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Theorem cnref1o 10984
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 9286), 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 cnref1o.1 . . . . 5  |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  (
_i  x.  y )
) )
2 ovex 6114 . . . . 5  |-  ( x  +  ( _i  x.  y ) )  e. 
_V
31, 2fnmpt2i 6641 . . . 4  |-  F  Fn  ( RR  X.  RR )
4 1st2nd2 6611 . . . . . . . . 9  |-  ( z  e.  ( RR  X.  RR )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
54fveq2d 5693 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
6 df-ov 6092 . . . . . . . 8  |-  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
75, 6syl6eqr 2491 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
) F ( 2nd `  z ) ) )
8 xp1st 6604 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
9 xp2nd 6605 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  RR )
10 oveq1 6096 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  ( x  +  ( _i  x.  y ) )  =  ( ( 1st `  z
)  +  ( _i  x.  y ) ) )
11 oveq2 6097 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  z
)  ->  ( _i  x.  y )  =  ( _i  x.  ( 2nd `  z ) ) )
1211oveq2d 6105 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  ( ( 1st `  z )  +  ( _i  x.  y
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
13 ovex 6114 . . . . . . . . 9  |-  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  e.  _V
1410, 12, 1, 13ovmpt2 6224 . . . . . . . 8  |-  ( ( ( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR )  -> 
( ( 1st `  z
) F ( 2nd `  z ) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
158, 9, 14syl2anc 661 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
167, 15eqtrd 2473 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  =  ( ( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) ) )
178recnd 9410 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  CC )
18 ax-icn 9339 . . . . . . . 8  |-  _i  e.  CC
199recnd 9410 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 2nd `  z )  e.  CC )
20 mulcl 9364 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  ( 2nd `  z )  e.  CC )  -> 
( _i  x.  ( 2nd `  z ) )  e.  CC )
2118, 19, 20sylancr 663 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( _i  x.  ( 2nd `  z
) )  e.  CC )
2217, 21addcld 9403 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  e.  CC )
2316, 22eqeltrd 2515 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  ( F `
 z )  e.  CC )
2423rgen 2779 . . . 4  |-  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC
25 ffnfv 5867 . . . 4  |-  ( F : ( RR  X.  RR ) --> CC  <->  ( F  Fn  ( RR  X.  RR )  /\  A. z  e.  ( RR  X.  RR ) ( F `  z )  e.  CC ) )
263, 24, 25mpbir2an 911 . . 3  |-  F :
( RR  X.  RR )
--> CC
278, 9jca 532 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z )  e.  RR ) )
28 xp1st 6604 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  ( 1st `  w )  e.  RR )
29 xp2nd 6605 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  ( 2nd `  w )  e.  RR )
3028, 29jca 532 . . . . . . 7  |-  ( w  e.  ( RR  X.  RR )  ->  ( ( 1st `  w )  e.  RR  /\  ( 2nd `  w )  e.  RR ) )
31 cru 10312 . . . . . . 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 ) ) ) )
3227, 30, 31syl2an 477 . . . . . 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
) ) ) )
33 fveq2 5689 . . . . . . . . 9  |-  ( z  =  w  ->  ( F `  z )  =  ( F `  w ) )
34 fveq2 5689 . . . . . . . . . 10  |-  ( z  =  w  ->  ( 1st `  z )  =  ( 1st `  w
) )
35 fveq2 5689 . . . . . . . . . . 11  |-  ( z  =  w  ->  ( 2nd `  z )  =  ( 2nd `  w
) )
3635oveq2d 6105 . . . . . . . . . 10  |-  ( z  =  w  ->  (
_i  x.  ( 2nd `  z ) )  =  ( _i  x.  ( 2nd `  w ) ) )
3734, 36oveq12d 6107 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  z
)  +  ( _i  x.  ( 2nd `  z
) ) )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
3833, 37eqeq12d 2455 . . . . . . . 8  |-  ( z  =  w  ->  (
( F `  z
)  =  ( ( 1st `  z )  +  ( _i  x.  ( 2nd `  z ) ) )  <->  ( F `  w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) ) )
3938, 16vtoclga 3034 . . . . . . 7  |-  ( w  e.  ( RR  X.  RR )  ->  ( F `
 w )  =  ( ( 1st `  w
)  +  ( _i  x.  ( 2nd `  w
) ) ) )
4016, 39eqeqan12d 2456 . . . . . 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
) ) ) ) )
41 1st2nd2 6611 . . . . . . . 8  |-  ( w  e.  ( RR  X.  RR )  ->  w  = 
<. ( 1st `  w
) ,  ( 2nd `  w ) >. )
424, 41eqeqan12d 2456 . . . . . . 7  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( z  =  w  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
43 fvex 5699 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
44 fvex 5699 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
4543, 44opth 4564 . . . . . . 7  |-  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. 
<->  ( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) )
4642, 45syl6bb 261 . . . . . 6  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( z  =  w  <-> 
( ( 1st `  z
)  =  ( 1st `  w )  /\  ( 2nd `  z )  =  ( 2nd `  w
) ) ) )
4732, 40, 463bitr4d 285 . . . . 5  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  <-> 
z  =  w ) )
4847biimpd 207 . . . 4  |-  ( ( z  e.  ( RR 
X.  RR )  /\  w  e.  ( RR  X.  RR ) )  -> 
( ( F `  z )  =  ( F `  w )  ->  z  =  w ) )
4948rgen2a 2780 . . 3  |-  A. z  e.  ( RR  X.  RR ) A. w  e.  ( RR  X.  RR ) ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
50 dff13 5969 . . 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 ) ) )
5126, 49, 50mpbir2an 911 . 2  |-  F :
( RR  X.  RR ) -1-1-> CC
52 cnre 9380 . . . . . 6  |-  ( w  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  w  =  ( u  +  ( _i  x.  v ) ) )
53 oveq1 6096 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  y
) ) )
54 oveq2 6097 . . . . . . . . . 10  |-  ( y  =  v  ->  (
_i  x.  y )  =  ( _i  x.  v ) )
5554oveq2d 6105 . . . . . . . . 9  |-  ( y  =  v  ->  (
u  +  ( _i  x.  y ) )  =  ( u  +  ( _i  x.  v
) ) )
56 ovex 6114 . . . . . . . . 9  |-  ( u  +  ( _i  x.  v ) )  e. 
_V
5753, 55, 1, 56ovmpt2 6224 . . . . . . . 8  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u F v )  =  ( u  +  ( _i  x.  v ) ) )
5857eqeq2d 2452 . . . . . . 7  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( w  =  ( u F v )  <-> 
w  =  ( u  +  ( _i  x.  v ) ) ) )
59582rexbiia 2747 . . . . . 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 ) ) )
6052, 59sylibr 212 . . . . 5  |-  ( w  e.  CC  ->  E. u  e.  RR  E. v  e.  RR  w  =  ( u F v ) )
61 fveq2 5689 . . . . . . . 8  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( F `  <. u ,  v >. )
)
62 df-ov 6092 . . . . . . . 8  |-  ( u F v )  =  ( F `  <. u ,  v >. )
6361, 62syl6eqr 2491 . . . . . . 7  |-  ( z  =  <. u ,  v
>.  ->  ( F `  z )  =  ( u F v ) )
6463eqeq2d 2452 . . . . . 6  |-  ( z  =  <. u ,  v
>.  ->  ( w  =  ( F `  z
)  <->  w  =  (
u F v ) ) )
6564rexxp 4980 . . . . 5  |-  ( E. z  e.  ( RR 
X.  RR ) w  =  ( F `  z )  <->  E. u  e.  RR  E. v  e.  RR  w  =  ( u F v ) )
6660, 65sylibr 212 . . . 4  |-  ( w  e.  CC  ->  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
)
6766rgen 2779 . . 3  |-  A. w  e.  CC  E. z  e.  ( RR  X.  RR ) w  =  ( F `  z )
68 dffo3 5856 . . 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 )
) )
6926, 67, 68mpbir2an 911 . 2  |-  F :
( RR  X.  RR ) -onto-> CC
70 df-f1o 5423 . 2  |-  ( F : ( RR  X.  RR ) -1-1-onto-> CC  <->  ( F :
( RR  X.  RR ) -1-1-> CC  /\  F :
( RR  X.  RR ) -onto-> CC ) )
7151, 69, 70mpbir2an 911 1  |-  F :
( RR  X.  RR )
-1-1-onto-> CC
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714   <.cop 3881    X. cxp 4836    Fn wfn 5411   -->wf 5412   -1-1->wf1 5413   -onto->wfo 5414   -1-1-onto->wf1o 5415   ` cfv 5416  (class class class)co 6089    e. cmpt2 6091   1stc1st 6573   2ndc2nd 6574   CCcc 9278   RRcr 9279   _ici 9282    + caddc 9283    x. cmul 9285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992
This theorem is referenced by:  cnexALT  10985  cnrecnv  12652  cpnnen  13509  cnheiborlem  20524  mbfimaopnlem  21131
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