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Theorem iccf1o 11776
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 9643 . . . . . . . . 9  |-  0  e.  RR
3 1re 9642 . . . . . . . . 9  |-  1  e.  RR
42, 3elicc2i 11700 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <_  1
) )
54simp1bi 1023 . . . . . . 7  |-  ( x  e.  ( 0 [,] 1 )  ->  x  e.  RR )
65adantl 468 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  RR )
76recnd 9669 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  CC )
8 simpl2 1012 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
98recnd 9669 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
107, 9mulcld 9663 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  B )  e.  CC )
11 ax-1cn 9597 . . . . . 6  |-  1  e.  CC
12 subcl 9874 . . . . . 6  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1  -  x
)  e.  CC )
1311, 7, 12sylancr 669 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  -  x )  e.  CC )
14 simpl1 1011 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1514recnd 9669 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
1613, 15mulcld 9663 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  e.  CC )
1710, 16addcomd 9835 . . 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 11775 . . 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 2529 . 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 463 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  ( A [,] B ) )
21 simpl1 1011 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  RR )
22 simpl2 1012 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  B  e.  RR )
23 elicc2 11699 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
24233adant3 1028 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  <->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
2524biimpa 487 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
2625simp1d 1020 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  RR )
27 eqid 2451 . . . . . . 7  |-  ( A  -  A )  =  ( A  -  A
)
28 eqid 2451 . . . . . . 7  |-  ( B  -  A )  =  ( B  -  A
)
2927, 28iccshftl 11768 . . . . . 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 1269 . . . . 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 214 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  ( ( A  -  A ) [,] ( B  -  A ) ) )
3226, 21resubcld 10047 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  RR )
3332recnd 9669 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  CC )
34 difrp 11337 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
3534biimp3a 1369 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
3635adantr 467 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  RR+ )
3736rpcnd 11343 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  CC )
3836rpne0d 11346 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  =/=  0 )
3933, 37, 38divcan1d 10384 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( ( y  -  A )  / 
( B  -  A
) )  x.  ( B  -  A )
)  =  ( y  -  A ) )
4037mul02d 9831 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  0 )
4121recnd 9669 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  CC )
4241subidd 9974 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( A  -  A
)  =  0 )
4340, 42eqtr4d 2488 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  ( A  -  A ) )
4437mulid2d 9661 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 1  x.  ( B  -  A )
)  =  ( B  -  A ) )
4543, 44oveq12d 6308 . . . 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 ) ) )
4631, 39, 453eltr4d 2544 . . 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
) ) ) )
47 0red 9644 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
0  e.  RR )
48 1red 9658 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
1  e.  RR )
4932, 36rerpdivcld 11369 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  RR )
50 eqid 2451 . . . . 5  |-  ( 0  x.  ( B  -  A ) )  =  ( 0  x.  ( B  -  A )
)
51 eqid 2451 . . . . 5  |-  ( 1  x.  ( B  -  A ) )  =  ( 1  x.  ( B  -  A )
)
5250, 51iccdil 11770 . . . 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 ) ) ) ) )
5347, 48, 49, 36, 52syl22anc 1269 . . 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
) ) ) ) )
5446, 53mpbird 236 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  ( 0 [,] 1 ) )
55 eqcom 2458 . . . 4  |-  ( x  =  ( ( y  -  A )  / 
( B  -  A
) )  <->  ( (
y  -  A )  /  ( B  -  A ) )  =  x )
5633adantrl 722 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
y  -  A )  e.  CC )
577adantrr 723 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  x  e.  CC )
5837adantrl 722 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A )  e.  CC )
5938adantrl 722 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A )  =/=  0 )
6056, 57, 58, 59divmul3d 10417 . . . 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 ) ) ) )
6155, 60syl5bb 261 . . 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 ) ) ) )
6226adantrl 722 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  RR )
6362recnd 9669 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  CC )
6441adantrl 722 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  A  e.  CC )
658, 14resubcld 10047 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
666, 65remulcld 9671 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  e.  RR )
6766adantrr 723 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
x  x.  ( B  -  A ) )  e.  RR )
6867recnd 9669 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
x  x.  ( B  -  A ) )  e.  CC )
6963, 64, 68subadd2d 10005 . . . 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 ) )
70 eqcom 2458 . . . 4  |-  ( ( ( x  x.  ( B  -  A )
)  +  A )  =  y  <->  y  =  ( ( x  x.  ( B  -  A
) )  +  A
) )
7169, 70syl6bb 265 . . 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
) ) )
727, 15mulcld 9663 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  A )  e.  CC )
7310, 72, 15subadd23d 10008 . . . . . 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 ) ) ) )
747, 9, 15subdid 10074 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  =  ( ( x  x.  B
)  -  ( x  x.  A ) ) )
7574oveq1d 6305 . . . . . 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 ) )
76 1cnd 9659 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
7776, 7, 15subdird 10075 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( ( 1  x.  A
)  -  ( x  x.  A ) ) )
7815mulid2d 9661 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
7978oveq1d 6305 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  x.  A )  -  ( x  x.  A
) )  =  ( A  -  ( x  x.  A ) ) )
8077, 79eqtrd 2485 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( A  -  ( x  x.  A ) ) )
8180oveq2d 6306 . . . . . 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 ) ) ) )
8273, 75, 813eqtr4d 2495 . . . . 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 ) ) )
8382adantrr 723 . . . 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
) ) )
8483eqeq2d 2461 . . 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 ) ) ) )
8561, 71, 843bitrd 283 . 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 ) ) ) )
861, 19, 54, 85f1ocnv2d 6520 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833   -1-1-onto->wf1o 5581  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   RR+crp 11302   [,]cicc 11638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-rp 11303  df-icc 11642
This theorem is referenced by:  iccen  11777  icchmeo  21969
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