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Theorem cnref1o 9652

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 7819), 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 7988	. . . . . . 7 $/\$
3		ax-icn 7908	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 7988	. . . . . . . 8 $/\$
7	4, 6	mulcld 7980	. . . . . . 7 $/\$
8	2, 7	addcld 7979	. . . . . 6 $/\$
9	8	rgen2a 2531	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6205	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6178	. . . . . . . . 9
14	13	fveq2d 5521	. . . . . . . 8
15		df-ov 5880	. . . . . . . 8
16	14, 15	eqtr4di 2228	. . . . . . 7
17		xp1st 6168	. . . . . . . 8
18		xp2nd 6169	. . . . . . . 8
19	17	recnd 7988	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 7988	. . . . . . . . . 10
22	20, 21	mulcld 7980	. . . . . . . . 9
23	19, 22	addcld 7979	. . . . . . . 8
24		oveq1 5884	. . . . . . . . 9
25		oveq2 5885	. . . . . . . . . 10
26	25	oveq2d 5893	. . . . . . . . 9
27	24, 26, 10	ovmpog 6011	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1238	. . . . . . 7
29	16, 28	eqtrd 2210	. . . . . 6
30	29, 23	eqeltrd 2254	. . . . 5
31	30	rgen 2530	. . . 4
32		ffnfv 5676	. . . 4 $/\$
33	12, 31, 32	mpbir2an 942	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6168	. . . . . . . 8
36		xp2nd 6169	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8561	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5517	. . . . . . . . 9
41		fveq2 5517	. . . . . . . . . 10
42		fveq2 5517	. . . . . . . . . . 11
43	42	oveq2d 5893	. . . . . . . . . 10
44	41, 43	oveq12d 5895	. . . . . . . . 9
45	40, 44	eqeq12d 2192	. . . . . . . 8
46	45, 29	vtoclga 2805	. . . . . . 7
47	29, 46	eqeqan12d 2193	. . . . . 6 $/\$
48		1st2nd2 6178	. . . . . . . 8
49	13, 48	eqeqan12d 2193	. . . . . . 7 $/\$
50		vex 2742	. . . . . . . . 9
51		1stexg 6170	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6171	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4239	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2531	. . 3
60		dff13 5771	. . 3 $/\$
61	33, 59, 60	mpbir2an 942	. 2
62		cnre 7955	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 7988	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 7988	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 7980	. . . . . . . . . 10 $/\$
69	65, 68	addcld 7979	. . . . . . . . 9 $/\$
70		oveq1 5884	. . . . . . . . . 10
71		oveq2 5885	. . . . . . . . . . 11
72	71	oveq2d 5893	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6011	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1238	. . . . . . . 8 $/\$
75	74	eqeq2d 2189	. . . . . . 7 $/\$
76	75	2rexbiia 2493	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5517	. . . . . . . 8
79		df-ov 5880	. . . . . . . 8
80	78, 79	eqtr4di 2228	. . . . . . 7
81	80	eqeq2d 2189	. . . . . 6
82	81	rexxp 4773	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2530	. . 3
85		dffo3 5665	. . 3 $/\$
86	33, 84, 85	mpbir2an 942	. 2
87		df-f1o 5225	. 2 $/\$
88	61, 86, 87	mpbir2an 942	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148

wral 2455

wrex 2456

cvv 2739

cop 3597

cxp 4626

wfn 5213

wf 5214

wf1 5215

wfo 5216

wf1o 5217

cfv 5218 (class class class)co 5877

cmpo 5879

c1st 6141

c2nd 6142

cc 7811

cr 7812

ci 7815

caddc 7816

cmul 7818

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-pnf 7996 df-mnf 7997 df-ltxr 7999 df-sub 8132 df-neg 8133 df-reap 8534

This theorem is referenced by: cnrecnv 10921

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