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Theorem cnref1o 8733

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 6987), 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 107	. . . . . . . 8 $/\$
2	1	recnd 7147	. . . . . . 7 $/\$
3		ax-icn 7071	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 108	. . . . . . . . 9 $/\$
6	5	recnd 7147	. . . . . . . 8 $/\$
7	4, 6	mulcld 7139	. . . . . . 7 $/\$
8	2, 7	addcld 7138	. . . . . 6 $/\$
9	8	rgen2a 2417	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpt2 5848	. . . . 5
12	9, 11	ax-mp 7	. . . 4
13		1st2nd2 5821	. . . . . . . . 9
14	13	fveq2d 5202	. . . . . . . 8
15		df-ov 5535	. . . . . . . 8
16	14, 15	syl6eqr 2131	. . . . . . 7
17		xp1st 5812	. . . . . . . 8
18		xp2nd 5813	. . . . . . . 8
19	17	recnd 7147	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 7147	. . . . . . . . . 10
22	20, 21	mulcld 7139	. . . . . . . . 9
23	19, 22	addcld 7138	. . . . . . . 8
24		oveq1 5539	. . . . . . . . 9
25		oveq2 5540	. . . . . . . . . 10
26	25	oveq2d 5548	. . . . . . . . 9
27	24, 26, 10	ovmpt2g 5655	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1169	. . . . . . 7
29	16, 28	eqtrd 2113	. . . . . 6
30	29, 23	eqeltrd 2155	. . . . 5
31	30	rgen 2416	. . . 4
32		ffnfv 5344	. . . 4 $/\$
33	12, 31, 32	mpbir2an 883	. . 3
34	17, 18	jca 300	. . . . . . 7 $/\$
35		xp1st 5812	. . . . . . . 8
36		xp2nd 5813	. . . . . . . 8
37	35, 36	jca 300	. . . . . . 7 $/\$
38		cru 7702	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 283	. . . . . 6 $/\$ $/\$
40		fveq2 5198	. . . . . . . . 9
41		fveq2 5198	. . . . . . . . . 10
42		fveq2 5198	. . . . . . . . . . 11
43	42	oveq2d 5548	. . . . . . . . . 10
44	41, 43	oveq12d 5550	. . . . . . . . 9
45	40, 44	eqeq12d 2095	. . . . . . . 8
46	45, 29	vtoclga 2664	. . . . . . 7
47	29, 46	eqeqan12d 2096	. . . . . 6 $/\$
48		1st2nd2 5821	. . . . . . . 8
49	13, 48	eqeqan12d 2096	. . . . . . 7 $/\$
50		vex 2604	. . . . . . . . 9
51		1stexg 5814	. . . . . . . . 9
52	50, 51	ax-mp 7	. . . . . . . 8
53		2ndexg 5815	. . . . . . . . 9
54	50, 53	ax-mp 7	. . . . . . . 8
55	52, 54	opth 3992	. . . . . . 7 $/\$
56	49, 55	syl6bb 194	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 218	. . . . 5 $/\$
58	57	biimpd 142	. . . 4 $/\$
59	58	rgen2a 2417	. . 3
60		dff13 5428	. . 3 $/\$
61	33, 59, 60	mpbir2an 883	. 2
62		cnre 7115	. . . . . 6
63		simpl 107	. . . . . . . . 9 $/\$
64		simpr 108	. . . . . . . . 9 $/\$
65	63	recnd 7147	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 7147	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 7139	. . . . . . . . . 10 $/\$
69	65, 68	addcld 7138	. . . . . . . . 9 $/\$
70		oveq1 5539	. . . . . . . . . 10
71		oveq2 5540	. . . . . . . . . . 11
72	71	oveq2d 5548	. . . . . . . . . 10
73	70, 72, 10	ovmpt2g 5655	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1169	. . . . . . . 8 $/\$
75	74	eqeq2d 2092	. . . . . . 7 $/\$
76	75	2rexbiia 2382	. . . . . 6
77	62, 76	sylibr 132	. . . . 5
78		fveq2 5198	. . . . . . . 8
79		df-ov 5535	. . . . . . . 8
80	78, 79	syl6eqr 2131	. . . . . . 7
81	80	eqeq2d 2092	. . . . . 6
82	81	rexxp 4498	. . . . 5
83	77, 82	sylibr 132	. . . 4
84	83	rgen 2416	. . 3
85		dffo3 5335	. . 3 $/\$
86	33, 84, 85	mpbir2an 883	. 2
87		df-f1o 4929	. 2 $/\$
88	61, 86, 87	mpbir2an 883	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wb 103

wceq 1284

wcel 1433

wral 2348

wrex 2349

cvv 2601

cop 3401

cxp 4361

wfn 4917

wf 4918

wf1 4919

wfo 4920

wf1o 4921

cfv 4922 (class class class)co 5532

cmpt2 5534

c1st 5785

c2nd 5786

cc 6979

cr 6980

ci 6983

caddc 6984

cmul 6986

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675

This theorem is referenced by: cnrecnv 9797

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