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Theorem rmxypairf1o 27099
Description: The function used to extract rational and irrational parts in df-rmx 27090 and df-rmy 27091 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 5899 . . . 4  |-  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  e.  _V
2 eqid 2296 . . . 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 5388 . . 3  |-  ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) ) )  Fn  ( NN0  X.  ZZ )
43a1i 10 . 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 2804 . . . . . . . . . 10  |-  c  e. 
_V
6 vex 2804 . . . . . . . . . 10  |-  d  e. 
_V
75, 6op1std 6146 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( 1st `  b
)  =  c )
85, 6op2ndd 6147 . . . . . . . . . 10  |-  ( b  =  <. c ,  d
>.  ->  ( 2nd `  b
)  =  d )
98oveq2d 5890 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )
107, 9oveq12d 5892 . . . . . . . 8  |-  ( b  =  <. c ,  d
>.  ->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) )
1110eqeq2d 2307 . . . . . . 7  |-  ( b  =  <. c ,  d
>.  ->  ( a  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  <->  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) ) )
1211rexxp 4844 . . . . . 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 193 . . . . 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 10 . . . 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 2410 . . 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 4941 . . 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 2347 . 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 5541 . . . . . . . 8  |-  ( b  =  c  ->  ( 1st `  b )  =  ( 1st `  c
) )
19 fveq2 5541 . . . . . . . . 9  |-  ( b  =  c  ->  ( 2nd `  b )  =  ( 2nd `  c
) )
2019oveq2d 5890 . . . . . . . 8  |-  ( b  =  c  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c ) ) )
2118, 20oveq12d 5892 . . . . . . 7  |-  ( b  =  c  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  c
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c
) ) ) )
22 ovex 5899 . . . . . . 7  |-  ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) )  e.  _V
2321, 2, 22fvmpt 5618 . . . . . 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 708 . . . . 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 5541 . . . . . . . 8  |-  ( b  =  d  ->  ( 1st `  b )  =  ( 1st `  d
) )
26 fveq2 5541 . . . . . . . . 9  |-  ( b  =  d  ->  ( 2nd `  b )  =  ( 2nd `  d
) )
2726oveq2d 5890 . . . . . . . 8  |-  ( b  =  d  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d ) ) )
2825, 27oveq12d 5892 . . . . . . 7  |-  ( b  =  d  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  d
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d
) ) ) )
29 ovex 5899 . . . . . . 7  |-  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) )  e.  _V
3028, 2, 29fvmpt 5618 . . . . . 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 709 . . . . 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 2310 . . . 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 27094 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  ( CC  \  QQ ) )
3433adantr 451 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( sqr `  (
( A ^ 2 )  -  1 ) )  e.  ( CC 
\  QQ ) )
35 nn0ssq 10340 . . . . . . . 8  |-  NN0  C_  QQ
36 xp1st 6165 . . . . . . . . 9  |-  ( c  e.  ( NN0  X.  ZZ )  ->  ( 1st `  c )  e.  NN0 )
3736ad2antrl 708 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  c
)  e.  NN0 )
3835, 37sseldi 3191 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  c
)  e.  QQ )
39 xp2nd 6166 . . . . . . . . 9  |-  ( c  e.  ( NN0  X.  ZZ )  ->  ( 2nd `  c )  e.  ZZ )
4039ad2antrl 708 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  c
)  e.  ZZ )
41 zq 10338 . . . . . . . 8  |-  ( ( 2nd `  c )  e.  ZZ  ->  ( 2nd `  c )  e.  QQ )
4240, 41syl 15 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  c
)  e.  QQ )
43 xp1st 6165 . . . . . . . . 9  |-  ( d  e.  ( NN0  X.  ZZ )  ->  ( 1st `  d )  e.  NN0 )
4443ad2antll 709 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  d
)  e.  NN0 )
4535, 44sseldi 3191 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  d
)  e.  QQ )
46 xp2nd 6166 . . . . . . . . 9  |-  ( d  e.  ( NN0  X.  ZZ )  ->  ( 2nd `  d )  e.  ZZ )
4746ad2antll 709 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  d
)  e.  ZZ )
48 zq 10338 . . . . . . . 8  |-  ( ( 2nd `  d )  e.  ZZ  ->  ( 2nd `  d )  e.  QQ )
4947, 48syl 15 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  d
)  e.  QQ )
50 qirropth 27096 . . . . . . 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 1191 . . . . . 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 198 . . . . 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 6177 . . . . . 6  |-  ( ( c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) )  -> 
( ( ( 1st `  c )  =  ( 1st `  d )  /\  ( 2nd `  c
)  =  ( 2nd `  d ) )  <->  c  =  d ) )
5453adantl 452 . . . . 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 205 . . . 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 206 . . 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 2648 . 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 5807 . 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 1136 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    \ cdif 3162   <.cop 3656    e. cmpt 4093    X. cxp 4703   ran crn 4706    Fn wfn 5266   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   QQcq 10332   ^cexp 11120   sqrcsqr 11734
This theorem is referenced by:  rmxyelxp  27100  rmxyval  27103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-numer 12822  df-denom 12823
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