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Theorem cnref1o 9875

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 8028), 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 8198	. . . . . . 7 $/\$
3		ax-icn 8117	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8198	. . . . . . . 8 $/\$
7	4, 6	mulcld 8190	. . . . . . 7 $/\$
8	2, 7	addcld 8189	. . . . . 6 $/\$
9	8	rgen2a 2584	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6362	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6333	. . . . . . . . 9
14	13	fveq2d 5639	. . . . . . . 8
15		df-ov 6016	. . . . . . . 8
16	14, 15	eqtr4di 2280	. . . . . . 7
17		xp1st 6323	. . . . . . . 8
18		xp2nd 6324	. . . . . . . 8
19	17	recnd 8198	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8198	. . . . . . . . . 10
22	20, 21	mulcld 8190	. . . . . . . . 9
23	19, 22	addcld 8189	. . . . . . . 8
24		oveq1 6020	. . . . . . . . 9
25		oveq2 6021	. . . . . . . . . 10
26	25	oveq2d 6029	. . . . . . . . 9
27	24, 26, 10	ovmpog 6151	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1271	. . . . . . 7
29	16, 28	eqtrd 2262	. . . . . 6
30	29, 23	eqeltrd 2306	. . . . 5
31	30	rgen 2583	. . . 4
32		ffnfv 5801	. . . 4 $/\$
33	12, 31, 32	mpbir2an 948	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6323	. . . . . . . 8
36		xp2nd 6324	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8772	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5635	. . . . . . . . 9
41		fveq2 5635	. . . . . . . . . 10
42		fveq2 5635	. . . . . . . . . . 11
43	42	oveq2d 6029	. . . . . . . . . 10
44	41, 43	oveq12d 6031	. . . . . . . . 9
45	40, 44	eqeq12d 2244	. . . . . . . 8
46	45, 29	vtoclga 2868	. . . . . . 7
47	29, 46	eqeqan12d 2245	. . . . . 6 $/\$
48		1st2nd2 6333	. . . . . . . 8
49	13, 48	eqeqan12d 2245	. . . . . . 7 $/\$
50		vex 2803	. . . . . . . . 9
51		1stexg 6325	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6326	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4327	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2584	. . 3
60		dff13 5904	. . 3 $/\$
61	33, 59, 60	mpbir2an 948	. 2
62		cnre 8165	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8198	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8198	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8190	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8189	. . . . . . . . 9 $/\$
70		oveq1 6020	. . . . . . . . . 10
71		oveq2 6021	. . . . . . . . . . 11
72	71	oveq2d 6029	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6151	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1271	. . . . . . . 8 $/\$
75	74	eqeq2d 2241	. . . . . . 7 $/\$
76	75	2rexbiia 2546	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5635	. . . . . . . 8
79		df-ov 6016	. . . . . . . 8
80	78, 79	eqtr4di 2280	. . . . . . 7
81	80	eqeq2d 2241	. . . . . 6
82	81	rexxp 4872	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2583	. . 3
85		dffo3 5790	. . 3 $/\$
86	33, 84, 85	mpbir2an 948	. 2
87		df-f1o 5331	. 2 $/\$
88	61, 86, 87	mpbir2an 948	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1395

wcel 2200

wral 2508

wrex 2509

cvv 2800

cop 3670

cxp 4721

wfn 5319

wf 5320

wf1 5321

wfo 5322

wf1o 5323

cfv 5324 (class class class)co 6013

cmpo 6015

c1st 6296

c2nd 6297

cc 8020

cr 8021

ci 8024

caddc 8025

cmul 8027

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-pnf 8206 df-mnf 8207 df-ltxr 8209 df-sub 8342 df-neg 8343 df-reap 8745

This theorem is referenced by: cnrecnv 11461

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