Theorem cnref1o 10051

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 8185), 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 109	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> x e. RR )
2	1	recnd 8354	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> x e. CC )
3		ax-icn 8274	. . . . . . . . 9 |- _i e. CC
4	3	a1i 9	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> _i e. CC )
5		simpr 110	. . . . . . . . 9 |- ( ( x e. RR /\ y e. RR ) -> y e. RR )
6	5	recnd 8354	. . . . . . . 8 |- ( ( x e. RR /\ y e. RR ) -> y e. CC )
7	4, 6	mulcld 8346	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( _i x. y ) e. CC )
8	2, 7	addcld 8345	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( x + ( _i x. y ) ) e. CC )
9	8	rgen2a 2604	. . . . 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	fnmpo 6438	. . . . 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 5	. . . 4 |- F Fn ( RR X. RR )
13		1st2nd2 6409	. . . . . . . . 9 |- ( z e. ( RR X. RR ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
14	13	fveq2d 5699	. . . . . . . 8 |- ( z e. ( RR X. RR ) -> ( F ` z ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
15		df-ov 6088	. . . . . . . 8 |- ( ( 1st ` z ) F ( 2nd ` z ) ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
16	14, 15	eqtr4di 2289	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( F ` z ) = ( ( 1st ` z ) F ( 2nd ` z ) ) )
17		xp1st 6399	. . . . . . . 8 |- ( z e. ( RR X. RR ) -> ( 1st ` z ) e. RR )
18		xp2nd 6400	. . . . . . . 8 |- ( z e. ( RR X. RR ) -> ( 2nd ` z ) e. RR )
19	17	recnd 8354	. . . . . . . . 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 8354	. . . . . . . . . 10 |- ( z e. ( RR X. RR ) -> ( 2nd ` z ) e. CC )
22	20, 21	mulcld 8346	. . . . . . . . 9 |- ( z e. ( RR X. RR ) -> ( _i x. ( 2nd ` z ) ) e. CC )
23	19, 22	addcld 8345	. . . . . . . 8 |- ( z e. ( RR X. RR ) -> ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) e. CC )
24		oveq1 6092	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> ( x + ( _i x. y ) ) = ( ( 1st ` z ) + ( _i x. y ) ) )
25		oveq2 6093	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( _i x. y ) = ( _i x. ( 2nd ` z ) ) )
26	25	oveq2d 6101	. . . . . . . . 9 |- ( y = ( 2nd ` z ) -> ( ( 1st ` z ) + ( _i x. y ) ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
27	24, 26, 10	ovmpog 6223	. . . . . . . 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 1278	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
29	16, 28	eqtrd 2271	. . . . . 6 |- ( z e. ( RR X. RR ) -> ( F ` z ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) )
30	29, 23	eqeltrd 2315	. . . . 5 |- ( z e. ( RR X. RR ) -> ( F ` z ) e. CC )
31	30	rgen 2603	. . . 4 |- A. z e. ( RR X. RR ) ( F ` z ) e. CC
32		ffnfv 5866	. . . 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 955	. . 3 |- F : ( RR X. RR ) --> CC
34	17, 18	jca 306	. . . . . . 7 |- ( z e. ( RR X. RR ) -> ( ( 1st ` z ) e. RR /\ ( 2nd ` z ) e. RR ) )
35		xp1st 6399	. . . . . . . 8 |- ( w e. ( RR X. RR ) -> ( 1st ` w ) e. RR )
36		xp2nd 6400	. . . . . . . 8 |- ( w e. ( RR X. RR ) -> ( 2nd ` w ) e. RR )
37	35, 36	jca 306	. . . . . . 7 |- ( w e. ( RR X. RR ) -> ( ( 1st ` w ) e. RR /\ ( 2nd ` w ) e. RR ) )
38		cru 8930	. . . . . . 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 289	. . . . . 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 5695	. . . . . . . . 9 |- ( z = w -> ( F ` z ) = ( F ` w ) )
41		fveq2 5695	. . . . . . . . . 10 |- ( z = w -> ( 1st ` z ) = ( 1st ` w ) )
42		fveq2 5695	. . . . . . . . . . 11 |- ( z = w -> ( 2nd ` z ) = ( 2nd ` w ) )
43	42	oveq2d 6101	. . . . . . . . . 10 |- ( z = w -> ( _i x. ( 2nd ` z ) ) = ( _i x. ( 2nd ` w ) ) )
44	41, 43	oveq12d 6103	. . . . . . . . 9 |- ( z = w -> ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) = ( ( 1st ` w ) + ( _i x. ( 2nd ` w ) ) ) )
45	40, 44	eqeq12d 2253	. . . . . . . 8 |- ( z = w -> ( ( F ` z ) = ( ( 1st ` z ) + ( _i x. ( 2nd ` z ) ) ) <-> ( F ` w ) = ( ( 1st ` w ) + ( _i x. ( 2nd ` w ) ) ) ) )
46	45, 29	vtoclga 2889	. . . . . . 7 |- ( w e. ( RR X. RR ) -> ( F ` w ) = ( ( 1st ` w ) + ( _i x. ( 2nd ` w ) ) ) )
47	29, 46	eqeqan12d 2254	. . . . . 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 6409	. . . . . . . 8 |- ( w e. ( RR X. RR ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
49	13, 48	eqeqan12d 2254	. . . . . . 7 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( z = w <-> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
50		vex 2824	. . . . . . . . 9 |- z e. _V
51		1stexg 6401	. . . . . . . . 9 |- ( z e. _V -> ( 1st ` z ) e. _V )
52	50, 51	ax-mp 5	. . . . . . . 8 |- ( 1st ` z ) e. _V
53		2ndexg 6402	. . . . . . . . 9 |- ( z e. _V -> ( 2nd ` z ) e. _V )
54	50, 53	ax-mp 5	. . . . . . . 8 |- ( 2nd ` z ) e. _V
55	52, 54	opth 4377	. . . . . . 7 |- ( <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. <-> ( ( 1st ` z ) = ( 1st ` w ) /\ ( 2nd ` z ) = ( 2nd ` w ) ) )
56	49, 55	bitrdi 196	. . . . . 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 220	. . . . 5 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( ( F ` z ) = ( F ` w ) <-> z = w ) )
58	57	biimpd 144	. . . 4 |- ( ( z e. ( RR X. RR ) /\ w e. ( RR X. RR ) ) -> ( ( F ` z ) = ( F ` w ) -> z = w ) )
59	58	rgen2a 2604	. . 3 |- A. z e. ( RR X. RR ) A. w e. ( RR X. RR ) ( ( F ` z ) = ( F ` w ) -> z = w )
60		dff13 5974	. . 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 955	. 2 |- F : ( RR X. RR ) -1-1-> CC
62		cnre 8322	. . . . . 6 |- ( w e. CC -> E. u e. RR E. v e. RR w = ( u + ( _i x. v ) ) )
63		simpl 109	. . . . . . . . 9 |- ( ( u e. RR /\ v e. RR ) -> u e. RR )
64		simpr 110	. . . . . . . . 9 |- ( ( u e. RR /\ v e. RR ) -> v e. RR )
65	63	recnd 8354	. . . . . . . . . 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 8354	. . . . . . . . . . 11 |- ( ( u e. RR /\ v e. RR ) -> v e. CC )
68	66, 67	mulcld 8346	. . . . . . . . . 10 |- ( ( u e. RR /\ v e. RR ) -> ( _i x. v ) e. CC )
69	65, 68	addcld 8345	. . . . . . . . 9 |- ( ( u e. RR /\ v e. RR ) -> ( u + ( _i x. v ) ) e. CC )
70		oveq1 6092	. . . . . . . . . 10 |- ( x = u -> ( x + ( _i x. y ) ) = ( u + ( _i x. y ) ) )
71		oveq2 6093	. . . . . . . . . . 11 |- ( y = v -> ( _i x. y ) = ( _i x. v ) )
72	71	oveq2d 6101	. . . . . . . . . 10 |- ( y = v -> ( u + ( _i x. y ) ) = ( u + ( _i x. v ) ) )
73	70, 72, 10	ovmpog 6223	. . . . . . . . 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 1278	. . . . . . . 8 |- ( ( u e. RR /\ v e. RR ) -> ( u F v ) = ( u + ( _i x. v ) ) )
75	74	eqeq2d 2250	. . . . . . 7 |- ( ( u e. RR /\ v e. RR ) -> ( w = ( u F v ) <-> w = ( u + ( _i x. v ) ) ) )
76	75	2rexbiia 2566	. . . . . 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 134	. . . . 5 |- ( w e. CC -> E. u e. RR E. v e. RR w = ( u F v ) )
78		fveq2 5695	. . . . . . . 8 |- ( z = <. u , v >. -> ( F ` z ) = ( F ` <. u , v >. ) )
79		df-ov 6088	. . . . . . . 8 |- ( u F v ) = ( F ` <. u , v >. )
80	78, 79	eqtr4di 2289	. . . . . . 7 |- ( z = <. u , v >. -> ( F ` z ) = ( u F v ) )
81	80	eqeq2d 2250	. . . . . 6 |- ( z = <. u , v >. -> ( w = ( F ` z ) <-> w = ( u F v ) ) )
82	81	rexxp 4924	. . . . 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 134	. . . 4 |- ( w e. CC -> E. z e. ( RR X. RR ) w = ( F ` z ) )
84	83	rgen 2603	. . 3 |- A. w e. CC E. z e. ( RR X. RR ) w = ( F ` z )
85		dffo3 5855	. . 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 955	. 2 |- F : ( RR X. RR ) -onto-> CC
87		df-f1o 5384	. 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 955	1 |- F : ( RR X. RR ) -1-1-onto-> CC

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   _Vcvv 2821   <.cop 3712   X. cxp 4772   Fn wfn 5372   -->wf 5373   -1-1->wf1 5374   -onto->wfo 5375   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373   CCcc 8177   RRcr 8178   _ici 8181   + caddc 8182   x. cmul 8184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-sub 8499  df-neg 8500  df-reap 8903

This theorem is used by:  cnrecnv  11676