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Theorem iccf1o 8973
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 7085 . . . . . . . . 9  |-  0  e.  RR
3 1re 7084 . . . . . . . . 9  |-  1  e.  RR
42, 3elicc2i 8909 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <_  1
) )
54simp1bi 930 . . . . . . 7  |-  ( x  e.  ( 0 [,] 1 )  ->  x  e.  RR )
65adantl 266 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  RR )
76recnd 7113 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  CC )
8 simpl2 919 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
98recnd 7113 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
107, 9mulcld 7105 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  B )  e.  CC )
11 ax-1cn 7035 . . . . . 6  |-  1  e.  CC
12 subcl 7273 . . . . . 6  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1  -  x
)  e.  CC )
1311, 7, 12sylancr 399 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  -  x )  e.  CC )
14 simpl1 918 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1514recnd 7113 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
1613, 15mulcld 7105 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  e.  CC )
1710, 16addcomd 7225 . . 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 8972 . . 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 2130 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( x  x.  B )  +  ( ( 1  -  x )  x.  A
) )  e.  ( A [,] B ) )
20 simpr 107 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  ( A [,] B ) )
21 simpl1 918 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  RR )
22 simpl2 919 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  B  e.  RR )
23 elicc2 8908 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
24233adant3 935 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  <->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
2524biimpa 284 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
2625simp1d 927 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  RR )
27 eqid 2056 . . . . . . 7  |-  ( A  -  A )  =  ( A  -  A
)
28 eqid 2056 . . . . . . 7  |-  ( B  -  A )  =  ( B  -  A
)
2927, 28iccshftl 8965 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( y  e.  RR  /\  A  e.  RR ) )  -> 
( y  e.  ( A [,] B )  <-> 
( y  -  A
)  e.  ( ( A  -  A ) [,] ( B  -  A ) ) ) )
3021, 22, 26, 21, 29syl22anc 1147 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  e.  ( A [,] B )  <-> 
( y  -  A
)  e.  ( ( A  -  A ) [,] ( B  -  A ) ) ) )
3120, 30mpbid 139 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  ( ( A  -  A ) [,] ( B  -  A ) ) )
3226, 21resubcld 7451 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  RR )
3332recnd 7113 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  CC )
34 difrp 8717 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
3534biimp3a 1251 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
3635adantr 265 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  RR+ )
3736rpcnd 8722 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  CC )
38 rpap0 8697 . . . . . 6  |-  ( ( B  -  A )  e.  RR+  ->  ( B  -  A ) #  0 )
3936, 38syl 14 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
) #  0 )
4033, 37, 39divcanap1d 7841 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( ( y  -  A )  / 
( B  -  A
) )  x.  ( B  -  A )
)  =  ( y  -  A ) )
4137mul02d 7461 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  0 )
4221recnd 7113 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  CC )
4342subidd 7373 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( A  -  A
)  =  0 )
4441, 43eqtr4d 2091 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  ( A  -  A ) )
4537mulid2d 7103 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 1  x.  ( B  -  A )
)  =  ( B  -  A ) )
4644, 45oveq12d 5558 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( 0  x.  ( B  -  A
) ) [,] (
1  x.  ( B  -  A ) ) )  =  ( ( A  -  A ) [,] ( B  -  A ) ) )
4731, 40, 463eltr4d 2137 . . 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
) ) ) )
48 0red 7086 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
0  e.  RR )
49 1red 7100 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
1  e.  RR )
5032, 36rerpdivcld 8752 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  RR )
51 eqid 2056 . . . . 5  |-  ( 0  x.  ( B  -  A ) )  =  ( 0  x.  ( B  -  A )
)
52 eqid 2056 . . . . 5  |-  ( 1  x.  ( B  -  A ) )  =  ( 1  x.  ( B  -  A )
)
5351, 52iccdil 8967 . . . 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, 36, 53syl22anc 1147 . . 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 160 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  ( 0 [,] 1 ) )
56 eqcom 2058 . . . 4  |-  ( x  =  ( ( y  -  A )  / 
( B  -  A
) )  <->  ( (
y  -  A )  /  ( B  -  A ) )  =  x )
5733adantrl 455 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
y  -  A )  e.  CC )
587adantrr 456 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  x  e.  CC )
5937adantrl 455 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A )  e.  CC )
6039adantrl 455 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A ) #  0 )
6157, 58, 59, 60divmulap3d 7874 . . . 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 185 . . 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 ) ) ) )
6326adantrl 455 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  RR )
6463recnd 7113 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  CC )
6542adantrl 455 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  A  e.  CC )
668, 14resubcld 7451 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
676, 66remulcld 7115 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  e.  RR )
6867adantrr 456 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
x  x.  ( B  -  A ) )  e.  RR )
6968recnd 7113 . . . . 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 7404 . . . 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 2058 . . . 4  |-  ( ( ( x  x.  ( B  -  A )
)  +  A )  =  y  <->  y  =  ( ( x  x.  ( B  -  A
) )  +  A
) )
7270, 71syl6bb 189 . . 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 7105 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  A )  e.  CC )
7410, 73, 15subadd23d 7407 . . . . . 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 7483 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  =  ( ( x  x.  B
)  -  ( x  x.  A ) ) )
7675oveq1d 5555 . . . . . 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 ) )
77 1cnd 7101 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
7877, 7, 15subdird 7484 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( ( 1  x.  A
)  -  ( x  x.  A ) ) )
7915mulid2d 7103 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
8079oveq1d 5555 . . . . . . . 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 2088 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( A  -  ( x  x.  A ) ) )
8281oveq2d 5556 . . . . . 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 2098 . . . . 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 456 . . . 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 2067 . . 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 207 . 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 5732 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    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   class class class wbr 3792    |-> cmpt 3846   `'ccnv 4372   -1-1-onto->wf1o 4929  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947   1c1 6948    + caddc 6950    x. cmul 6952    < clt 7119    <_ cle 7120    - cmin 7245   # cap 7646    / cdiv 7725   RR+crp 8681   [,]cicc 8861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-rp 8682  df-icc 8865
This theorem is referenced by: (None)
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