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Theorem cnref1o 9807

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 7966), 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 8136	. . . . . . 7 $/\$
3		ax-icn 8055	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8136	. . . . . . . 8 $/\$
7	4, 6	mulcld 8128	. . . . . . 7 $/\$
8	2, 7	addcld 8127	. . . . . 6 $/\$
9	8	rgen2a 2562	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6311	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6284	. . . . . . . . 9
14	13	fveq2d 5603	. . . . . . . 8
15		df-ov 5970	. . . . . . . 8
16	14, 15	eqtr4di 2258	. . . . . . 7
17		xp1st 6274	. . . . . . . 8
18		xp2nd 6275	. . . . . . . 8
19	17	recnd 8136	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8136	. . . . . . . . . 10
22	20, 21	mulcld 8128	. . . . . . . . 9
23	19, 22	addcld 8127	. . . . . . . 8
24		oveq1 5974	. . . . . . . . 9
25		oveq2 5975	. . . . . . . . . 10
26	25	oveq2d 5983	. . . . . . . . 9
27	24, 26, 10	ovmpog 6103	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1250	. . . . . . 7
29	16, 28	eqtrd 2240	. . . . . 6
30	29, 23	eqeltrd 2284	. . . . 5
31	30	rgen 2561	. . . 4
32		ffnfv 5761	. . . 4 $/\$
33	12, 31, 32	mpbir2an 945	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6274	. . . . . . . 8
36		xp2nd 6275	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8710	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5599	. . . . . . . . 9
41		fveq2 5599	. . . . . . . . . 10
42		fveq2 5599	. . . . . . . . . . 11
43	42	oveq2d 5983	. . . . . . . . . 10
44	41, 43	oveq12d 5985	. . . . . . . . 9
45	40, 44	eqeq12d 2222	. . . . . . . 8
46	45, 29	vtoclga 2844	. . . . . . 7
47	29, 46	eqeqan12d 2223	. . . . . 6 $/\$
48		1st2nd2 6284	. . . . . . . 8
49	13, 48	eqeqan12d 2223	. . . . . . 7 $/\$
50		vex 2779	. . . . . . . . 9
51		1stexg 6276	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6277	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4299	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2562	. . 3
60		dff13 5860	. . 3 $/\$
61	33, 59, 60	mpbir2an 945	. 2
62		cnre 8103	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8136	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8136	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8128	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8127	. . . . . . . . 9 $/\$
70		oveq1 5974	. . . . . . . . . 10
71		oveq2 5975	. . . . . . . . . . 11
72	71	oveq2d 5983	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6103	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1250	. . . . . . . 8 $/\$
75	74	eqeq2d 2219	. . . . . . 7 $/\$
76	75	2rexbiia 2524	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5599	. . . . . . . 8
79		df-ov 5970	. . . . . . . 8
80	78, 79	eqtr4di 2258	. . . . . . 7
81	80	eqeq2d 2219	. . . . . 6
82	81	rexxp 4840	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2561	. . 3
85		dffo3 5750	. . 3 $/\$
86	33, 84, 85	mpbir2an 945	. 2
87		df-f1o 5297	. 2 $/\$
88	61, 86, 87	mpbir2an 945	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1373

wcel 2178

wral 2486

wrex 2487

cvv 2776

cop 3646

cxp 4691

wfn 5285

wf 5286

wf1 5287

wfo 5288

wf1o 5289

cfv 5290 (class class class)co 5967

cmpo 5969

c1st 6247

c2nd 6248

cc 7958

cr 7959

ci 7962

caddc 7963

cmul 7965

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-id 4358 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-pnf 8144 df-mnf 8145 df-ltxr 8147 df-sub 8280 df-neg 8281 df-reap 8683

This theorem is referenced by: cnrecnv 11336

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