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Theorem cnref1o 9588

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 7759), 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 108	. . . . . . . 8 $/\$
2	1	recnd 7927	. . . . . . 7 $/\$
3		ax-icn 7848	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 109	. . . . . . . . 9 $/\$
6	5	recnd 7927	. . . . . . . 8 $/\$
7	4, 6	mulcld 7919	. . . . . . 7 $/\$
8	2, 7	addcld 7918	. . . . . 6 $/\$
9	8	rgen2a 2520	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6170	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6143	. . . . . . . . 9
14	13	fveq2d 5490	. . . . . . . 8
15		df-ov 5845	. . . . . . . 8
16	14, 15	eqtr4di 2217	. . . . . . 7
17		xp1st 6133	. . . . . . . 8
18		xp2nd 6134	. . . . . . . 8
19	17	recnd 7927	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 7927	. . . . . . . . . 10
22	20, 21	mulcld 7919	. . . . . . . . 9
23	19, 22	addcld 7918	. . . . . . . 8
24		oveq1 5849	. . . . . . . . 9
25		oveq2 5850	. . . . . . . . . 10
26	25	oveq2d 5858	. . . . . . . . 9
27	24, 26, 10	ovmpog 5976	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1228	. . . . . . 7
29	16, 28	eqtrd 2198	. . . . . 6
30	29, 23	eqeltrd 2243	. . . . 5
31	30	rgen 2519	. . . 4
32		ffnfv 5643	. . . 4 $/\$
33	12, 31, 32	mpbir2an 932	. . 3
34	17, 18	jca 304	. . . . . . 7 $/\$
35		xp1st 6133	. . . . . . . 8
36		xp2nd 6134	. . . . . . . 8
37	35, 36	jca 304	. . . . . . 7 $/\$
38		cru 8500	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 287	. . . . . 6 $/\$ $/\$
40		fveq2 5486	. . . . . . . . 9
41		fveq2 5486	. . . . . . . . . 10
42		fveq2 5486	. . . . . . . . . . 11
43	42	oveq2d 5858	. . . . . . . . . 10
44	41, 43	oveq12d 5860	. . . . . . . . 9
45	40, 44	eqeq12d 2180	. . . . . . . 8
46	45, 29	vtoclga 2792	. . . . . . 7
47	29, 46	eqeqan12d 2181	. . . . . 6 $/\$
48		1st2nd2 6143	. . . . . . . 8
49	13, 48	eqeqan12d 2181	. . . . . . 7 $/\$
50		vex 2729	. . . . . . . . 9
51		1stexg 6135	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6136	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4215	. . . . . . 7 $/\$
56	49, 55	bitrdi 195	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 219	. . . . 5 $/\$
58	57	biimpd 143	. . . 4 $/\$
59	58	rgen2a 2520	. . 3
60		dff13 5736	. . 3 $/\$
61	33, 59, 60	mpbir2an 932	. 2
62		cnre 7895	. . . . . 6
63		simpl 108	. . . . . . . . 9 $/\$
64		simpr 109	. . . . . . . . 9 $/\$
65	63	recnd 7927	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 7927	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 7919	. . . . . . . . . 10 $/\$
69	65, 68	addcld 7918	. . . . . . . . 9 $/\$
70		oveq1 5849	. . . . . . . . . 10
71		oveq2 5850	. . . . . . . . . . 11
72	71	oveq2d 5858	. . . . . . . . . 10
73	70, 72, 10	ovmpog 5976	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1228	. . . . . . . 8 $/\$
75	74	eqeq2d 2177	. . . . . . 7 $/\$
76	75	2rexbiia 2482	. . . . . 6
77	62, 76	sylibr 133	. . . . 5
78		fveq2 5486	. . . . . . . 8
79		df-ov 5845	. . . . . . . 8
80	78, 79	eqtr4di 2217	. . . . . . 7
81	80	eqeq2d 2177	. . . . . 6
82	81	rexxp 4748	. . . . 5
83	77, 82	sylibr 133	. . . 4
84	83	rgen 2519	. . 3
85		dffo3 5632	. . 3 $/\$
86	33, 84, 85	mpbir2an 932	. 2
87		df-f1o 5195	. 2 $/\$
88	61, 86, 87	mpbir2an 932	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104

wceq 1343

wcel 2136

wral 2444

wrex 2445

cvv 2726

cop 3579

cxp 4602

wfn 5183

wf 5184

wf1 5185

wfo 5186

wf1o 5187

cfv 5188 (class class class)co 5842

cmpo 5844

c1st 6106

c2nd 6107

cc 7751

cr 7752

ci 7755

caddc 7756

cmul 7758

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-pnf 7935 df-mnf 7936 df-ltxr 7938 df-sub 8071 df-neg 8072 df-reap 8473

This theorem is referenced by: cnrecnv 10852

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