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Theorem cnref1o 9772

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 7931), 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 8101	. . . . . . 7 $/\$
3		ax-icn 8020	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8101	. . . . . . . 8 $/\$
7	4, 6	mulcld 8093	. . . . . . 7 $/\$
8	2, 7	addcld 8092	. . . . . 6 $/\$
9	8	rgen2a 2560	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6288	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6261	. . . . . . . . 9
14	13	fveq2d 5580	. . . . . . . 8
15		df-ov 5947	. . . . . . . 8
16	14, 15	eqtr4di 2256	. . . . . . 7
17		xp1st 6251	. . . . . . . 8
18		xp2nd 6252	. . . . . . . 8
19	17	recnd 8101	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8101	. . . . . . . . . 10
22	20, 21	mulcld 8093	. . . . . . . . 9
23	19, 22	addcld 8092	. . . . . . . 8
24		oveq1 5951	. . . . . . . . 9
25		oveq2 5952	. . . . . . . . . 10
26	25	oveq2d 5960	. . . . . . . . 9
27	24, 26, 10	ovmpog 6080	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1250	. . . . . . 7
29	16, 28	eqtrd 2238	. . . . . 6
30	29, 23	eqeltrd 2282	. . . . 5
31	30	rgen 2559	. . . 4
32		ffnfv 5738	. . . 4 $/\$
33	12, 31, 32	mpbir2an 945	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6251	. . . . . . . 8
36		xp2nd 6252	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8675	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5576	. . . . . . . . 9
41		fveq2 5576	. . . . . . . . . 10
42		fveq2 5576	. . . . . . . . . . 11
43	42	oveq2d 5960	. . . . . . . . . 10
44	41, 43	oveq12d 5962	. . . . . . . . 9
45	40, 44	eqeq12d 2220	. . . . . . . 8
46	45, 29	vtoclga 2839	. . . . . . 7
47	29, 46	eqeqan12d 2221	. . . . . 6 $/\$
48		1st2nd2 6261	. . . . . . . 8
49	13, 48	eqeqan12d 2221	. . . . . . 7 $/\$
50		vex 2775	. . . . . . . . 9
51		1stexg 6253	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6254	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4281	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2560	. . 3
60		dff13 5837	. . 3 $/\$
61	33, 59, 60	mpbir2an 945	. 2
62		cnre 8068	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8101	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8101	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8093	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8092	. . . . . . . . 9 $/\$
70		oveq1 5951	. . . . . . . . . 10
71		oveq2 5952	. . . . . . . . . . 11
72	71	oveq2d 5960	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6080	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1250	. . . . . . . 8 $/\$
75	74	eqeq2d 2217	. . . . . . 7 $/\$
76	75	2rexbiia 2522	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5576	. . . . . . . 8
79		df-ov 5947	. . . . . . . 8
80	78, 79	eqtr4di 2256	. . . . . . 7
81	80	eqeq2d 2217	. . . . . 6
82	81	rexxp 4822	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2559	. . 3
85		dffo3 5727	. . 3 $/\$
86	33, 84, 85	mpbir2an 945	. 2
87		df-f1o 5278	. 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 2176

wral 2484

wrex 2485

cvv 2772

cop 3636

cxp 4673

wfn 5266

wf 5267

wf1 5268

wfo 5269

wf1o 5270

cfv 5271 (class class class)co 5944

cmpo 5946

c1st 6224

c2nd 6225

cc 7923

cr 7924

ci 7927

caddc 7928

cmul 7930

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-pnf 8109 df-mnf 8110 df-ltxr 8112 df-sub 8245 df-neg 8246 df-reap 8648

This theorem is referenced by: cnrecnv 11221

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