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Theorem cnref1o 8814
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 7049), 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 107 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  RR )
21recnd 7209 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  CC )
3 ax-icn 7133 . . . . . . . . 9  |-  _i  e.  CC
43a1i 9 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  _i  e.  CC )
5 simpr 108 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  RR )
65recnd 7209 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
74, 6mulcld 7201 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( _i  x.  y
)  e.  CC )
82, 7addcld 7200 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  e.  CC )
98rgen2a 2418 . . . . 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 5859 . . . . 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 5832 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1413fveq2d 5213 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
15 df-ov 5546 . . . . . . . 8  |-  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
1614, 15syl6eqr 2132 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
17 xp1st 5823 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
18 xp2nd 5824 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
1917recnd 7209 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  CC )
203a1i 9 . . . . . . . . . 10  |-  ( z  e.  ( RR  X.  RR )  ->  _i  e.  CC )
2118recnd 7209 . . . . . . . . . 10  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  CC )
2220, 21mulcld 7201 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  ( _i  x.  ( 2nd `  z
) )  e.  CC )
2319, 22addcld 7200 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  e.  CC )
24 oveq1 5550 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  ( _i  x.  y ) )  =  ( ( 1st `  z
)  +  ( _i  x.  y ) ) )
25 oveq2 5551 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( _i  x.  y )  =  ( _i  x.  ( 2nd `  z ) ) )
2625oveq2d 5559 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z )  +  ( _i  x.  y
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
2724, 26, 10ovmpt2g 5666 . . . . . . . 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 1170 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
2916, 28eqtrd 2114 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
3029, 23eqeltrd 2156 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  e.  CC )
3130rgen 2417 . . . 4  |-  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC
32 ffnfv 5355 . . . 4  |-  ( F : ( RR  X.  RR ) --> CC  <->  ( F  Fn  ( RR  X.  RR )  /\  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC ) )
3312, 31, 32mpbir2an 884 . . 3  |-  F :
( RR  X.  RR )
--> CC
3417, 18jca 300 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z )  e.  RR ) )
35 xp1st 5823 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  ( 1st `  w )  e.  RR )
36 xp2nd 5824 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  ( 2nd `  w )  e.  RR )
3735, 36jca 300 . . . . . . 7  |-  ( w  e.  ( RR  X.  RR )  ->  ( ( 1st `  w )  e.  RR  /\  ( 2nd `  w )  e.  RR ) )
38 cru 7769 . . . . . . 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 283 . . . . . 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 5209 . . . . . . . . 9  |-  ( z  =  w  ->  ( F `  z )  =  ( F `  w ) )
41 fveq2 5209 . . . . . . . . . 10  |-  ( z  =  w  ->  ( 1st `  z )  =  ( 1st `  w
) )
42 fveq2 5209 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( 2nd `  z )  =  ( 2nd `  w
) )
4342oveq2d 5559 . . . . . . . . . 10  |-  ( z  =  w  ->  (
_i  x.  ( 2nd `  z ) )  =  ( _i  x.  ( 2nd `  w ) ) )
4441, 43oveq12d 5561 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
4540, 44eqeq12d 2096 . . . . . . . 8  |-  ( z  =  w  ->  (
( F `  z
)  =  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  <->  ( F `  w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) ) )
4645, 29vtoclga 2665 . . . . . . 7  |-  ( w  e.  ( RR  X.  RR )  ->  ( F `
 w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
4729, 46eqeqan12d 2097 . . . . . 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 5832 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  w  = 
<. ( 1st `  w
) ,  ( 2nd `  w ) >. )
4913, 48eqeqan12d 2097 . . . . . . 7  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( z  =  w  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
50 vex 2605 . . . . . . . . 9  |-  z  e. 
_V
51 1stexg 5825 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 1st `  z )  e. 
_V )
5250, 51ax-mp 7 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
53 2ndexg 5826 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 2nd `  z )  e. 
_V )
5450, 53ax-mp 7 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
5552, 54opth 4000 . . . . . . 7  |-  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. 
<->  ( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) )
5649, 55syl6bb 194 . . . . . 6  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( z  =  w  <-> 
( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) ) )
5739, 47, 563bitr4d 218 . . . . 5  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  <-> 
z  =  w ) )
5857biimpd 142 . . . 4  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  ->  z  =  w ) )
5958rgen2a 2418 . . 3  |-  A. z  e.  ( RR  X.  RR ) A. w  e.  ( RR  X.  RR ) ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
60 dff13 5439 . . 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 884 . 2  |-  F :
( RR  X.  RR ) -1-1-> CC
62 cnre 7177 . . . . . 6  |-  ( w  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  w  =  ( u  +  ( _i  x.  v ) ) )
63 simpl 107 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  u  e.  RR )
64 simpr 108 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  v  e.  RR )
6563recnd 7209 . . . . . . . . . 10  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  u  e.  CC )
663a1i 9 . . . . . . . . . . 11  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  _i  e.  CC )
6764recnd 7209 . . . . . . . . . . 11  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  v  e.  CC )
6866, 67mulcld 7201 . . . . . . . . . 10  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( _i  x.  v
)  e.  CC )
6965, 68addcld 7200 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u  +  ( _i  x.  v ) )  e.  CC )
70 oveq1 5550 . . . . . . . . . 10  |-  ( x  =  u  ->  (
x  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  y
) ) )
71 oveq2 5551 . . . . . . . . . . 11  |-  ( y  =  v  ->  (
_i  x.  y )  =  ( _i  x.  v ) )
7271oveq2d 5559 . . . . . . . . . 10  |-  ( y  =  v  ->  (
u  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  v
) ) )
7370, 72, 10ovmpt2g 5666 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR  /\  (
u  +  ( _i  x.  v ) )  e.  CC )  -> 
( u F v )  =  ( u  +  ( _i  x.  v ) ) )
7463, 64, 69, 73syl3anc 1170 . . . . . . . 8  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u F v )  =  ( u  +  ( _i  x.  v ) ) )
7574eqeq2d 2093 . . . . . . 7  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( w  =  ( u F v )  <-> 
w  =  ( u  +  ( _i  x.  v ) ) ) )
76752rexbiia 2383 . . . . . 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 132 . . . . 5  |-  ( w  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  w  =  ( u F v ) )
78 fveq2 5209 . . . . . . . 8  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( F `  <. u ,  v >. )
)
79 df-ov 5546 . . . . . . . 8  |-  ( u F v )  =  ( F `  <. u ,  v >. )
8078, 79syl6eqr 2132 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( u F v ) )
8180eqeq2d 2093 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  ( w  =  ( F `  z
)  <->  w  =  (
u F v ) ) )
8281rexxp 4508 . . . . 5  |-  ( E. z  e.  ( RR 
X.  RR ) w  =  ( F `  z )  <->  E. u  e.  RR  E. v  e.  RR  w  =  ( u F v ) )
8377, 82sylibr 132 . . . 4  |-  ( w  e.  CC  ->  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
)
8483rgen 2417 . . 3  |-  A. w  e.  CC  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
85 dffo3 5346 . . 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 884 . 2  |-  F :
( RR  X.  RR ) -onto-> CC
87 df-f1o 4939 . 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 884 1  |-  F :
( RR  X.  RR )
-1-1-onto-> CC
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   _Vcvv 2602   <.cop 3409    X. cxp 4369    Fn wfn 4927   -->wf 4928   -1-1->wf1 4929   -onto->wfo 4930   -1-1-onto->wf1o 4931   ` cfv 4932  (class class class)co 5543    |-> cmpt2 5545   1stc1st 5796   2ndc2nd 5797   CCcc 7041   RRcr 7042   _ici 7045    + caddc 7046    x. cmul 7048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-pnf 7217  df-mnf 7218  df-ltxr 7220  df-sub 7348  df-neg 7349  df-reap 7742
This theorem is referenced by:  cnrecnv  9935
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