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Theorem rmxypairf1o 29250
Description: The function used to extract rational and irrational parts in df-rmx 29241 and df-rmy 29242 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 6115 . . . 4  |-  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  e.  _V
2 eqid 2442 . . . 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 5538 . . 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 2974 . . . . . . . . . 10  |-  c  e. 
_V
6 vex 2974 . . . . . . . . . 10  |-  d  e. 
_V
75, 6op1std 6586 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( 1st `  b
)  =  c )
85, 6op2ndd 6587 . . . . . . . . . 10  |-  ( b  =  <. c ,  d
>.  ->  ( 2nd `  b
)  =  d )
98oveq2d 6106 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )
107, 9oveq12d 6108 . . . . . . . 8  |-  ( b  =  <. c ,  d
>.  ->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) )
1110eqeq2d 2453 . . . . . . 7  |-  ( b  =  <. c ,  d
>.  ->  ( a  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  <->  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) ) )
1211rexxp 4981 . . . . . 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 2556 . . 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 5084 . . 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 2493 . 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 5690 . . . . . . . 8  |-  ( b  =  c  ->  ( 1st `  b )  =  ( 1st `  c
) )
19 fveq2 5690 . . . . . . . . 9  |-  ( b  =  c  ->  ( 2nd `  b )  =  ( 2nd `  c
) )
2019oveq2d 6106 . . . . . . . 8  |-  ( b  =  c  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c ) ) )
2118, 20oveq12d 6108 . . . . . . 7  |-  ( b  =  c  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  c
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c
) ) ) )
22 ovex 6115 . . . . . . 7  |-  ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) )  e.  _V
2321, 2, 22fvmpt 5773 . . . . . 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 727 . . . . 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 5690 . . . . . . . 8  |-  ( b  =  d  ->  ( 1st `  b )  =  ( 1st `  d
) )
26 fveq2 5690 . . . . . . . . 9  |-  ( b  =  d  ->  ( 2nd `  b )  =  ( 2nd `  d
) )
2726oveq2d 6106 . . . . . . . 8  |-  ( b  =  d  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d ) ) )
2825, 27oveq12d 6108 . . . . . . 7  |-  ( b  =  d  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  d
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d
) ) ) )
29 ovex 6115 . . . . . . 7  |-  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) )  e.  _V
3028, 2, 29fvmpt 5773 . . . . . 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 728 . . . . 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 2456 . . . 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 rmspecsqrnq 29245 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  ( CC  \  QQ ) )
3433adantr 465 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( sqr `  (
( A ^ 2 )  -  1 ) )  e.  ( CC 
\  QQ ) )
35 nn0ssq 10960 . . . . . . . 8  |-  NN0  C_  QQ
36 xp1st 6605 . . . . . . . . 9  |-  ( c  e.  ( NN0  X.  ZZ )  ->  ( 1st `  c )  e.  NN0 )
3736ad2antrl 727 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  c
)  e.  NN0 )
3835, 37sseldi 3353 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  c
)  e.  QQ )
39 xp2nd 6606 . . . . . . . . 9  |-  ( c  e.  ( NN0  X.  ZZ )  ->  ( 2nd `  c )  e.  ZZ )
4039ad2antrl 727 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  c
)  e.  ZZ )
41 zq 10958 . . . . . . . 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 6605 . . . . . . . . 9  |-  ( d  e.  ( NN0  X.  ZZ )  ->  ( 1st `  d )  e.  NN0 )
4443ad2antll 728 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  d
)  e.  NN0 )
4535, 44sseldi 3353 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  d
)  e.  QQ )
46 xp2nd 6606 . . . . . . . . 9  |-  ( d  e.  ( NN0  X.  ZZ )  ->  ( 2nd `  d )  e.  ZZ )
4746ad2antll 728 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  d
)  e.  ZZ )
48 zq 10958 . . . . . . . 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 29247 . . . . . . 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 1227 . . . . . 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 6614 . . . . . 6  |-  ( ( c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) )  -> 
( ( ( 1st `  c )  =  ( 1st `  d )  /\  ( 2nd `  c
)  =  ( 2nd `  d ) )  <->  c  =  d ) )
5453adantl 466 . . . . 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 2807 . 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 5981 . 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 1172 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 369    = wceq 1369    e. wcel 1756   {cab 2428   A.wral 2714   E.wrex 2715    \ cdif 3324   <.cop 3882    e. cmpt 4349    X. cxp 4837   ran crn 4840    Fn wfn 5412   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   CCcc 9279   1c1 9282    + caddc 9284    x. cmul 9286    - cmin 9594   2c2 10370   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860   QQcq 10952   ^cexp 11864   sqrcsqr 12721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-q 10953  df-rp 10991  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-dvds 13535  df-gcd 13690  df-numer 13812  df-denom 13813
This theorem is referenced by:  rmxyelxp  29251  rmxyval  29254
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