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Theorem iccf1o 9794
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 7773 . . . . . . . . 9  |-  0  e.  RR
3 1re 7772 . . . . . . . . 9  |-  1  e.  RR
42, 3elicc2i 9729 . . . . . . . 8  |-  ( x  e.  ( 0 [,] 1 )  <->  ( x  e.  RR  /\  0  <_  x  /\  x  <_  1
) )
54simp1bi 996 . . . . . . 7  |-  ( x  e.  ( 0 [,] 1 )  ->  x  e.  RR )
65adantl 275 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  RR )
76recnd 7801 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  x  e.  CC )
8 simpl2 985 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
98recnd 7801 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
107, 9mulcld 7793 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  B )  e.  CC )
11 ax-1cn 7720 . . . . . 6  |-  1  e.  CC
12 subcl 7968 . . . . . 6  |-  ( ( 1  e.  CC  /\  x  e.  CC )  ->  ( 1  -  x
)  e.  CC )
1311, 7, 12sylancr 410 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  -  x )  e.  CC )
14 simpl1 984 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1514recnd 7801 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
1613, 15mulcld 7793 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  e.  CC )
1710, 16addcomd 7920 . . 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 9793 . . 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 2216 . 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 109 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  ( A [,] B ) )
21 simpl1 984 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  RR )
22 simpl2 985 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  B  e.  RR )
23 elicc2 9728 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
24233adant3 1001 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
y  e.  ( A [,] B )  <->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) ) )
2524biimpa 294 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) )
2625simp1d 993 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
y  e.  RR )
27 eqid 2139 . . . . . . 7  |-  ( A  -  A )  =  ( A  -  A
)
28 eqid 2139 . . . . . . 7  |-  ( B  -  A )  =  ( B  -  A
)
2927, 28iccshftl 9786 . . . . . 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 1217 . . . . 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 146 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  ( ( A  -  A ) [,] ( B  -  A ) ) )
3226, 21resubcld 8150 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  RR )
3332recnd 7801 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( y  -  A
)  e.  CC )
34 difrp 9487 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
3534biimp3a 1323 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
3635adantr 274 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  RR+ )
3736rpcnd 9492 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( B  -  A
)  e.  CC )
38 rpap0 9465 . . . . . 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 8558 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( ( y  -  A )  / 
( B  -  A
) )  x.  ( B  -  A )
)  =  ( y  -  A ) )
4137mul02d 8161 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  0 )
4221recnd 7801 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  ->  A  e.  CC )
4342subidd 8068 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( A  -  A
)  =  0 )
4441, 43eqtr4d 2175 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 0  x.  ( B  -  A )
)  =  ( A  -  A ) )
4537mulid2d 7791 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( 1  x.  ( B  -  A )
)  =  ( B  -  A ) )
4644, 45oveq12d 5792 . . . 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 2223 . . 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 7774 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
0  e.  RR )
49 1red 7788 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
1  e.  RR )
5032, 36rerpdivcld 9522 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  RR )
51 eqid 2139 . . . . 5  |-  ( 0  x.  ( B  -  A ) )  =  ( 0  x.  ( B  -  A )
)
52 eqid 2139 . . . . 5  |-  ( 1  x.  ( B  -  A ) )  =  ( 1  x.  ( B  -  A )
)
5351, 52iccdil 9788 . . . 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 1217 . . 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 166 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  y  e.  ( A [,] B ) )  -> 
( ( y  -  A )  /  ( B  -  A )
)  e.  ( 0 [,] 1 ) )
56 eqcom 2141 . . . 4  |-  ( x  =  ( ( y  -  A )  / 
( B  -  A
) )  <->  ( (
y  -  A )  /  ( B  -  A ) )  =  x )
5733adantrl 469 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
y  -  A )  e.  CC )
587adantrr 470 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  x  e.  CC )
5937adantrl 469 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A )  e.  CC )
6039adantrl 469 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  ( B  -  A ) #  0 )
6157, 58, 59, 60divmulap3d 8592 . . . 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 191 . . 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 469 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  RR )
6463recnd 7801 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  y  e.  CC )
6542adantrl 469 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  A  e.  CC )
668, 14resubcld 8150 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
676, 66remulcld 7803 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  e.  RR )
6867adantrr 470 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( x  e.  (
0 [,] 1 )  /\  y  e.  ( A [,] B ) ) )  ->  (
x  x.  ( B  -  A ) )  e.  RR )
6968recnd 7801 . . . . 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 8099 . . . 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 2141 . . . 4  |-  ( ( ( x  x.  ( B  -  A )
)  +  A )  =  y  <->  y  =  ( ( x  x.  ( B  -  A
) )  +  A
) )
7270, 71syl6bb 195 . . 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 7793 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  A )  e.  CC )
7410, 73, 15subadd23d 8102 . . . . . 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 8183 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( x  x.  ( B  -  A
) )  =  ( ( x  x.  B
)  -  ( x  x.  A ) ) )
7675oveq1d 5789 . . . . . 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 7789 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
7877, 7, 15subdird 8184 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( ( 1  x.  A
)  -  ( x  x.  A ) ) )
7915mulid2d 7791 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
8079oveq1d 5789 . . . . . . . 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 2172 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  x )  x.  A )  =  ( A  -  ( x  x.  A ) ) )
8281oveq2d 5790 . . . . . 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 2182 . . . . 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 470 . . . 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 2151 . . 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 213 . 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 5974 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 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929    |-> cmpt 3989   `'ccnv 4538   -1-1-onto->wf1o 5122  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627   1c1 7628    + caddc 7630    x. cmul 7632    < clt 7807    <_ cle 7808    - cmin 7940   # cap 8350    / cdiv 8439   RR+crp 9448   [,]cicc 9681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-rp 9449  df-icc 9685
This theorem is referenced by: (None)
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