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Theorem cnref1o 8580
Description: There is a natural one-to-one mapping from  ( RR  X.  RR ) to  CC, where we map  <. x ,  y
>. to  ( x  +  ( _i  x.  y ) ). In our construction of the complex numbers, this is in fact our definition of  CC (see df-c 6893), but in the axiomatic treatment we can only show that there is the expected mapping between these two sets. (Contributed by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro, 17-Feb-2014.)
Hypothesis
Ref Expression
cnref1o.1  |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  (
_i  x.  y )
) )
Assertion
Ref Expression
cnref1o  |-  F :
( RR  X.  RR )
-1-1-onto-> CC
Distinct variable group:    x, y
Allowed substitution hints:    F( x, y)

Proof of Theorem cnref1o
Dummy variables  u  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 102 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  RR )
21recnd 7052 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  CC )
3 ax-icn 6977 . . . . . . . . 9  |-  _i  e.  CC
43a1i 9 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  _i  e.  CC )
5 simpr 103 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  RR )
65recnd 7052 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
74, 6mulcld 7045 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( _i  x.  y
)  e.  CC )
82, 7addcld 7044 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  e.  CC )
98rgen2a 2375 . . . . 5  |-  A. x  e.  RR  A. y  e.  RR  ( x  +  ( _i  x.  y
) )  e.  CC
10 cnref1o.1 . . . . . 6  |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  (
_i  x.  y )
) )
1110fnmpt2 5828 . . . . 5  |-  ( A. x  e.  RR  A. y  e.  RR  ( x  +  ( _i  x.  y
) )  e.  CC  ->  F  Fn  ( RR 
X.  RR ) )
129, 11ax-mp 7 . . . 4  |-  F  Fn  ( RR  X.  RR )
13 1st2nd2 5801 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1413fveq2d 5182 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
15 df-ov 5515 . . . . . . . 8  |-  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
1614, 15syl6eqr 2090 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
17 xp1st 5792 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
18 xp2nd 5793 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
1917recnd 7052 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  CC )
203a1i 9 . . . . . . . . . 10  |-  ( z  e.  ( RR  X.  RR )  ->  _i  e.  CC )
2118recnd 7052 . . . . . . . . . 10  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  CC )
2220, 21mulcld 7045 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  ( _i  x.  ( 2nd `  z
) )  e.  CC )
2319, 22addcld 7044 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  e.  CC )
24 oveq1 5519 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  ( _i  x.  y ) )  =  ( ( 1st `  z
)  +  ( _i  x.  y ) ) )
25 oveq2 5520 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( _i  x.  y )  =  ( _i  x.  ( 2nd `  z ) ) )
2625oveq2d 5528 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z )  +  ( _i  x.  y
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
2724, 26, 10ovmpt2g 5635 . . . . . . . 8  |-  ( ( ( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR  /\  (
( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) )  e.  CC )  ->  (
( 1st `  z
) F ( 2nd `  z ) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
2817, 18, 23, 27syl3anc 1135 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
2916, 28eqtrd 2072 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
3029, 23eqeltrd 2114 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  e.  CC )
3130rgen 2374 . . . 4  |-  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC
32 ffnfv 5323 . . . 4  |-  ( F : ( RR  X.  RR ) --> CC  <->  ( F  Fn  ( RR  X.  RR )  /\  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC ) )
3312, 31, 32mpbir2an 849 . . 3  |-  F :
( RR  X.  RR )
--> CC
3417, 18jca 290 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z )  e.  RR ) )
35 xp1st 5792 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  ( 1st `  w )  e.  RR )
36 xp2nd 5793 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  ( 2nd `  w )  e.  RR )
3735, 36jca 290 . . . . . . 7  |-  ( w  e.  ( RR  X.  RR )  ->  ( ( 1st `  w )  e.  RR  /\  ( 2nd `  w )  e.  RR ) )
38 cru 7591 . . . . . . 7  |-  ( ( ( ( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR )  /\  ( ( 1st `  w
)  e.  RR  /\  ( 2nd `  w )  e.  RR ) )  ->  ( ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) )  <->  ( ( 1st `  z )  =  ( 1st `  w
)  /\  ( 2nd `  z )  =  ( 2nd `  w ) ) ) )
3934, 37, 38syl2an 273 . . . . . 6  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  =  ( ( 1st `  w )  +  ( _i  x.  ( 2nd `  w ) ) )  <-> 
( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) ) )
40 fveq2 5178 . . . . . . . . 9  |-  ( z  =  w  ->  ( F `  z )  =  ( F `  w ) )
41 fveq2 5178 . . . . . . . . . 10  |-  ( z  =  w  ->  ( 1st `  z )  =  ( 1st `  w
) )
42 fveq2 5178 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( 2nd `  z )  =  ( 2nd `  w
) )
4342oveq2d 5528 . . . . . . . . . 10  |-  ( z  =  w  ->  (
_i  x.  ( 2nd `  z ) )  =  ( _i  x.  ( 2nd `  w ) ) )
4441, 43oveq12d 5530 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
4540, 44eqeq12d 2054 . . . . . . . 8  |-  ( z  =  w  ->  (
( F `  z
)  =  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  <->  ( F `  w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) ) )
4645, 29vtoclga 2619 . . . . . . 7  |-  ( w  e.  ( RR  X.  RR )  ->  ( F `
 w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
4729, 46eqeqan12d 2055 . . . . . 6  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  <-> 
( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) ) )
48 1st2nd2 5801 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  w  = 
<. ( 1st `  w
) ,  ( 2nd `  w ) >. )
4913, 48eqeqan12d 2055 . . . . . . 7  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( z  =  w  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
50 vex 2560 . . . . . . . . 9  |-  z  e. 
_V
51 1stexg 5794 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 1st `  z )  e. 
_V )
5250, 51ax-mp 7 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
53 2ndexg 5795 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 2nd `  z )  e. 
_V )
5450, 53ax-mp 7 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
5552, 54opth 3974 . . . . . . 7  |-  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. 
<->  ( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) )
5649, 55syl6bb 185 . . . . . 6  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( z  =  w  <-> 
( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) ) )
5739, 47, 563bitr4d 209 . . . . 5  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  <-> 
z  =  w ) )
5857biimpd 132 . . . 4  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  ->  z  =  w ) )
5958rgen2a 2375 . . 3  |-  A. z  e.  ( RR  X.  RR ) A. w  e.  ( RR  X.  RR ) ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
60 dff13 5407 . . 3  |-  ( F : ( RR  X.  RR ) -1-1-> CC  <->  ( F :
( RR  X.  RR )
--> CC  /\  A. z  e.  ( RR  X.  RR ) A. w  e.  ( RR  X.  RR ) ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) ) )
6133, 59, 60mpbir2an 849 . 2  |-  F :
( RR  X.  RR ) -1-1-> CC
62 cnre 7021 . . . . . 6  |-  ( w  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  w  =  ( u  +  ( _i  x.  v ) ) )
63 simpl 102 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  u  e.  RR )
64 simpr 103 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  v  e.  RR )
6563recnd 7052 . . . . . . . . . 10  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  u  e.  CC )
663a1i 9 . . . . . . . . . . 11  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  _i  e.  CC )
6764recnd 7052 . . . . . . . . . . 11  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  v  e.  CC )
6866, 67mulcld 7045 . . . . . . . . . 10  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( _i  x.  v
)  e.  CC )
6965, 68addcld 7044 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u  +  ( _i  x.  v ) )  e.  CC )
70 oveq1 5519 . . . . . . . . . 10  |-  ( x  =  u  ->  (
x  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  y
) ) )
71 oveq2 5520 . . . . . . . . . . 11  |-  ( y  =  v  ->  (
_i  x.  y )  =  ( _i  x.  v ) )
7271oveq2d 5528 . . . . . . . . . 10  |-  ( y  =  v  ->  (
u  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  v
) ) )
7370, 72, 10ovmpt2g 5635 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR  /\  (
u  +  ( _i  x.  v ) )  e.  CC )  -> 
( u F v )  =  ( u  +  ( _i  x.  v ) ) )
7463, 64, 69, 73syl3anc 1135 . . . . . . . 8  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u F v )  =  ( u  +  ( _i  x.  v ) ) )
7574eqeq2d 2051 . . . . . . 7  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( w  =  ( u F v )  <-> 
w  =  ( u  +  ( _i  x.  v ) ) ) )
76752rexbiia 2340 . . . . . 6  |-  ( E. u  e.  RR  E. v  e.  RR  w  =  ( u F v )  <->  E. u  e.  RR  E. v  e.  RR  w  =  ( u  +  ( _i  x.  v ) ) )
7762, 76sylibr 137 . . . . 5  |-  ( w  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  w  =  ( u F v ) )
78 fveq2 5178 . . . . . . . 8  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( F `  <. u ,  v >. )
)
79 df-ov 5515 . . . . . . . 8  |-  ( u F v )  =  ( F `  <. u ,  v >. )
8078, 79syl6eqr 2090 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( u F v ) )
8180eqeq2d 2051 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  ( w  =  ( F `  z
)  <->  w  =  (
u F v ) ) )
8281rexxp 4480 . . . . 5  |-  ( E. z  e.  ( RR 
X.  RR ) w  =  ( F `  z )  <->  E. u  e.  RR  E. v  e.  RR  w  =  ( u F v ) )
8377, 82sylibr 137 . . . 4  |-  ( w  e.  CC  ->  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
)
8483rgen 2374 . . 3  |-  A. w  e.  CC  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
85 dffo3 5314 . . 3  |-  ( F : ( RR  X.  RR ) -onto-> CC  <->  ( F :
( RR  X.  RR )
--> CC  /\  A. w  e.  CC  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
) )
8633, 84, 85mpbir2an 849 . 2  |-  F :
( RR  X.  RR ) -onto-> CC
87 df-f1o 4909 . 2  |-  ( F : ( RR  X.  RR ) -1-1-onto-> CC  <->  ( F :
( RR  X.  RR ) -1-1-> CC  /\  F :
( RR  X.  RR ) -onto-> CC ) )
8861, 86, 87mpbir2an 849 1  |-  F :
( RR  X.  RR )
-1-1-onto-> CC
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306   E.wrex 2307   _Vcvv 2557   <.cop 3378    X. cxp 4343    Fn wfn 4897   -->wf 4898   -1-1->wf1 4899   -onto->wfo 4900   -1-1-onto->wf1o 4901   ` cfv 4902  (class class class)co 5512    |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766   CCcc 6885   RRcr 6886   _ici 6889    + caddc 6890    x. cmul 6892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6973  ax-resscn 6974  ax-1cn 6975  ax-1re 6976  ax-icn 6977  ax-addcl 6978  ax-addrcl 6979  ax-mulcl 6980  ax-mulrcl 6981  ax-addcom 6982  ax-mulcom 6983  ax-addass 6984  ax-mulass 6985  ax-distr 6986  ax-i2m1 6987  ax-1rid 6989  ax-0id 6990  ax-rnegex 6991  ax-precex 6992  ax-cnre 6993  ax-pre-ltirr 6994  ax-pre-lttrn 6996  ax-pre-apti 6997  ax-pre-ltadd 6998  ax-pre-mulgt0 6999
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-iltp 6566  df-enr 6809  df-nr 6810  df-ltr 6813  df-0r 6814  df-1r 6815  df-0 6894  df-1 6895  df-r 6897  df-lt 6900  df-pnf 7060  df-mnf 7061  df-ltxr 7063  df-sub 7182  df-neg 7183  df-reap 7564
This theorem is referenced by:  cnrecnv  9484
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