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Theorem cnref1o 9336

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 7547), 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 7712	. . . . . . 7 $/\$
3		ax-icn 7634	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 109	. . . . . . . . 9 $/\$
6	5	recnd 7712	. . . . . . . 8 $/\$
7	4, 6	mulcld 7704	. . . . . . 7 $/\$
8	2, 7	addcld 7703	. . . . . 6 $/\$
9	8	rgen2a 2458	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6052	. . . . 5
12	9, 11	ax-mp 7	. . . 4
13		1st2nd2 6025	. . . . . . . . 9
14	13	fveq2d 5377	. . . . . . . 8
15		df-ov 5729	. . . . . . . 8
16	14, 15	syl6eqr 2163	. . . . . . 7
17		xp1st 6015	. . . . . . . 8
18		xp2nd 6016	. . . . . . . 8
19	17	recnd 7712	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 7712	. . . . . . . . . 10
22	20, 21	mulcld 7704	. . . . . . . . 9
23	19, 22	addcld 7703	. . . . . . . 8
24		oveq1 5733	. . . . . . . . 9
25		oveq2 5734	. . . . . . . . . 10
26	25	oveq2d 5742	. . . . . . . . 9
27	24, 26, 10	ovmpog 5857	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1197	. . . . . . 7
29	16, 28	eqtrd 2145	. . . . . 6
30	29, 23	eqeltrd 2189	. . . . 5
31	30	rgen 2457	. . . 4
32		ffnfv 5530	. . . 4 $/\$
33	12, 31, 32	mpbir2an 907	. . 3
34	17, 18	jca 302	. . . . . . 7 $/\$
35		xp1st 6015	. . . . . . . 8
36		xp2nd 6016	. . . . . . . 8
37	35, 36	jca 302	. . . . . . 7 $/\$
38		cru 8276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 285	. . . . . 6 $/\$ $/\$
40		fveq2 5373	. . . . . . . . 9
41		fveq2 5373	. . . . . . . . . 10
42		fveq2 5373	. . . . . . . . . . 11
43	42	oveq2d 5742	. . . . . . . . . 10
44	41, 43	oveq12d 5744	. . . . . . . . 9
45	40, 44	eqeq12d 2127	. . . . . . . 8
46	45, 29	vtoclga 2721	. . . . . . 7
47	29, 46	eqeqan12d 2128	. . . . . 6 $/\$
48		1st2nd2 6025	. . . . . . . 8
49	13, 48	eqeqan12d 2128	. . . . . . 7 $/\$
50		vex 2658	. . . . . . . . 9
51		1stexg 6017	. . . . . . . . 9
52	50, 51	ax-mp 7	. . . . . . . 8
53		2ndexg 6018	. . . . . . . . 9
54	50, 53	ax-mp 7	. . . . . . . 8
55	52, 54	opth 4117	. . . . . . 7 $/\$
56	49, 55	syl6bb 195	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 219	. . . . 5 $/\$
58	57	biimpd 143	. . . 4 $/\$
59	58	rgen2a 2458	. . 3
60		dff13 5621	. . 3 $/\$
61	33, 59, 60	mpbir2an 907	. 2
62		cnre 7680	. . . . . 6
63		simpl 108	. . . . . . . . 9 $/\$
64		simpr 109	. . . . . . . . 9 $/\$
65	63	recnd 7712	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 7712	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 7704	. . . . . . . . . 10 $/\$
69	65, 68	addcld 7703	. . . . . . . . 9 $/\$
70		oveq1 5733	. . . . . . . . . 10
71		oveq2 5734	. . . . . . . . . . 11
72	71	oveq2d 5742	. . . . . . . . . 10
73	70, 72, 10	ovmpog 5857	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1197	. . . . . . . 8 $/\$
75	74	eqeq2d 2124	. . . . . . 7 $/\$
76	75	2rexbiia 2423	. . . . . 6
77	62, 76	sylibr 133	. . . . 5
78		fveq2 5373	. . . . . . . 8
79		df-ov 5729	. . . . . . . 8
80	78, 79	syl6eqr 2163	. . . . . . 7
81	80	eqeq2d 2124	. . . . . 6
82	81	rexxp 4641	. . . . 5
83	77, 82	sylibr 133	. . . 4
84	83	rgen 2457	. . 3
85		dffo3 5519	. . 3 $/\$
86	33, 84, 85	mpbir2an 907	. 2
87		df-f1o 5086	. 2 $/\$
88	61, 86, 87	mpbir2an 907	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1312

wcel 1461

wral 2388

wrex 2389

cvv 2655

cop 3494

cxp 4495

wfn 5074

wf 5075

wf1 5076

wfo 5077

wf1o 5078

cfv 5079 (class class class)co 5726

cmpo 5728

c1st 5988

c2nd 5989

cc 7539

cr 7540

ci 7543

caddc 7544

cmul 7546

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-1rid 7646 ax-0id 7647 ax-rnegex 7648 ax-precex 7649 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656

This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-pnf 7720 df-mnf 7721 df-ltxr 7723 df-sub 7852 df-neg 7853 df-reap 8249

This theorem is referenced by: cnrecnv 10569

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