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Theorem iccf1o 8872
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 7027 . . . . . . . . 9  |-  0  e.  RR
3 1re 7026 . . . . . . . . 9  |-  1  e.  RR
42, 3elicc2i 8808 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <_  1
) )
54simp1bi 919 . . . . . . 7  |-  ( x  e.  ( 0 [,] 1 )  ->  x  e.  RR )
65adantl 262 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  RR )
76recnd 7054 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  CC )
8 simpl2 908 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
98recnd 7054 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
107, 9mulcld 7047 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  B )  e.  CC )
11 ax-1cn 6977 . . . . . 6  |-  1  e.  CC
12 subcl 7210 . . . . . 6  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1  -  x
)  e.  CC )
1311, 7, 12sylancr 393 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  -  x )  e.  CC )
14 simpl1 907 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1514recnd 7054 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
1613, 15mulcld 7047 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  e.  CC )
1710, 16addcomd 7164 . . 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 8871 . . 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 2114 . 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 103 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  ( A [,] B ) )
21 simpl1 907 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  RR )
22 simpl2 908 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  B  e.  RR )
23 elicc2 8807 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
24233adant3 924 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  <->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
2524biimpa 280 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
2625simp1d 916 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  RR )
27 eqid 2040 . . . . . . 7  |-  ( A  -  A )  =  ( A  -  A
)
28 eqid 2040 . . . . . . 7  |-  ( B  -  A )  =  ( B  -  A
)
2927, 28iccshftl 8864 . . . . . 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 1136 . . . . 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 135 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  ( ( A  -  A ) [,] ( B  -  A ) ) )
3226, 21resubcld 7379 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  RR )
3332recnd 7054 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  CC )
34 difrp 8619 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
3534biimp3a 1235 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
3635adantr 261 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  RR+ )
3736rpcnd 8624 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  CC )
38 rpap0 8599 . . . . . 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 7766 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( ( y  -  A )  / 
( B  -  A
) )  x.  ( B  -  A )
)  =  ( y  -  A ) )
4137mul02d 7389 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  0 )
4221recnd 7054 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  CC )
4342subidd 7310 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( A  -  A
)  =  0 )
4441, 43eqtr4d 2075 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  ( A  -  A ) )
4537mulid2d 7045 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 1  x.  ( B  -  A )
)  =  ( B  -  A ) )
4644, 45oveq12d 5530 . . . 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 2121 . . 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 7028 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
0  e.  RR )
49 1red 7042 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
1  e.  RR )
5032, 36rerpdivcld 8654 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  RR )
51 eqid 2040 . . . . 5  |-  ( 0  x.  ( B  -  A ) )  =  ( 0  x.  ( B  -  A )
)
52 eqid 2040 . . . . 5  |-  ( 1  x.  ( B  -  A ) )  =  ( 1  x.  ( B  -  A )
)
5351, 52iccdil 8866 . . . 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 1136 . . 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 156 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  ( 0 [,] 1 ) )
56 eqcom 2042 . . . 4  |-  ( x  =  ( ( y  -  A )  / 
( B  -  A
) )  <->  ( (
y  -  A )  /  ( B  -  A ) )  =  x )
5733adantrl 447 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
y  -  A )  e.  CC )
587adantrr 448 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  x  e.  CC )
5937adantrl 447 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A )  e.  CC )
6039adantrl 447 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A ) #  0 )
6157, 58, 59, 60divmulap3d 7799 . . . 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 181 . . 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 447 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  RR )
6463recnd 7054 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  CC )
6542adantrl 447 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  A  e.  CC )
668, 14resubcld 7379 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
676, 66remulcld 7056 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  e.  RR )
6867adantrr 448 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
x  x.  ( B  -  A ) )  e.  RR )
6968recnd 7054 . . . . 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 7341 . . . 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 2042 . . . 4  |-  ( ( ( x  x.  ( B  -  A )
)  +  A )  =  y  <->  y  =  ( ( x  x.  ( B  -  A
) )  +  A
) )
7270, 71syl6bb 185 . . 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 7047 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  A )  e.  CC )
7410, 73, 15subadd23d 7344 . . . . . 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 7411 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  =  ( ( x  x.  B
)  -  ( x  x.  A ) ) )
7675oveq1d 5527 . . . . . 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 7043 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
7877, 7, 15subdird 7412 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( ( 1  x.  A
)  -  ( x  x.  A ) ) )
7915mulid2d 7045 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
8079oveq1d 5527 . . . . . . . 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 2072 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( A  -  ( x  x.  A ) ) )
8281oveq2d 5528 . . . . . 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 2082 . . . . 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 448 . . . 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 2051 . . 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 203 . 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 5704 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 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   class class class wbr 3764    |-> cmpt 3818   `'ccnv 4344   -1-1-onto->wf1o 4901  (class class class)co 5512   CCcc 6887   RRcr 6888   0cc0 6889   1c1 6890    + caddc 6892    x. cmul 6894    < clt 7060    <_ cle 7061    - cmin 7182   # cap 7572    / cdiv 7651   RR+crp 8583   [,]cicc 8760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-rp 8584  df-icc 8764
This theorem is referenced by: (None)
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