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Theorem cnref1o 9846

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 8005), 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 8175	. . . . . . 7 $/\$
3		ax-icn 8094	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8175	. . . . . . . 8 $/\$
7	4, 6	mulcld 8167	. . . . . . 7 $/\$
8	2, 7	addcld 8166	. . . . . 6 $/\$
9	8	rgen2a 2584	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6348	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6321	. . . . . . . . 9
14	13	fveq2d 5631	. . . . . . . 8
15		df-ov 6004	. . . . . . . 8
16	14, 15	eqtr4di 2280	. . . . . . 7
17		xp1st 6311	. . . . . . . 8
18		xp2nd 6312	. . . . . . . 8
19	17	recnd 8175	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8175	. . . . . . . . . 10
22	20, 21	mulcld 8167	. . . . . . . . 9
23	19, 22	addcld 8166	. . . . . . . 8
24		oveq1 6008	. . . . . . . . 9
25		oveq2 6009	. . . . . . . . . 10
26	25	oveq2d 6017	. . . . . . . . 9
27	24, 26, 10	ovmpog 6139	. . . . . . . 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 5793	. . . 4 $/\$
33	12, 31, 32	mpbir2an 948	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6311	. . . . . . . 8
36		xp2nd 6312	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8749	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5627	. . . . . . . . 9
41		fveq2 5627	. . . . . . . . . 10
42		fveq2 5627	. . . . . . . . . . 11
43	42	oveq2d 6017	. . . . . . . . . 10
44	41, 43	oveq12d 6019	. . . . . . . . 9
45	40, 44	eqeq12d 2244	. . . . . . . 8
46	45, 29	vtoclga 2867	. . . . . . 7
47	29, 46	eqeqan12d 2245	. . . . . 6 $/\$
48		1st2nd2 6321	. . . . . . . 8
49	13, 48	eqeqan12d 2245	. . . . . . 7 $/\$
50		vex 2802	. . . . . . . . 9
51		1stexg 6313	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6314	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4323	. . . . . . 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 5892	. . 3 $/\$
61	33, 59, 60	mpbir2an 948	. 2
62		cnre 8142	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8175	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8175	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8167	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8166	. . . . . . . . 9 $/\$
70		oveq1 6008	. . . . . . . . . 10
71		oveq2 6009	. . . . . . . . . . 11
72	71	oveq2d 6017	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6139	. . . . . . . . 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 5627	. . . . . . . 8
79		df-ov 6004	. . . . . . . 8
80	78, 79	eqtr4di 2280	. . . . . . 7
81	80	eqeq2d 2241	. . . . . 6
82	81	rexxp 4866	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2583	. . 3
85		dffo3 5782	. . 3 $/\$
86	33, 84, 85	mpbir2an 948	. 2
87		df-f1o 5325	. 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 2799

cop 3669

cxp 4717

wfn 5313

wf 5314

wf1 5315

wfo 5316

wf1o 5317

cfv 5318 (class class class)co 6001

cmpo 6003

c1st 6284

c2nd 6285

cc 7997

cr 7998

ci 8001

caddc 8002

cmul 8004

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 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116

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 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-pnf 8183 df-mnf 8184 df-ltxr 8186 df-sub 8319 df-neg 8320 df-reap 8722

This theorem is referenced by: cnrecnv 11421

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