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Theorem cnref1o 9440

Description: There is a natural one-to-one mapping from

, where we map

. In our construction of the complex numbers, this is in fact our definition of

(see df-c 7626), 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

**Assertion**
Ref	Expression
cnref1o

Distinct variable group:

Allowed substitution hints:

(

)

Proof of Theorem cnref1o Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 108	. . . . . . . 8 $/\$
2	1	recnd 7794	. . . . . . 7 $/\$
3		ax-icn 7715	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 109	. . . . . . . . 9 $/\$
6	5	recnd 7794	. . . . . . . 8 $/\$
7	4, 6	mulcld 7786	. . . . . . 7 $/\$
8	2, 7	addcld 7785	. . . . . 6 $/\$
9	8	rgen2a 2486	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6100	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6073	. . . . . . . . 9
14	13	fveq2d 5425	. . . . . . . 8
15		df-ov 5777	. . . . . . . 8
16	14, 15	syl6eqr 2190	. . . . . . 7
17		xp1st 6063	. . . . . . . 8
18		xp2nd 6064	. . . . . . . 8
19	17	recnd 7794	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 7794	. . . . . . . . . 10
22	20, 21	mulcld 7786	. . . . . . . . 9
23	19, 22	addcld 7785	. . . . . . . 8
24		oveq1 5781	. . . . . . . . 9
25		oveq2 5782	. . . . . . . . . 10
26	25	oveq2d 5790	. . . . . . . . 9
27	24, 26, 10	ovmpog 5905	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1216	. . . . . . 7
29	16, 28	eqtrd 2172	. . . . . 6
30	29, 23	eqeltrd 2216	. . . . 5
31	30	rgen 2485	. . . 4
32		ffnfv 5578	. . . 4 $/\$
33	12, 31, 32	mpbir2an 926	. . 3
34	17, 18	jca 304	. . . . . . 7 $/\$
35		xp1st 6063	. . . . . . . 8
36		xp2nd 6064	. . . . . . . 8
37	35, 36	jca 304	. . . . . . 7 $/\$
38		cru 8364	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 287	. . . . . 6 $/\$ $/\$
40		fveq2 5421	. . . . . . . . 9
41		fveq2 5421	. . . . . . . . . 10
42		fveq2 5421	. . . . . . . . . . 11
43	42	oveq2d 5790	. . . . . . . . . 10
44	41, 43	oveq12d 5792	. . . . . . . . 9
45	40, 44	eqeq12d 2154	. . . . . . . 8
46	45, 29	vtoclga 2752	. . . . . . 7
47	29, 46	eqeqan12d 2155	. . . . . 6 $/\$
48		1st2nd2 6073	. . . . . . . 8
49	13, 48	eqeqan12d 2155	. . . . . . 7 $/\$
50		vex 2689	. . . . . . . . 9
51		1stexg 6065	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6066	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4159	. . . . . . 7 $/\$
56	49, 55	syl6bb 195	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 219	. . . . 5 $/\$
58	57	biimpd 143	. . . 4 $/\$
59	58	rgen2a 2486	. . 3
60		dff13 5669	. . 3 $/\$
61	33, 59, 60	mpbir2an 926	. 2
62		cnre 7762	. . . . . 6
63		simpl 108	. . . . . . . . 9 $/\$
64		simpr 109	. . . . . . . . 9 $/\$
65	63	recnd 7794	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 7794	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 7786	. . . . . . . . . 10 $/\$
69	65, 68	addcld 7785	. . . . . . . . 9 $/\$
70		oveq1 5781	. . . . . . . . . 10
71		oveq2 5782	. . . . . . . . . . 11
72	71	oveq2d 5790	. . . . . . . . . 10
73	70, 72, 10	ovmpog 5905	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1216	. . . . . . . 8 $/\$
75	74	eqeq2d 2151	. . . . . . 7 $/\$
76	75	2rexbiia 2451	. . . . . 6
77	62, 76	sylibr 133	. . . . 5
78		fveq2 5421	. . . . . . . 8
79		df-ov 5777	. . . . . . . 8
80	78, 79	syl6eqr 2190	. . . . . . 7
81	80	eqeq2d 2151	. . . . . 6
82	81	rexxp 4683	. . . . 5
83	77, 82	sylibr 133	. . . 4
84	83	rgen 2485	. . 3
85		dffo3 5567	. . 3 $/\$
86	33, 84, 85	mpbir2an 926	. 2
87		df-f1o 5130	. 2 $/\$
88	61, 86, 87	mpbir2an 926	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1331

wcel 1480

wral 2416

wrex 2417

cvv 2686

cop 3530

cxp 4537

wfn 5118

wf 5119

wf1 5120

wfo 5121

wf1o 5122

cfv 5123 (class class class)co 5774

cmpo 5776

c1st 6036

c2nd 6037

cc 7618

cr 7619

ci 7622

caddc 7623

cmul 7625

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337

This theorem is referenced by: cnrecnv 10682

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