-omul 6731  df-er 6907  df-map 7022  df-pm 7023  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-acn 7831  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-n0 10224  df-z 10285  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-fl 11204  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-clim 12284  df-rlim 12285  df-sum 12482  df-rest 13652  df-topgen 13669  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-top 16965  df-bases 16967  df-topon 16968  df-cmp 17452  df-ovol 19363  df-vol 19364
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