MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iccf1o Unicode version

Theorem iccf1o 10794
Description: Describe a bijection from  [ 0 ,  1 ] to an arbitrary nontrivial closed interval  [ A ,  B ]. (Contributed by Mario Carneiro, 8-Sep-2015.)
Hypothesis
Ref Expression
iccf1o.1  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  ( ( x  x.  B )  +  ( ( 1  -  x
)  x.  A ) ) )
Assertion
Ref Expression
iccf1o  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  /\  `' F  =  ( y  e.  ( A [,] B
)  |->  ( ( y  -  A )  / 
( B  -  A
) ) ) ) )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    F( x, y)

Proof of Theorem iccf1o
StepHypRef Expression
1 iccf1o.1 . 2  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  ( ( x  x.  B )  +  ( ( 1  -  x
)  x.  A ) ) )
2 0re 8854 . . . . . . . . 9  |-  0  e.  RR
3 1re 8853 . . . . . . . . 9  |-  1  e.  RR
42, 3elicc2i 10732 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <_  1
) )
54simp1bi 970 . . . . . . 7  |-  ( x  e.  ( 0 [,] 1 )  ->  x  e.  RR )
65adantl 452 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  RR )
76recnd 8877 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  CC )
8 simpl2 959 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
98recnd 8877 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
107, 9mulcld 8871 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  B )  e.  CC )
11 ax-1cn 8811 . . . . . 6  |-  1  e.  CC
12 subcl 9067 . . . . . 6  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1  -  x
)  e.  CC )
1311, 7, 12sylancr 644 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  -  x )  e.  CC )
14 simpl1 958 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1514recnd 8877 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
1613, 15mulcld 8871 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  e.  CC )
1710, 16addcomd 9030 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A
) )  =  ( ( ( 1  -  x )  x.  A
)  +  ( x  x.  B ) ) )
18 lincmb01cmp 10793 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  x )  x.  A )  +  ( x  x.  B
) )  e.  ( A [,] B ) )
1917, 18eqeltrd 2370 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A
) )  e.  ( A [,] B ) )
20 elicc2 10731 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
21203adant3 975 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  <->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
2221biimpa 470 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
2322simp1d 967 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  RR )
24 simpl1 958 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  RR )
2523, 24resubcld 9227 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  RR )
2625recnd 8877 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  CC )
27 difrp 10403 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
2827biimp3a 1281 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
2928adantr 451 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  RR+ )
3029rpcnd 10408 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  CC )
3129rpne0d 10411 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  =/=  0 )
3226, 30, 31divcan1d 9553 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( ( y  -  A )  / 
( B  -  A
) )  x.  ( B  -  A )
)  =  ( y  -  A ) )
33 simpr 447 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  ( A [,] B ) )
34 simpl2 959 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  B  e.  RR )
35 eqid 2296 . . . . . . . 8  |-  ( A  -  A )  =  ( A  -  A
)
36 eqid 2296 . . . . . . . 8  |-  ( B  -  A )  =  ( B  -  A
)
3735, 36iccshftl 10787 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( y  e.  RR  /\  A  e.  RR ) )  -> 
( y  e.  ( A [,] B )  <-> 
( y  -  A
)  e.  ( ( A  -  A ) [,] ( B  -  A ) ) ) )
3824, 34, 23, 24, 37syl22anc 1183 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  e.  ( A [,] B )  <-> 
( y  -  A
)  e.  ( ( A  -  A ) [,] ( B  -  A ) ) ) )
3933, 38mpbid 201 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  ( ( A  -  A ) [,] ( B  -  A ) ) )
4030mul02d 9026 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  0 )
4124recnd 8877 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  CC )
4241subidd 9161 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( A  -  A
)  =  0 )
4340, 42eqtr4d 2331 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  ( A  -  A ) )
4430mulid2d 8869 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 1  x.  ( B  -  A )
)  =  ( B  -  A ) )
4543, 44oveq12d 5892 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( 0  x.  ( B  -  A
) ) [,] (
1  x.  ( B  -  A ) ) )  =  ( ( A  -  A ) [,] ( B  -  A ) ) )
4639, 45eleqtrrd 2373 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  ( ( 0  x.  ( B  -  A ) ) [,] ( 1  x.  ( B  -  A
) ) ) )
4732, 46eqeltrd 2370 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( ( y  -  A )  / 
( B  -  A
) )  x.  ( B  -  A )
)  e.  ( ( 0  x.  ( B  -  A ) ) [,] ( 1  x.  ( B  -  A
) ) ) )
482a1i 10 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
0  e.  RR )
493a1i 10 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
1  e.  RR )
5025, 29rerpdivcld 10433 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  RR )
51 eqid 2296 . . . . 5  |-  ( 0  x.  ( B  -  A ) )  =  ( 0  x.  ( B  -  A )
)
52 eqid 2296 . . . . 5  |-  ( 1  x.  ( B  -  A ) )  =  ( 1  x.  ( B  -  A )
)
5351, 52iccdil 10789 . . . 4  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( ( ( y  -  A )  /  ( B  -  A ) )  e.  RR  /\  ( B  -  A )  e.  RR+ ) )  ->  (
( ( y  -  A )  /  ( B  -  A )
)  e.  ( 0 [,] 1 )  <->  ( (
( y  -  A
)  /  ( B  -  A ) )  x.  ( B  -  A ) )  e.  ( ( 0  x.  ( B  -  A
) ) [,] (
1  x.  ( B  -  A ) ) ) ) )
5448, 49, 50, 29, 53syl22anc 1183 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( ( y  -  A )  / 
( B  -  A
) )  e.  ( 0 [,] 1 )  <-> 
( ( ( y  -  A )  / 
( B  -  A
) )  x.  ( B  -  A )
)  e.  ( ( 0  x.  ( B  -  A ) ) [,] ( 1  x.  ( B  -  A
) ) ) ) )
5547, 54mpbird 223 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  ( 0 [,] 1 ) )
56 eqcom 2298 . . . 4  |-  ( x  =  ( ( y  -  A )  / 
( B  -  A
) )  <->  ( (
y  -  A )  /  ( B  -  A ) )  =  x )
5726adantrl 696 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
y  -  A )  e.  CC )
587adantrr 697 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  x  e.  CC )
5930adantrl 696 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A )  e.  CC )
6031adantrl 696 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A )  =/=  0 )
6157, 58, 59, 60divmul3d 9586 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
( ( y  -  A )  /  ( B  -  A )
)  =  x  <->  ( y  -  A )  =  ( x  x.  ( B  -  A ) ) ) )
6256, 61syl5bb 248 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
x  =  ( ( y  -  A )  /  ( B  -  A ) )  <->  ( y  -  A )  =  ( x  x.  ( B  -  A ) ) ) )
6323adantrl 696 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  RR )
6463recnd 8877 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  CC )
6541adantrl 696 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  A  e.  CC )
668, 14resubcld 9227 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
676, 66remulcld 8879 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  e.  RR )
6867adantrr 697 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
x  x.  ( B  -  A ) )  e.  RR )
6968recnd 8877 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
x  x.  ( B  -  A ) )  e.  CC )
7064, 65, 69subadd2d 9192 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
( y  -  A
)  =  ( x  x.  ( B  -  A ) )  <->  ( (
x  x.  ( B  -  A ) )  +  A )  =  y ) )
71 eqcom 2298 . . . 4  |-  ( ( ( x  x.  ( B  -  A )
)  +  A )  =  y  <->  y  =  ( ( x  x.  ( B  -  A
) )  +  A
) )
7270, 71syl6bb 252 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
( y  -  A
)  =  ( x  x.  ( B  -  A ) )  <->  y  =  ( ( x  x.  ( B  -  A
) )  +  A
) ) )
737, 15mulcld 8871 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  A )  e.  CC )
7410, 73, 15subadd23d 9195 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( ( x  x.  B )  -  ( x  x.  A ) )  +  A )  =  ( ( x  x.  B
)  +  ( A  -  ( x  x.  A ) ) ) )
757, 9, 15subdid 9251 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  =  ( ( x  x.  B
)  -  ( x  x.  A ) ) )
7675oveq1d 5889 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( x  x.  ( B  -  A ) )  +  A )  =  ( ( ( x  x.  B )  -  (
x  x.  A ) )  +  A ) )
7711a1i 10 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
7877, 7, 15subdird 9252 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( ( 1  x.  A
)  -  ( x  x.  A ) ) )
7915mulid2d 8869 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
8079oveq1d 5889 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  x.  A )  -  ( x  x.  A
) )  =  ( A  -  ( x  x.  A ) ) )
8178, 80eqtrd 2328 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( A  -  ( x  x.  A ) ) )
8281oveq2d 5890 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A
) )  =  ( ( x  x.  B
)  +  ( A  -  ( x  x.  A ) ) ) )
8374, 76, 823eqtr4d 2338 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( x  x.  ( B  -  A ) )  +  A )  =  ( ( x  x.  B
)  +  ( ( 1  -  x )  x.  A ) ) )
8483adantrr 697 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
( x  x.  ( B  -  A )
)  +  A )  =  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A
) ) )
8584eqeq2d 2307 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
y  =  ( ( x  x.  ( B  -  A ) )  +  A )  <->  y  =  ( ( x  x.  B )  +  ( ( 1  -  x
)  x.  A ) ) ) )
8662, 72, 853bitrd 270 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
x  =  ( ( y  -  A )  /  ( B  -  A ) )  <->  y  =  ( ( x  x.  B )  +  ( ( 1  -  x
)  x.  A ) ) ) )
871, 19, 55, 86f1ocnv2d 6084 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( F : ( 0 [,] 1 ) -1-1-onto-> ( A [,] B
)  /\  `' F  =  ( y  e.  ( A [,] B
)  |->  ( ( y  -  A )  / 
( B  -  A
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   -1-1-onto->wf1o 5270  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   RR+crp 10370   [,]cicc 10675
This theorem is referenced by:  iccen  10795  icchmeo  18455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-rp 10371  df-icc 10679
  Copyright terms: Public domain W3C validator