Theorem cnref1o 11827

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 9942), 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		cnref1o.1	. . . . 5 |- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) )
2		ovex 6678	. . . . 5 |- ( x + ( _i x. y ) ) e. _V
3	1, 2	fnmpt2i 7239	. . . 4 |- F Fn ( RR X. RR )
4		1st2nd2 7205	. . . . . . . . 9 |- ( z e. ( RR X. RR ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
5	4	fveq2d 6195	. . . . . . . 8 |- ( z e. ( RR X. RR ) -> ( F ` z ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
6		df-ov 6653	. . . . . . . 8 |- ( ( 1st ` z ) F ( 2nd ` z ) ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
7	5, 6	syl6eqr 2674	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( F ` z ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
8		xp1st 7198	. . . . . . . 8 |- ( z e. ( RR X. RR ) -> ( 1st ` z ) e. RR )
9		xp2nd 7199	. . . . . . . 8 |- ( z e. ( RR X. RR ) -> ( 2nd ` z ) e. RR )
10		oveq1 6657	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> ( x + ( _i x. y ) ) = ( ( 1st ` z ) + ( _i x. y ) ) )
11		oveq2 6658	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( _i x. y ) = ( _i x. ( 2nd ` z ) ) )
12	11	oveq2d 6666	. . . . . . . . 9 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) + ( _i x. y ) ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
13		ovex 6678	. . . . . . . . 9 |- ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) e. _V
14	10, 12, 1, 13	ovmpt2 6796	. . . . . . . 8 |- ( ( ( 1st ` z ) e. RR /\ ( 2nd ` z ) e. RR ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
15	8, 9, 14	syl2anc 693	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
16	7, 15	eqtrd 2656	. . . . . 6 |- ( z e. ( RR X. RR ) -> ( F ` z ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
17	8	recnd 10068	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( 1st ` z ) e. CC )
18		ax-icn 9995	. . . . . . . 8 |- _i e. CC
19	9	recnd 10068	. . . . . . . 8 |- ( z e. ( RR X. RR ) -> ( 2nd ` z ) e. CC )
20		mulcl 10020	. . . . . . . 8 |- ( ( _i e. CC /\ ( 2nd ` z ) e. CC ) -> ( _i x. ( 2nd ` z ) ) e. CC )
21	18, 19, 20	sylancr 695	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( _i x. ( 2nd ` z ) ) e. CC )
22	17, 21	addcld 10059	. . . . . 6 |- ( z e. ( RR X. RR ) -> ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) e. CC )
23	16, 22	eqeltrd 2701	. . . . 5 |- ( z e. ( RR X. RR ) -> ( F ` z ) e. CC )
24	23	rgen 2922	. . . 4 |- A. z e. ( RR X. RR ) ( F ` z ) e. CC
25		ffnfv 6388	. . . 4 |- ( F : ( RR X. RR ) --> CC <-> ( F Fn ( RR X. RR ) /\ A. z e. ( RR X. RR ) ( F ` z ) e. CC ) )
26	3, 24, 25	mpbir2an 955	. . 3 |- F : ( RR X. RR ) --> CC
27	8, 9	jca 554	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( ( 1st ` z ) e. RR /\ ( 2nd ` z ) e. RR ) )
28		xp1st 7198	. . . . . . . 8 |- ( w e. ( RR X. RR ) -> ( 1st ` w ) e. RR )
29		xp2nd 7199	. . . . . . . 8 |- ( w e. ( RR X. RR ) -> ( 2nd ` w ) e. RR )
30	28, 29	jca 554	. . . . . . 7 |- ( w e. ( RR X. RR ) -> ( ( 1st ` w ) e. RR /\ ( 2nd ` w ) e. RR ) )
31		cru 11012	. . . . . . 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 ) ) ) )
32	27, 30, 31	syl2an 494	. . . . . 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 6191	. . . . . . . . 9 |- ( z = w -> ( F ` z ) = ( F ` w ) )
34		fveq2 6191	. . . . . . . . . 10 |- ( z = w -> ( 1st ` z ) = ( 1st ` w ) )
35		fveq2 6191	. . . . . . . . . . 11 |- ( z = w -> ( 2nd ` z ) = ( 2nd ` w ) )
36	35	oveq2d 6666	. . . . . . . . . 10 |- ( z = w -> ( _i x. ( 2nd ` z ) ) = ( _i x. ( 2nd ` w ) ) )
37	34, 36	oveq12d 6668	. . . . . . . . 9 |- ( z = w -> ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) = ( ( 1st ` w ) + ( _i x. ( 2nd ` w ) ) ) )
38	33, 37	eqeq12d 2637	. . . . . . . 8 |- ( z = w -> ( ( F ` z ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) <-> ( F ` w ) = ( ( 1st ` w ) + ( _i x. ( 2nd ` w ) ) ) ) )
39	38, 16	vtoclga 3272	. . . . . . 7 |- ( w e. ( RR X. RR ) -> ( F ` w ) = ( ( 1st ` w ) + ( _i x. ( 2nd ` w ) ) ) )
40	16, 39	eqeqan12d 2638	. . . . . 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 7205	. . . . . . . 8 |- ( w e. ( RR X. RR ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
42	4, 41	eqeqan12d 2638	. . . . . . 7 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( z = w <-> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
43		fvex 6201	. . . . . . . 8 |- ( 1st ` z ) e. _V
44		fvex 6201	. . . . . . . 8 |- ( 2nd ` z ) e. _V
45	43, 44	opth 4945	. . . . . . 7 |- ( <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. <-> ( ( 1st ` z ) = ( 1st ` w ) /\ ( 2nd ` z ) = ( 2nd ` w ) ) )
46	42, 45	syl6bb 276	. . . . . 6 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( z = w <-> ( ( 1st ` z ) = ( 1st ` w ) /\ ( 2nd ` z ) = ( 2nd ` w ) ) ) )
47	32, 40, 46	3bitr4d 300	. . . . 5 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( ( F ` z ) = ( F ` w ) <-> z = w ) )
48	47	biimpd 219	. . . 4 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( ( F ` z ) = ( F ` w ) -> z = w ) )
49	48	rgen2a 2977	. . 3 |- A. z e. ( RR X. RR ) A. w e. ( RR X. RR ) ( ( F ` z ) = ( F ` w ) -> z = w )
50		dff13 6512	. . 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 ) ) )
51	26, 49, 50	mpbir2an 955	. 2 |- F : ( RR X. RR ) -1-1-> CC
52		cnre 10036	. . . . . 6 |- ( w e. CC -> E. u e. RR E. v e. RR w = ( u + ( _i x. v ) ) )
53		oveq1 6657	. . . . . . . . 9 |- ( x = u -> ( x + ( _i x. y ) ) = ( u + ( _i x. y ) ) )
54		oveq2 6658	. . . . . . . . . 10 |- ( y = v -> ( _i x. y ) = ( _i x. v ) )
55	54	oveq2d 6666	. . . . . . . . 9 |- ( y = v -> ( u + ( _i x. y ) ) = ( u + ( _i x. v ) ) )
56		ovex 6678	. . . . . . . . 9 |- ( u + ( _i x. v ) ) e. _V
57	53, 55, 1, 56	ovmpt2 6796	. . . . . . . 8 |- ( ( u e. RR /\ v e. RR ) -> ( u F v ) = ( u + ( _i x. v ) ) )
58	57	eqeq2d 2632	. . . . . . 7 |- ( ( u e. RR /\ v e. RR ) -> ( w = ( u F v ) <-> w = ( u + ( _i x. v ) ) ) )
59	58	2rexbiia 3055	. . . . . 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 ) ) )
60	52, 59	sylibr 224	. . . . 5 |- ( w e. CC -> E. u e. RR E. v e. RR w = ( u F v ) )
61		fveq2 6191	. . . . . . . 8 |- ( z = <. u , v >. -> ( F ` z ) = ( F ` <. u , v >. ) )
62		df-ov 6653	. . . . . . . 8 |- ( u F v ) = ( F ` <. u , v >. )
63	61, 62	syl6eqr 2674	. . . . . . 7 |- ( z = <. u , v >. -> ( F ` z ) = ( u F v ) )
64	63	eqeq2d 2632	. . . . . 6 |- ( z = <. u , v >. -> ( w = ( F ` z ) <-> w = ( u F v ) ) )
65	64	rexxp 5264	. . . . 5 |- ( E. z e. ( RR X. RR ) w = ( F ` z ) <-> E. u e. RR E. v e. RR w = ( u F v ) )
66	60, 65	sylibr 224	. . . 4 |- ( w e. CC -> E. z e. ( RR X. RR ) w = ( F ` z ) )
67	66	rgen 2922	. . 3 |- A. w e. CC E. z e. ( RR X. RR ) w = ( F ` z )
68		dffo3 6374	. . 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 ) ) )
69	26, 67, 68	mpbir2an 955	. 2 |- F : ( RR X. RR ) -onto-> CC
70		df-f1o 5895	. 2 |- ( F : ( RR X. RR ) -1-1-onto-> CC <-> ( F : ( RR X. RR ) -1-1-> CC /\ F : ( RR X. RR ) -onto-> CC ) )
71	51, 69, 70	mpbir2an 955	1 |- F : ( RR X. RR ) -1-1-onto-> CC

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   <.cop 4183   X. cxp 5112   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   CCcc 9934   RRcr 9935   _ici 9938   + caddc 9939   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685

This theorem is referenced by:  cnexALT  11828  cnrecnv  13905  cpnnen  14958  cnheiborlem  22753  mbfimaopnlem  23422