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Theorem cnref1o 9719

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 7880), 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 8050	. . . . . . 7 $/\$
3		ax-icn 7969	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8050	. . . . . . . 8 $/\$
7	4, 6	mulcld 8042	. . . . . . 7 $/\$
8	2, 7	addcld 8041	. . . . . 6 $/\$
9	8	rgen2a 2548	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6257	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6230	. . . . . . . . 9
14	13	fveq2d 5559	. . . . . . . 8
15		df-ov 5922	. . . . . . . 8
16	14, 15	eqtr4di 2244	. . . . . . 7
17		xp1st 6220	. . . . . . . 8
18		xp2nd 6221	. . . . . . . 8
19	17	recnd 8050	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8050	. . . . . . . . . 10
22	20, 21	mulcld 8042	. . . . . . . . 9
23	19, 22	addcld 8041	. . . . . . . 8
24		oveq1 5926	. . . . . . . . 9
25		oveq2 5927	. . . . . . . . . 10
26	25	oveq2d 5935	. . . . . . . . 9
27	24, 26, 10	ovmpog 6054	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1249	. . . . . . 7
29	16, 28	eqtrd 2226	. . . . . 6
30	29, 23	eqeltrd 2270	. . . . 5
31	30	rgen 2547	. . . 4
32		ffnfv 5717	. . . 4 $/\$
33	12, 31, 32	mpbir2an 944	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6220	. . . . . . . 8
36		xp2nd 6221	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8623	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5555	. . . . . . . . 9
41		fveq2 5555	. . . . . . . . . 10
42		fveq2 5555	. . . . . . . . . . 11
43	42	oveq2d 5935	. . . . . . . . . 10
44	41, 43	oveq12d 5937	. . . . . . . . 9
45	40, 44	eqeq12d 2208	. . . . . . . 8
46	45, 29	vtoclga 2827	. . . . . . 7
47	29, 46	eqeqan12d 2209	. . . . . 6 $/\$
48		1st2nd2 6230	. . . . . . . 8
49	13, 48	eqeqan12d 2209	. . . . . . 7 $/\$
50		vex 2763	. . . . . . . . 9
51		1stexg 6222	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6223	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4267	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2548	. . 3
60		dff13 5812	. . 3 $/\$
61	33, 59, 60	mpbir2an 944	. 2
62		cnre 8017	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8050	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8050	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8042	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8041	. . . . . . . . 9 $/\$
70		oveq1 5926	. . . . . . . . . 10
71		oveq2 5927	. . . . . . . . . . 11
72	71	oveq2d 5935	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6054	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1249	. . . . . . . 8 $/\$
75	74	eqeq2d 2205	. . . . . . 7 $/\$
76	75	2rexbiia 2510	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5555	. . . . . . . 8
79		df-ov 5922	. . . . . . . 8
80	78, 79	eqtr4di 2244	. . . . . . 7
81	80	eqeq2d 2205	. . . . . 6
82	81	rexxp 4807	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2547	. . 3
85		dffo3 5706	. . 3 $/\$
86	33, 84, 85	mpbir2an 944	. 2
87		df-f1o 5262	. 2 $/\$
88	61, 86, 87	mpbir2an 944	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2164

wral 2472

wrex 2473

cvv 2760

cop 3622

cxp 4658

wfn 5250

wf 5251

wf1 5252

wfo 5253

wf1o 5254

cfv 5255 (class class class)co 5919

cmpo 5921

c1st 6193

c2nd 6194

cc 7872

cr 7873

ci 7876

caddc 7877

cmul 7879

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-pnf 8058 df-mnf 8059 df-ltxr 8061 df-sub 8194 df-neg 8195 df-reap 8596

This theorem is referenced by: cnrecnv 11057

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