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Theorem rmxypairf1o 30440
Description: The function used to extract rational and irrational parts in df-rmx 30431 and df-rmy 30432 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 6302 . . . 4  |-  ( ( 1st `  b )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  e.  _V
2 eqid 2462 . . . 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 5702 . . 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 3111 . . . . . . . . . 10  |-  c  e. 
_V
6 vex 3111 . . . . . . . . . 10  |-  d  e. 
_V
75, 6op1std 6786 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( 1st `  b
)  =  c )
85, 6op2ndd 6787 . . . . . . . . . 10  |-  ( b  =  <. c ,  d
>.  ->  ( 2nd `  b
)  =  d )
98oveq2d 6293 . . . . . . . . 9  |-  ( b  =  <. c ,  d
>.  ->  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )
107, 9oveq12d 6295 . . . . . . . 8  |-  ( b  =  <. c ,  d
>.  ->  ( ( 1st `  b )  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) ) )  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) )
1110eqeq2d 2476 . . . . . . 7  |-  ( b  =  <. c ,  d
>.  ->  ( a  =  ( ( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  <->  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) ) )
1211rexxp 5138 . . . . . 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 2598 . . 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 5241 . . 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 2522 . 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 5859 . . . . . . . 8  |-  ( b  =  c  ->  ( 1st `  b )  =  ( 1st `  c
) )
19 fveq2 5859 . . . . . . . . 9  |-  ( b  =  c  ->  ( 2nd `  b )  =  ( 2nd `  c
) )
2019oveq2d 6293 . . . . . . . 8  |-  ( b  =  c  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c ) ) )
2118, 20oveq12d 6295 . . . . . . 7  |-  ( b  =  c  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  c
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  c
) ) ) )
22 ovex 6302 . . . . . . 7  |-  ( ( 1st `  c )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  c ) ) )  e.  _V
2321, 2, 22fvmpt 5943 . . . . . 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 5859 . . . . . . . 8  |-  ( b  =  d  ->  ( 1st `  b )  =  ( 1st `  d
) )
26 fveq2 5859 . . . . . . . . 9  |-  ( b  =  d  ->  ( 2nd `  b )  =  ( 2nd `  d
) )
2726oveq2d 6293 . . . . . . . 8  |-  ( b  =  d  ->  (
( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b ) )  =  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d ) ) )
2825, 27oveq12d 6295 . . . . . . 7  |-  ( b  =  d  ->  (
( 1st `  b
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
) ) )  =  ( ( 1st `  d
)  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  d
) ) ) )
29 ovex 6302 . . . . . . 7  |-  ( ( 1st `  d )  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  ( 2nd `  d ) ) )  e.  _V
3028, 2, 29fvmpt 5943 . . . . . 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 2484 . . . 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 30435 . . . . . . . 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 11181 . . . . . . . 8  |-  NN0  C_  QQ
36 xp1st 6806 . . . . . . . . 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 3497 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  c
)  e.  QQ )
39 xp2nd 6807 . . . . . . . . 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 11179 . . . . . . . 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 6806 . . . . . . . . 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 3497 . . . . . . 7  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  (
c  e.  ( NN0 
X.  ZZ )  /\  d  e.  ( NN0  X.  ZZ ) ) )  ->  ( 1st `  d
)  e.  QQ )
46 xp2nd 6807 . . . . . . . . 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 11179 . . . . . . . 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 30437 . . . . . . 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 1232 . . . . . 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 6815 . . . . . 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 2880 . 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 6162 . 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 1175 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 1374    e. wcel 1762   {cab 2447   A.wral 2809   E.wrex 2810    \ cdif 3468   <.cop 4028    |-> cmpt 4500    X. cxp 4992   ran crn 4995    Fn wfn 5576   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   1stc1st 6774   2ndc2nd 6775   CCcc 9481   1c1 9484    + caddc 9486    x. cmul 9488    - cmin 9796   2c2 10576   NN0cn0 10786   ZZcz 10855   ZZ>=cuz 11073   QQcq 11173   ^cexp 12124   sqrcsqr 13018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-dvds 13839  df-gcd 13995  df-numer 14118  df-denom 14119
This theorem is referenced by:  rmxyelxp  30441  rmxyval  30444
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