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Theorem cnref1o 9946

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 8098), 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 109	. . . . . . . 8 $/\$
2	1	recnd 8267	. . . . . . 7 $/\$
3		ax-icn 8187	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8267	. . . . . . . 8 $/\$
7	4, 6	mulcld 8259	. . . . . . 7 $/\$
8	2, 7	addcld 8258	. . . . . 6 $/\$
9	8	rgen2a 2587	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6376	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6347	. . . . . . . . 9
14	13	fveq2d 5652	. . . . . . . 8
15		df-ov 6031	. . . . . . . 8
16	14, 15	eqtr4di 2282	. . . . . . 7
17		xp1st 6337	. . . . . . . 8
18		xp2nd 6338	. . . . . . . 8
19	17	recnd 8267	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8267	. . . . . . . . . 10
22	20, 21	mulcld 8259	. . . . . . . . 9
23	19, 22	addcld 8258	. . . . . . . 8
24		oveq1 6035	. . . . . . . . 9
25		oveq2 6036	. . . . . . . . . 10
26	25	oveq2d 6044	. . . . . . . . 9
27	24, 26, 10	ovmpog 6166	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1274	. . . . . . 7
29	16, 28	eqtrd 2264	. . . . . 6
30	29, 23	eqeltrd 2308	. . . . 5
31	30	rgen 2586	. . . 4
32		ffnfv 5813	. . . 4 $/\$
33	12, 31, 32	mpbir2an 951	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6337	. . . . . . . 8
36		xp2nd 6338	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8841	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5648	. . . . . . . . 9
41		fveq2 5648	. . . . . . . . . 10
42		fveq2 5648	. . . . . . . . . . 11
43	42	oveq2d 6044	. . . . . . . . . 10
44	41, 43	oveq12d 6046	. . . . . . . . 9
45	40, 44	eqeq12d 2246	. . . . . . . 8
46	45, 29	vtoclga 2871	. . . . . . 7
47	29, 46	eqeqan12d 2247	. . . . . 6 $/\$
48		1st2nd2 6347	. . . . . . . 8
49	13, 48	eqeqan12d 2247	. . . . . . 7 $/\$
50		vex 2806	. . . . . . . . 9
51		1stexg 6339	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6340	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4335	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2587	. . 3
60		dff13 5919	. . 3 $/\$
61	33, 59, 60	mpbir2an 951	. 2
62		cnre 8235	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8267	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8267	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8259	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8258	. . . . . . . . 9 $/\$
70		oveq1 6035	. . . . . . . . . 10
71		oveq2 6036	. . . . . . . . . . 11
72	71	oveq2d 6044	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6166	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1274	. . . . . . . 8 $/\$
75	74	eqeq2d 2243	. . . . . . 7 $/\$
76	75	2rexbiia 2549	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5648	. . . . . . . 8
79		df-ov 6031	. . . . . . . 8
80	78, 79	eqtr4di 2282	. . . . . . 7
81	80	eqeq2d 2243	. . . . . 6
82	81	rexxp 4880	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2586	. . 3
85		dffo3 5802	. . 3 $/\$
86	33, 84, 85	mpbir2an 951	. 2
87		df-f1o 5340	. 2 $/\$
88	61, 86, 87	mpbir2an 951	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2202

wral 2511

wrex 2512

cvv 2803

cop 3676

cxp 4729

wfn 5328

wf 5329

wf1 5330

wfo 5331

wf1o 5332

cfv 5333 (class class class)co 6028

cmpo 6030

c1st 6310

c2nd 6311

cc 8090

cr 8091

ci 8094

caddc 8095

cmul 8097

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-pnf 8275 df-mnf 8276 df-ltxr 8278 df-sub 8411 df-neg 8412 df-reap 8814

This theorem is referenced by: cnrecnv 11550

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