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Theorem cnref1o 9716

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 7878), 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 8048	. . . . . . 7 $/\$
3		ax-icn 7967	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8048	. . . . . . . 8 $/\$
7	4, 6	mulcld 8040	. . . . . . 7 $/\$
8	2, 7	addcld 8039	. . . . . 6 $/\$
9	8	rgen2a 2548	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6255	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6228	. . . . . . . . 9
14	13	fveq2d 5558	. . . . . . . 8
15		df-ov 5921	. . . . . . . 8
16	14, 15	eqtr4di 2244	. . . . . . 7
17		xp1st 6218	. . . . . . . 8
18		xp2nd 6219	. . . . . . . 8
19	17	recnd 8048	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8048	. . . . . . . . . 10
22	20, 21	mulcld 8040	. . . . . . . . 9
23	19, 22	addcld 8039	. . . . . . . 8
24		oveq1 5925	. . . . . . . . 9
25		oveq2 5926	. . . . . . . . . 10
26	25	oveq2d 5934	. . . . . . . . 9
27	24, 26, 10	ovmpog 6053	. . . . . . . 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 5716	. . . 4 $/\$
33	12, 31, 32	mpbir2an 944	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6218	. . . . . . . 8
36		xp2nd 6219	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8621	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5554	. . . . . . . . 9
41		fveq2 5554	. . . . . . . . . 10
42		fveq2 5554	. . . . . . . . . . 11
43	42	oveq2d 5934	. . . . . . . . . 10
44	41, 43	oveq12d 5936	. . . . . . . . 9
45	40, 44	eqeq12d 2208	. . . . . . . 8
46	45, 29	vtoclga 2826	. . . . . . 7
47	29, 46	eqeqan12d 2209	. . . . . 6 $/\$
48		1st2nd2 6228	. . . . . . . 8
49	13, 48	eqeqan12d 2209	. . . . . . 7 $/\$
50		vex 2763	. . . . . . . . 9
51		1stexg 6220	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6221	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4266	. . . . . . 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 5811	. . 3 $/\$
61	33, 59, 60	mpbir2an 944	. 2
62		cnre 8015	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8048	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8048	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8040	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8039	. . . . . . . . 9 $/\$
70		oveq1 5925	. . . . . . . . . 10
71		oveq2 5926	. . . . . . . . . . 11
72	71	oveq2d 5934	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6053	. . . . . . . . 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 5554	. . . . . . . 8
79		df-ov 5921	. . . . . . . 8
80	78, 79	eqtr4di 2244	. . . . . . 7
81	80	eqeq2d 2205	. . . . . 6
82	81	rexxp 4806	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2547	. . 3
85		dffo3 5705	. . 3 $/\$
86	33, 84, 85	mpbir2an 944	. 2
87		df-f1o 5261	. 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 3621

cxp 4657

wfn 5249

wf 5250

wf1 5251

wfo 5252

wf1o 5253

cfv 5254 (class class class)co 5918

cmpo 5920

c1st 6191

c2nd 6192

cc 7870

cr 7871

ci 7874

caddc 7875

cmul 7877

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 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989

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 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-pnf 8056 df-mnf 8057 df-ltxr 8059 df-sub 8192 df-neg 8193 df-reap 8594

This theorem is referenced by: cnrecnv 11054

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