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Theorem cnref1o 8357
Description: There is a natural one-to-one mapping from  RR  X.  RR to  CC, where we map  <. , 
>. to  +  _i  x. . In our construction of the complex numbers, this is in fact our definition of  CC (see df-c 6717), 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  RR ,  RR  |->  +  _i  x.
Assertion
Ref Expression
cnref1o  F : RR  X.  RR
-1-1-onto-> CC
Distinct variable group:   ,
Allowed substitution hints:    F(,)

Proof of Theorem cnref1o
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 102 . . . . . . . 8  RR  RR  RR
21recnd 6851 . . . . . . 7  RR  RR  CC
3 ax-icn 6778 . . . . . . . . 9  _i  CC
43a1i 9 . . . . . . . 8  RR  RR  _i  CC
5 simpr 103 . . . . . . . . 9  RR  RR  RR
65recnd 6851 . . . . . . . 8  RR  RR  CC
74, 6mulcld 6845 . . . . . . 7  RR  RR  _i  x.  CC
82, 7addcld 6844 . . . . . 6  RR  RR  +  _i  x.  CC
98rgen2a 2369 . . . . 5  RR  RR  +  _i  x.  CC
10 cnref1o.1 . . . . . 6  F  RR ,  RR  |->  +  _i  x.
1110fnmpt2 5770 . . . . 5  RR  RR  +  _i  x.  CC  F  Fn  RR 
X.  RR
129, 11ax-mp 7 . . . 4  F  Fn  RR  X.  RR
13 1st2nd2 5743 . . . . . . . . 9  RR  X.  RR  <. 1st `  ,  2nd `  >.
1413fveq2d 5125 . . . . . . . 8  RR  X.  RR  F `  F `  <. 1st `  ,  2nd `  >.
15 df-ov 5458 . . . . . . . 8  1st `  F 2nd `  F `  <. 1st `  ,  2nd ` 
>.
1614, 15syl6eqr 2087 . . . . . . 7  RR  X.  RR  F `  1st `  F 2nd `
17 xp1st 5734 . . . . . . . 8  RR  X.  RR  1st `  RR
18 xp2nd 5735 . . . . . . . 8  RR  X.  RR  2nd `  RR
1917recnd 6851 . . . . . . . . 9  RR  X.  RR  1st `  CC
203a1i 9 . . . . . . . . . 10  RR  X.  RR  _i  CC
2118recnd 6851 . . . . . . . . . 10  RR  X.  RR  2nd `  CC
2220, 21mulcld 6845 . . . . . . . . 9  RR  X.  RR  _i  x.  2nd `  CC
2319, 22addcld 6844 . . . . . . . 8  RR  X.  RR  1st `  +  _i  x.  2nd `  CC
24 oveq1 5462 . . . . . . . . 9  1st `  +  _i  x.  1st `  +  _i  x.
25 oveq2 5463 . . . . . . . . . 10  2nd `  _i  x.  _i  x.  2nd `
2625oveq2d 5471 . . . . . . . . 9  2nd `  1st `  +  _i  x.  1st `  +  _i  x.  2nd `
2724, 26, 10ovmpt2g 5577 . . . . . . . 8  1st `  RR  2nd `  RR  1st `  +  _i  x.  2nd `  CC  1st `  F 2nd `  1st `  +  _i  x.  2nd `
2817, 18, 23, 27syl3anc 1134 . . . . . . 7  RR  X.  RR  1st `  F 2nd `  1st `  +  _i  x.  2nd `
2916, 28eqtrd 2069 . . . . . 6  RR  X.  RR  F `  1st `  +  _i  x.  2nd `
3029, 23eqeltrd 2111 . . . . 5  RR  X.  RR  F `  CC
3130rgen 2368 . . . 4  RR  X.  RR F `  CC
32 ffnfv 5266 . . . 4  F : RR  X.  RR --> CC  F  Fn  RR  X.  RR  RR  X.  RR F `  CC
3312, 31, 32mpbir2an 848 . . 3  F : RR  X.  RR
--> CC
3417, 18jca 290 . . . . . . 7  RR  X.  RR  1st `  RR  2nd `  RR
35 xp1st 5734 . . . . . . . 8  RR  X.  RR  1st `  RR
36 xp2nd 5735 . . . . . . . 8  RR  X.  RR  2nd `  RR
3735, 36jca 290 . . . . . . 7  RR  X.  RR  1st `  RR  2nd `  RR
38 cru 7386 . . . . . . 7  1st `  RR  2nd `  RR  1st `  RR  2nd `  RR  1st `  +  _i  x.  2nd `  1st `  +  _i  x.  2nd `  1st `  1st `  2nd `  2nd `
3934, 37, 38syl2an 273 . . . . . 6  RR 
X.  RR  RR  X.  RR  1st `  +  _i  x.  2nd `  1st `  +  _i  x.  2nd `  1st `  1st `  2nd `  2nd `
40 fveq2 5121 . . . . . . . . 9  F `  F `
41 fveq2 5121 . . . . . . . . . 10  1st `  1st `
42 fveq2 5121 . . . . . . . . . . 11  2nd `  2nd `
4342oveq2d 5471 . . . . . . . . . 10  _i  x.  2nd `  _i  x.  2nd `
4441, 43oveq12d 5473 . . . . . . . . 9  1st `  +  _i  x.  2nd `  1st `  +  _i  x.  2nd `
4540, 44eqeq12d 2051 . . . . . . . 8  F `  1st `  +  _i  x.  2nd `  F `  1st `  +  _i  x.  2nd `
4645, 29vtoclga 2613 . . . . . . 7  RR  X.  RR  F `  1st `  +  _i  x.  2nd `
4729, 46eqeqan12d 2052 . . . . . 6  RR 
X.  RR  RR  X.  RR  F `  F `  1st `  +  _i  x.  2nd `  1st `  +  _i  x.  2nd `
48 1st2nd2 5743 . . . . . . . 8  RR  X.  RR  <. 1st `  ,  2nd `  >.
4913, 48eqeqan12d 2052 . . . . . . 7  RR 
X.  RR  RR  X.  RR  <. 1st `  ,  2nd `  >.  <. 1st `  ,  2nd `  >.
50 vex 2554 . . . . . . . . 9 
_V
51 1stexg 5736 . . . . . . . . 9  _V  1st ` 
_V
5250, 51ax-mp 7 . . . . . . . 8  1st `  _V
53 2ndexg 5737 . . . . . . . . 9  _V  2nd ` 
_V
5450, 53ax-mp 7 . . . . . . . 8  2nd `  _V
5552, 54opth 3965 . . . . . . 7  <. 1st `  ,  2nd `  >.  <. 1st `  ,  2nd `  >.  1st `  1st `  2nd `  2nd `
5649, 55syl6bb 185 . . . . . 6  RR 
X.  RR  RR  X.  RR  1st `  1st `  2nd `  2nd `
5739, 47, 563bitr4d 209 . . . . 5  RR 
X.  RR  RR  X.  RR  F `  F `
5857biimpd 132 . . . 4  RR 
X.  RR  RR  X.  RR  F `  F `
5958rgen2a 2369 . . 3  RR  X.  RR  RR  X.  RR F `  F `
60 dff13 5350 . . 3  F : RR  X.  RR -1-1-> CC  F : RR  X.  RR --> CC  RR  X.  RR  RR  X.  RR F `  F `
6133, 59, 60mpbir2an 848 . 2  F : RR  X.  RR -1-1-> CC
62 cnre 6821 . . . . . 6  CC  RR  RR  +  _i  x.
63 simpl 102 . . . . . . . . 9  RR  RR  RR
64 simpr 103 . . . . . . . . 9  RR  RR  RR
6563recnd 6851 . . . . . . . . . 10  RR  RR  CC
663a1i 9 . . . . . . . . . . 11  RR  RR  _i  CC
6764recnd 6851 . . . . . . . . . . 11  RR  RR  CC
6866, 67mulcld 6845 . . . . . . . . . 10  RR  RR  _i  x.  CC
6965, 68addcld 6844 . . . . . . . . 9  RR  RR  +  _i  x.  CC
70 oveq1 5462 . . . . . . . . . 10  +  _i  x.  +  _i  x.
71 oveq2 5463 . . . . . . . . . . 11  _i  x.  _i  x.
7271oveq2d 5471 . . . . . . . . . 10  +  _i  x.  +  _i  x.
7370, 72, 10ovmpt2g 5577 . . . . . . . . 9  RR  RR  +  _i  x.  CC  F  +  _i  x.
7463, 64, 69, 73syl3anc 1134 . . . . . . . 8  RR  RR  F  +  _i  x.
7574eqeq2d 2048 . . . . . . 7  RR  RR  F  +  _i  x.
76752rexbiia 2334 . . . . . 6  RR  RR  F  RR  RR  +  _i  x.
7762, 76sylibr 137 . . . . 5  CC  RR  RR  F
78 fveq2 5121 . . . . . . . 8  <. , 
>.  F `  F `  <. ,  >.
79 df-ov 5458 . . . . . . . 8  F  F `  <. ,  >.
8078, 79syl6eqr 2087 . . . . . . 7  <. , 
>.  F `  F
8180eqeq2d 2048 . . . . . 6  <. , 
>.  F `  F
8281rexxp 4423 . . . . 5  RR  X.  RR  F `  RR  RR  F
8377, 82sylibr 137 . . . 4  CC  RR  X.  RR  F `
8483rgen 2368 . . 3  CC  RR  X.  RR  F `
85 dffo3 5257 . . 3  F : RR  X.  RR -onto-> CC  F : RR  X.  RR --> CC  CC  RR  X.  RR  F `
8633, 84, 85mpbir2an 848 . 2  F : RR  X.  RR -onto-> CC
87 df-f1o 4852 . 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 848 1  F : RR  X.  RR
-1-1-onto-> CC
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wral 2300  wrex 2301   _Vcvv 2551   <.cop 3370    X. cxp 4286    Fn wfn 4840   -->wf 4841   -1-1->wf1 4842   -onto->wfo 4843   -1-1-onto->wf1o 4844   ` cfv 4845  (class class class)co 5455    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708   CCcc 6709   RRcr 6710   _ici 6713    + caddc 6714    x. cmul 6716
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulrcl 6782  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-precex 6793  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799  ax-pre-mulgt0 6800
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-ltxr 6862  df-sub 6981  df-neg 6982  df-reap 7359
This theorem is referenced by:  cnrecnv  9138
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