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Theorem cnref1o 9102
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 7335), 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 7495 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  x  e.  CC )
3 ax-icn 7419 . . . . . . . . 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 7495 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  y  e.  CC )
74, 6mulcld 7487 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( _i  x.  y
)  e.  CC )
82, 7addcld 7486 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  e.  CC )
98rgen2a 2429 . . . . 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 5954 . . . . 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 5927 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1413fveq2d 5293 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
15 df-ov 5637 . . . . . . . 8  |-  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
1614, 15syl6eqr 2138 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
17 xp1st 5918 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
18 xp2nd 5919 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
1917recnd 7495 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  CC )
203a1i 9 . . . . . . . . . 10  |-  ( z  e.  ( RR  X.  RR )  ->  _i  e.  CC )
2118recnd 7495 . . . . . . . . . 10  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  CC )
2220, 21mulcld 7487 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  ( _i  x.  ( 2nd `  z
) )  e.  CC )
2319, 22addcld 7486 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  e.  CC )
24 oveq1 5641 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  ( _i  x.  y ) )  =  ( ( 1st `  z
)  +  ( _i  x.  y ) ) )
25 oveq2 5642 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( _i  x.  y )  =  ( _i  x.  ( 2nd `  z ) ) )
2625oveq2d 5650 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z )  +  ( _i  x.  y
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
2724, 26, 10ovmpt2g 5761 . . . . . . . 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 1174 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
2916, 28eqtrd 2120 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
3029, 23eqeltrd 2164 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  e.  CC )
3130rgen 2428 . . . 4  |-  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC
32 ffnfv 5440 . . . 4  |-  ( F : ( RR  X.  RR ) --> CC  <->  ( F  Fn  ( RR  X.  RR )  /\  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC ) )
3312, 31, 32mpbir2an 888 . . 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 5918 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  ( 1st `  w )  e.  RR )
36 xp2nd 5919 . . . . . . . 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 8055 . . . . . . 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 5289 . . . . . . . . 9  |-  ( z  =  w  ->  ( F `  z )  =  ( F `  w ) )
41 fveq2 5289 . . . . . . . . . 10  |-  ( z  =  w  ->  ( 1st `  z )  =  ( 1st `  w
) )
42 fveq2 5289 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( 2nd `  z )  =  ( 2nd `  w
) )
4342oveq2d 5650 . . . . . . . . . 10  |-  ( z  =  w  ->  (
_i  x.  ( 2nd `  z ) )  =  ( _i  x.  ( 2nd `  w ) ) )
4441, 43oveq12d 5652 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
4540, 44eqeq12d 2102 . . . . . . . 8  |-  ( z  =  w  ->  (
( F `  z
)  =  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  <->  ( F `  w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) ) )
4645, 29vtoclga 2685 . . . . . . 7  |-  ( w  e.  ( RR  X.  RR )  ->  ( F `
 w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
4729, 46eqeqan12d 2103 . . . . . 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 5927 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  w  = 
<. ( 1st `  w
) ,  ( 2nd `  w ) >. )
4913, 48eqeqan12d 2103 . . . . . . 7  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( z  =  w  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
50 vex 2622 . . . . . . . . 9  |-  z  e. 
_V
51 1stexg 5920 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 1st `  z )  e. 
_V )
5250, 51ax-mp 7 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
53 2ndexg 5921 . . . . . . . . 9  |-  ( z  e.  _V  ->  ( 2nd `  z )  e. 
_V )
5450, 53ax-mp 7 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
5552, 54opth 4055 . . . . . . 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 2429 . . 3  |-  A. z  e.  ( RR  X.  RR ) A. w  e.  ( RR  X.  RR ) ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
60 dff13 5529 . . 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 888 . 2  |-  F :
( RR  X.  RR ) -1-1-> CC
62 cnre 7463 . . . . . 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 7495 . . . . . . . . . 10  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  u  e.  CC )
663a1i 9 . . . . . . . . . . 11  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  _i  e.  CC )
6764recnd 7495 . . . . . . . . . . 11  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  v  e.  CC )
6866, 67mulcld 7487 . . . . . . . . . 10  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( _i  x.  v
)  e.  CC )
6965, 68addcld 7486 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u  +  ( _i  x.  v ) )  e.  CC )
70 oveq1 5641 . . . . . . . . . 10  |-  ( x  =  u  ->  (
x  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  y
) ) )
71 oveq2 5642 . . . . . . . . . . 11  |-  ( y  =  v  ->  (
_i  x.  y )  =  ( _i  x.  v ) )
7271oveq2d 5650 . . . . . . . . . 10  |-  ( y  =  v  ->  (
u  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  v
) ) )
7370, 72, 10ovmpt2g 5761 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  v  e.  RR  /\  (
u  +  ( _i  x.  v ) )  e.  CC )  -> 
( u F v )  =  ( u  +  ( _i  x.  v ) ) )
7463, 64, 69, 73syl3anc 1174 . . . . . . . 8  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u F v )  =  ( u  +  ( _i  x.  v ) ) )
7574eqeq2d 2099 . . . . . . 7  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( w  =  ( u F v )  <-> 
w  =  ( u  +  ( _i  x.  v ) ) ) )
76752rexbiia 2394 . . . . . 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 5289 . . . . . . . 8  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( F `  <. u ,  v >. )
)
79 df-ov 5637 . . . . . . . 8  |-  ( u F v )  =  ( F `  <. u ,  v >. )
8078, 79syl6eqr 2138 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( u F v ) )
8180eqeq2d 2099 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  ( w  =  ( F `  z
)  <->  w  =  (
u F v ) ) )
8281rexxp 4568 . . . . 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 2428 . . 3  |-  A. w  e.  CC  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
85 dffo3 5430 . . 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 888 . 2  |-  F :
( RR  X.  RR ) -onto-> CC
87 df-f1o 5009 . 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 888 1  |-  F :
( RR  X.  RR )
-1-1-onto-> CC
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   _Vcvv 2619   <.cop 3444    X. cxp 4426    Fn wfn 4997   -->wf 4998   -1-1->wf1 4999   -onto->wfo 5000   -1-1-onto->wf1o 5001   ` cfv 5002  (class class class)co 5634    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892   CCcc 7327   RRcr 7328   _ici 7331    + caddc 7332    x. cmul 7334
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-pnf 7503  df-mnf 7504  df-ltxr 7506  df-sub 7634  df-neg 7635  df-reap 8028
This theorem is referenced by:  cnrecnv  10309
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