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Theorem cnref1o 9983

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 8133), 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 8302	. . . . . . 7 $/\$
3		ax-icn 8222	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8302	. . . . . . . 8 $/\$
7	4, 6	mulcld 8294	. . . . . . 7 $/\$
8	2, 7	addcld 8293	. . . . . 6 $/\$
9	8	rgen2a 2596	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6398	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6369	. . . . . . . . 9
14	13	fveq2d 5674	. . . . . . . 8
15		df-ov 6053	. . . . . . . 8
16	14, 15	eqtr4di 2283	. . . . . . 7
17		xp1st 6359	. . . . . . . 8
18		xp2nd 6360	. . . . . . . 8
19	17	recnd 8302	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8302	. . . . . . . . . 10
22	20, 21	mulcld 8294	. . . . . . . . 9
23	19, 22	addcld 8293	. . . . . . . 8
24		oveq1 6057	. . . . . . . . 9
25		oveq2 6058	. . . . . . . . . 10
26	25	oveq2d 6066	. . . . . . . . 9
27	24, 26, 10	ovmpog 6188	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1274	. . . . . . 7
29	16, 28	eqtrd 2265	. . . . . 6
30	29, 23	eqeltrd 2309	. . . . 5
31	30	rgen 2595	. . . 4
32		ffnfv 5835	. . . 4 $/\$
33	12, 31, 32	mpbir2an 951	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6359	. . . . . . . 8
36		xp2nd 6360	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8876	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5670	. . . . . . . . 9
41		fveq2 5670	. . . . . . . . . 10
42		fveq2 5670	. . . . . . . . . . 11
43	42	oveq2d 6066	. . . . . . . . . 10
44	41, 43	oveq12d 6068	. . . . . . . . 9
45	40, 44	eqeq12d 2247	. . . . . . . 8
46	45, 29	vtoclga 2881	. . . . . . 7
47	29, 46	eqeqan12d 2248	. . . . . 6 $/\$
48		1st2nd2 6369	. . . . . . . 8
49	13, 48	eqeqan12d 2248	. . . . . . 7 $/\$
50		vex 2816	. . . . . . . . 9
51		1stexg 6361	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6362	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4353	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2596	. . 3
60		dff13 5941	. . 3 $/\$
61	33, 59, 60	mpbir2an 951	. 2
62		cnre 8270	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8302	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8302	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8294	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8293	. . . . . . . . 9 $/\$
70		oveq1 6057	. . . . . . . . . 10
71		oveq2 6058	. . . . . . . . . . 11
72	71	oveq2d 6066	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6188	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1274	. . . . . . . 8 $/\$
75	74	eqeq2d 2244	. . . . . . 7 $/\$
76	75	2rexbiia 2558	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5670	. . . . . . . 8
79		df-ov 6053	. . . . . . . 8
80	78, 79	eqtr4di 2283	. . . . . . 7
81	80	eqeq2d 2244	. . . . . 6
82	81	rexxp 4899	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2595	. . 3
85		dffo3 5824	. . 3 $/\$
86	33, 84, 85	mpbir2an 951	. 2
87		df-f1o 5359	. 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 2203

wral 2520

wrex 2521

cvv 2813

cop 3692

cxp 4747

wfn 5347

wf 5348

wf1 5349

wfo 5350

wf1o 5351

cfv 5352 (class class class)co 6050

cmpo 6052

c1st 6332

c2nd 6333

cc 8125

cr 8126

ci 8129

caddc 8130

cmul 8132

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 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-pnf 8310 df-mnf 8311 df-ltxr 8313 df-sub 8446 df-neg 8447 df-reap 8849

This theorem is referenced by: cnrecnv 11595

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