Theorem cnref1o 8582

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 6895), 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.

Step	Hyp	Ref	Expression
1		simpl 102	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> x e. RR )
2	1	recnd 7054	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> x e. CC )
3		ax-icn 6979	. . . . . . . . 9 |- _i e. CC
4	3	a1i 9	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> _i e. CC )
5		simpr 103	. . . . . . . . 9 |- ( ( x e. RR /\ y e. RR ) -> y e. RR )
6	5	recnd 7054	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> y e. CC )
7	4, 6	mulcld 7047	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( _i x. y ) e. CC )
8	2, 7	addcld 7046	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( x + ( _i x. y ) ) e. CC )
9	8	rgen2a 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 ) ) )
11	10	fnmpt2 5828	. . . . 5 |- ( A. x e. RR A. y e. RR ( x + ( _i x. y ) ) e. CC -> F Fn ( RR X. RR ) )
12	9, 11	ax-mp 7	. . . 4 |- F Fn ( RR X. RR )
13		1st2nd2 5801	. . . . . . . . 9 |- ( z e. ( RR X. RR ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
14	13	fveq2d 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 ) >. )
16	14, 15	syl6eqr 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 )
19	17	recnd 7054	. . . . . . . . 9 |- ( z e. ( RR X. RR ) -> ( 1st ` z ) e. CC )
20	3	a1i 9	. . . . . . . . . 10 |- ( z e. ( RR X. RR ) -> _i e. CC )
21	18	recnd 7054	. . . . . . . . . 10 |- ( z e. ( RR X. RR ) -> ( 2nd ` z ) e. CC )
22	20, 21	mulcld 7047	. . . . . . . . 9 |- ( z e. ( RR X. RR ) -> ( _i x. ( 2nd ` z ) ) e. CC )
23	19, 22	addcld 7046	. . . . . . . 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 ) ) )
26	25	oveq2d 5528	. . . . . . . . 9 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) + ( _i x. y ) ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
27	24, 26, 10	ovmpt2g 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 ) ) ) )
28	17, 18, 23, 27	syl3anc 1135	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
29	16, 28	eqtrd 2072	. . . . . 6 |- ( z e. ( RR X. RR ) -> ( F ` z ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
30	29, 23	eqeltrd 2114	. . . . 5 |- ( z e. ( RR X. RR ) -> ( F ` z ) e. CC )
31	30	rgen 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 ) )
33	12, 31, 32	mpbir2an 849	. . 3 |- F : ( RR X. RR ) --> CC
34	17, 18	jca 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 )
37	35, 36	jca 290	. . . . . . 7 |- ( w e. ( RR X. RR ) -> ( ( 1st ` w ) e. RR /\ ( 2nd ` w ) e. RR ) )
38		cru 7593	. . . . . . 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 ) ) ) )
39	34, 37, 38	syl2an 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 ) )
43	42	oveq2d 5528	. . . . . . . . . 10 |- ( z = w -> ( _i x. ( 2nd ` z ) ) = ( _i x. ( 2nd ` w ) ) )
44	41, 43	oveq12d 5530	. . . . . . . . 9 |- ( z = w -> ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) = ( ( 1st ` w ) + ( _i x. ( 2nd ` w ) ) ) )
45	40, 44	eqeq12d 2054	. . . . . . . 8 |- ( z = w -> ( ( F ` z ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) <-> ( F ` w ) = ( ( 1st ` w ) + ( _i x. ( 2nd ` w ) ) ) ) )
46	45, 29	vtoclga 2619	. . . . . . 7 |- ( w e. ( RR X. RR ) -> ( F ` w ) = ( ( 1st ` w ) + ( _i x. ( 2nd ` w ) ) ) )
47	29, 46	eqeqan12d 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 ) >. )
49	13, 48	eqeqan12d 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 )
52	50, 51	ax-mp 7	. . . . . . . 8 |- ( 1st ` z ) e. _V
53		2ndexg 5795	. . . . . . . . 9 |- ( z e. _V -> ( 2nd ` z ) e. _V )
54	50, 53	ax-mp 7	. . . . . . . 8 |- ( 2nd ` z ) e. _V
55	52, 54	opth 3974	. . . . . . 7 |- ( <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. <-> ( ( 1st ` z ) = ( 1st ` w ) /\ ( 2nd ` z ) = ( 2nd ` w ) ) )
56	49, 55	syl6bb 185	. . . . . 6 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( z = w <-> ( ( 1st ` z ) = ( 1st ` w ) /\ ( 2nd ` z ) = ( 2nd ` w ) ) ) )
57	39, 47, 56	3bitr4d 209	. . . . 5 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( ( F ` z ) = ( F ` w ) <-> z = w ) )
58	57	biimpd 132	. . . 4 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( ( F ` z ) = ( F ` w ) -> z = w ) )
59	58	rgen2a 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 ) ) )
61	33, 59, 60	mpbir2an 849	. 2 |- F : ( RR X. RR ) -1-1-> CC
62		cnre 7023	. . . . . 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 )
65	63	recnd 7054	. . . . . . . . . 10 |- ( ( u e. RR /\ v e. RR ) -> u e. CC )
66	3	a1i 9	. . . . . . . . . . 11 |- ( ( u e. RR /\ v e. RR ) -> _i e. CC )
67	64	recnd 7054	. . . . . . . . . . 11 |- ( ( u e. RR /\ v e. RR ) -> v e. CC )
68	66, 67	mulcld 7047	. . . . . . . . . 10 |- ( ( u e. RR /\ v e. RR ) -> ( _i x. v ) e. CC )
69	65, 68	addcld 7046	. . . . . . . . 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 ) )
72	71	oveq2d 5528	. . . . . . . . . 10 |- ( y = v -> ( u + ( _i x. y ) ) = ( u + ( _i x. v ) ) )
73	70, 72, 10	ovmpt2g 5635	. . . . . . . . 9 |- ( ( u e. RR /\ v e. RR /\ ( u + ( _i x. v ) ) e. CC ) -> ( u F v ) = ( u + ( _i x. v ) ) )
74	63, 64, 69, 73	syl3anc 1135	. . . . . . . 8 |- ( ( u e. RR /\ v e. RR ) -> ( u F v ) = ( u + ( _i x. v ) ) )
75	74	eqeq2d 2051	. . . . . . 7 |- ( ( u e. RR /\ v e. RR ) -> ( w = ( u F v ) <-> w = ( u + ( _i x. v ) ) ) )
76	75	2rexbiia 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 ) ) )
77	62, 76	sylibr 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 >. )
80	78, 79	syl6eqr 2090	. . . . . . 7 |- ( z = <. u , v >. -> ( F ` z ) = ( u F v ) )
81	80	eqeq2d 2051	. . . . . 6 |- ( z = <. u , v >. -> ( w = ( F ` z ) <-> w = ( u F v ) ) )
82	81	rexxp 4480	. . . . 5 |- ( E. z e. ( RR X. RR ) w = ( F ` z ) <-> E. u e. RR E. v e. RR w = ( u F v ) )
83	77, 82	sylibr 137	. . . 4 |- ( w e. CC -> E. z e. ( RR X. RR ) w = ( F ` z ) )
84	83	rgen 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 ) ) )
86	33, 84, 85	mpbir2an 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 ) )
88	61, 86, 87	mpbir2an 849	1 |- F : ( RR X. RR ) -1-1-onto-> CC

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 6887   RRcr 6888   _ici 6891   + caddc 6892   x. cmul 6894

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 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001

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 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-ltxr 7065  df-sub 7184  df-neg 7185  df-reap 7566

This theorem is referenced by:  cnrecnv  9510