Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rmxypairf1o Structured version   Unicode version

Theorem rmxypairf1o 31012
Description: The function used to extract rational and irrational parts in df-rmx 31003 and df-rmy 31004 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Assertion
Ref Expression
rmxypairf1o  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) : ( NN0 
X.  ZZ ) -1-1-onto-> { a  |  E. c  e. 
NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) ) } )
Distinct variable group:    b, c, d, a, A

Proof of Theorem rmxypairf1o
StepHypRef Expression
1 ovex 6224 . . . 4  |-  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  e.  _V
2 eqid 2382 . . . 4  |-  ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) )  =  ( b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )
31, 2fnmpti 5617 . . 3  |-  ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) )  Fn  ( NN0  X.  ZZ )
43a1i 11 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) )  Fn  ( NN0 
X.  ZZ ) )
5 vex 3037 . . . . . . . . . 10  |-  c  e. 
_V
6 vex 3037 . . . . . . . . . 10  |-  d  e. 
_V
75, 6op1std 6709 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( 1st `  b
)  =  c )
85, 6op2ndd 6710 . . . . . . . . . 10  |-  ( b  =  <. c ,  d
>.  ->  ( 2nd `  b
)  =  d )
98oveq2d 6212 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )
107, 9oveq12d 6214 . . . . . . . 8  |-  ( b  =  <. c ,  d
>.  ->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) )
1110eqeq2d 2396 . . . . . . 7  |-  ( b  =  <. c ,  d
>.  ->  ( a  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  <->  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) ) )
1211rexxp 5058 . . . . . 6  |-  ( E. b  e.  ( NN0 
X.  ZZ ) a  =  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  <->  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) )
1312bicomi 202 . . . . 5  |-  ( E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) )  <->  E. b  e.  ( NN0  X.  ZZ ) a  =  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) )
1413a1i 11 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) )  <->  E. b  e.  ( NN0  X.  ZZ ) a  =  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) )
1514abbidv 2518 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  { a  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) }  =  {
a  |  E. b  e.  ( NN0  X.  ZZ ) a  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) } )
162rnmpt 5161 . . 3  |-  ran  (
b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )  =  { a  |  E. b  e.  ( NN0  X.  ZZ ) a  =  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) }
1715, 16syl6reqr 2442 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ran  ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) )  =  {
a  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) } )
18 fveq2 5774 . . . . . . . 8  |-  ( b  =  c  ->  ( 1st `  b )  =  ( 1st `  c
) )
19 fveq2 5774 . . . . . . . . 9  |-  ( b  =  c  ->  ( 2nd `  b )  =  ( 2nd `  c
) )
2019oveq2d 6212 . . . . . . . 8  |-  ( b  =  c  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c ) ) )
2118, 20oveq12d 6214 . . . . . . 7  |-  ( b  =  c  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  c
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c
) ) ) )
22 ovex 6224 . . . . . . 7  |-  ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) )  e.  _V
2321, 2, 22fvmpt 5857 . . . . . 6  |-  ( c  e.  ( NN0  X.  ZZ )  ->  ( ( b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 c )  =  ( ( 1st `  c
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c
) ) ) )
2423ad2antrl 725 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  c
)  =  ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) ) )
25 fveq2 5774 . . . . . . . 8  |-  ( b  =  d  ->  ( 1st `  b )  =  ( 1st `  d
) )
26 fveq2 5774 . . . . . . . . 9  |-  ( b  =  d  ->  ( 2nd `  b )  =  ( 2nd `  d
) )
2726oveq2d 6212 . . . . . . . 8  |-  ( b  =  d  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d ) ) )
2825, 27oveq12d 6214 . . . . . . 7  |-  ( b  =  d  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  d
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d
) ) ) )
29 ovex 6224 . . . . . . 7  |-  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) )  e.  _V
3028, 2, 29fvmpt 5857 . . . . . 6  |-  ( d  e.  ( NN0  X.  ZZ )  ->  ( ( b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 d )  =  ( ( 1st `  d
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d
) ) ) )
3130ad2antll 726 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  d
)  =  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) ) )
3224, 31eqeq12d 2404 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( ( ( b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 c )  =  ( ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  d )  <-> 
( ( 1st `  c
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c
) ) )  =  ( ( 1st `  d
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d
) ) ) ) )
33 rmspecsqrtnq 31007 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  ( CC  \  QQ ) )
3433adantr 463 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( sqr `  (
( A ^ 2 )  -  1 ) )  e.  ( CC 
\  QQ ) )
35 nn0ssq 11109 . . . . . . . 8  |-  NN0  C_  QQ
36 xp1st 6729 . . . . . . . . 9  |-  ( c  e.  ( NN0  X.  ZZ )  ->  ( 1st `  c )  e.  NN0 )
3736ad2antrl 725 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  c
)  e.  NN0 )
3835, 37sseldi 3415 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  c
)  e.  QQ )
39 xp2nd 6730 . . . . . . . . 9  |-  ( c  e.  ( NN0  X.  ZZ )  ->  ( 2nd `  c )  e.  ZZ )
4039ad2antrl 725 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  c
)  e.  ZZ )
41 zq 11107 . . . . . . . 8  |-  ( ( 2nd `  c )  e.  ZZ  ->  ( 2nd `  c )  e.  QQ )
4240, 41syl 16 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  c
)  e.  QQ )
43 xp1st 6729 . . . . . . . . 9  |-  ( d  e.  ( NN0  X.  ZZ )  ->  ( 1st `  d )  e.  NN0 )
4443ad2antll 726 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  d
)  e.  NN0 )
4535, 44sseldi 3415 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  d
)  e.  QQ )
46 xp2nd 6730 . . . . . . . . 9  |-  ( d  e.  ( NN0  X.  ZZ )  ->  ( 2nd `  d )  e.  ZZ )
4746ad2antll 726 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  d
)  e.  ZZ )
48 zq 11107 . . . . . . . 8  |-  ( ( 2nd `  d )  e.  ZZ  ->  ( 2nd `  d )  e.  QQ )
4947, 48syl 16 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  d
)  e.  QQ )
50 qirropth 31009 . . . . . . 7  |-  ( ( ( sqr `  (
( A ^ 2 )  -  1 ) )  e.  ( CC 
\  QQ )  /\  ( ( 1st `  c
)  e.  QQ  /\  ( 2nd `  c )  e.  QQ )  /\  ( ( 1st `  d
)  e.  QQ  /\  ( 2nd `  d )  e.  QQ ) )  ->  ( ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) )  =  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) )  <->  ( ( 1st `  c )  =  ( 1st `  d )  /\  ( 2nd `  c
)  =  ( 2nd `  d ) ) ) )
5134, 38, 42, 45, 49, 50syl122anc 1235 . . . . . 6  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) )  =  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) )  <->  ( ( 1st `  c )  =  ( 1st `  d )  /\  ( 2nd `  c
)  =  ( 2nd `  d ) ) ) )
5251biimpd 207 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) )  =  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) )  ->  ( ( 1st `  c )  =  ( 1st `  d
)  /\  ( 2nd `  c )  =  ( 2nd `  d ) ) ) )
53 xpopth 6738 . . . . . 6  |-  ( ( c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) )  -> 
( ( ( 1st `  c )  =  ( 1st `  d )  /\  ( 2nd `  c
)  =  ( 2nd `  d ) )  <->  c  =  d ) )
5453adantl 464 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( ( ( 1st `  c )  =  ( 1st `  d
)  /\  ( 2nd `  c )  =  ( 2nd `  d ) )  <->  c  =  d ) )
5552, 54sylibd 214 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) )  =  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) )  ->  c  =  d ) )
5632, 55sylbid 215 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( ( ( b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 c )  =  ( ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  d )  ->  c  =  d ) )
5756ralrimivva 2803 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  A. c  e.  ( NN0  X.  ZZ ) A. d  e.  ( NN0  X.  ZZ ) ( ( ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  c
)  =  ( ( b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 d )  -> 
c  =  d ) )
58 dff1o6 6082 . 2  |-  ( ( b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) : ( NN0  X.  ZZ ) -1-1-onto-> { a  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) }  <->  ( ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) )  Fn  ( NN0  X.  ZZ )  /\  ran  ( b  e.  ( NN0  X.  ZZ ) 
|->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) )  =  { a  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) }  /\  A. c  e.  ( NN0  X.  ZZ ) A. d  e.  ( NN0  X.  ZZ ) ( ( ( b  e.  ( NN0 
X.  ZZ )  |->  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) ) ) `
 c )  =  ( ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) `  d )  ->  c  =  d ) ) )
594, 17, 57, 58syl3anbrc 1178 1  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) ) : ( NN0 
X.  ZZ ) -1-1-onto-> { a  |  E. c  e. 
NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   {cab 2367   A.wral 2732   E.wrex 2733    \ cdif 3386   <.cop 3950    |-> cmpt 4425    X. cxp 4911   ran crn 4914    Fn wfn 5491   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   1stc1st 6697   2ndc2nd 6698   CCcc 9401   1c1 9404    + caddc 9406    x. cmul 9408    - cmin 9718   2c2 10502   NN0cn0 10712   ZZcz 10781   ZZ>=cuz 11001   QQcq 11101   ^cexp 12069   sqrcsqrt 13068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-dvds 13989  df-gcd 14147  df-numer 14270  df-denom 14271
This theorem is referenced by:  rmxyelxp  31013  rmxyval  31016
  Copyright terms: Public domain W3C validator