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Theorem rmxypairf1o 26864
Description: The function used to extract rational and irrational parts in df-rmx 26855 and df-rmy 26856 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 6065 . . . 4  |-  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  e.  _V
2 eqid 2404 . . . 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 5532 . . 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 2919 . . . . . . . . . 10  |-  c  e. 
_V
6 vex 2919 . . . . . . . . . 10  |-  d  e. 
_V
75, 6op1std 6316 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( 1st `  b
)  =  c )
85, 6op2ndd 6317 . . . . . . . . . 10  |-  ( b  =  <. c ,  d
>.  ->  ( 2nd `  b
)  =  d )
98oveq2d 6056 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )
107, 9oveq12d 6058 . . . . . . . 8  |-  ( b  =  <. c ,  d
>.  ->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) )
1110eqeq2d 2415 . . . . . . 7  |-  ( b  =  <. c ,  d
>.  ->  ( a  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  <->  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) ) )
1211rexxp 4976 . . . . . 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 194 . . . . 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 5075 . . 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 2455 . 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 5687 . . . . . . . 8  |-  ( b  =  c  ->  ( 1st `  b )  =  ( 1st `  c
) )
19 fveq2 5687 . . . . . . . . 9  |-  ( b  =  c  ->  ( 2nd `  b )  =  ( 2nd `  c
) )
2019oveq2d 6056 . . . . . . . 8  |-  ( b  =  c  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c ) ) )
2118, 20oveq12d 6058 . . . . . . 7  |-  ( b  =  c  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  c
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c
) ) ) )
22 ovex 6065 . . . . . . 7  |-  ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) )  e.  _V
2321, 2, 22fvmpt 5765 . . . . . 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 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 ) ) ) ) `  c
)  =  ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) ) )
25 fveq2 5687 . . . . . . . 8  |-  ( b  =  d  ->  ( 1st `  b )  =  ( 1st `  d
) )
26 fveq2 5687 . . . . . . . . 9  |-  ( b  =  d  ->  ( 2nd `  b )  =  ( 2nd `  d
) )
2726oveq2d 6056 . . . . . . . 8  |-  ( b  =  d  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d ) ) )
2825, 27oveq12d 6058 . . . . . . 7  |-  ( b  =  d  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  d
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d
) ) ) )
29 ovex 6065 . . . . . . 7  |-  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) )  e.  _V
3028, 2, 29fvmpt 5765 . . . . . 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 710 . . . . 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 2418 . . . 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 26859 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  ( CC  \  QQ ) )
3433adantr 452 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( sqr `  (
( A ^ 2 )  -  1 ) )  e.  ( CC 
\  QQ ) )
35 nn0ssq 10538 . . . . . . . 8  |-  NN0  C_  QQ
36 xp1st 6335 . . . . . . . . 9  |-  ( c  e.  ( NN0  X.  ZZ )  ->  ( 1st `  c )  e.  NN0 )
3736ad2antrl 709 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  c
)  e.  NN0 )
3835, 37sseldi 3306 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  c
)  e.  QQ )
39 xp2nd 6336 . . . . . . . . 9  |-  ( c  e.  ( NN0  X.  ZZ )  ->  ( 2nd `  c )  e.  ZZ )
4039ad2antrl 709 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  c
)  e.  ZZ )
41 zq 10536 . . . . . . . 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 6335 . . . . . . . . 9  |-  ( d  e.  ( NN0  X.  ZZ )  ->  ( 1st `  d )  e.  NN0 )
4443ad2antll 710 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  d
)  e.  NN0 )
4535, 44sseldi 3306 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  d
)  e.  QQ )
46 xp2nd 6336 . . . . . . . . 9  |-  ( d  e.  ( NN0  X.  ZZ )  ->  ( 2nd `  d )  e.  ZZ )
4746ad2antll 710 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 2nd `  d
)  e.  ZZ )
48 zq 10536 . . . . . . . 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 26861 . . . . . . 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 1193 . . . . . 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 199 . . . . 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 6347 . . . . . 6  |-  ( ( c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) )  -> 
( ( ( 1st `  c )  =  ( 1st `  d )  /\  ( 2nd `  c
)  =  ( 2nd `  d ) )  <->  c  =  d ) )
5453adantl 453 . . . . 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 206 . . . 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 207 . . 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 2758 . 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 5972 . 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 1138 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667    \ cdif 3277   <.cop 3777    e. cmpt 4226    X. cxp 4835   ran crn 4838    Fn wfn 5408   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   QQcq 10530   ^cexp 11337   sqrcsqr 11993
This theorem is referenced by:  rmxyelxp  26865  rmxyval  26868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083
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