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Theorem cnref1o 9884

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 8037), 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 8207	. . . . . . 7 $/\$
3		ax-icn 8126	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8207	. . . . . . . 8 $/\$
7	4, 6	mulcld 8199	. . . . . . 7 $/\$
8	2, 7	addcld 8198	. . . . . 6 $/\$
9	8	rgen2a 2586	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6366	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6337	. . . . . . . . 9
14	13	fveq2d 5643	. . . . . . . 8
15		df-ov 6020	. . . . . . . 8
16	14, 15	eqtr4di 2282	. . . . . . 7
17		xp1st 6327	. . . . . . . 8
18		xp2nd 6328	. . . . . . . 8
19	17	recnd 8207	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8207	. . . . . . . . . 10
22	20, 21	mulcld 8199	. . . . . . . . 9
23	19, 22	addcld 8198	. . . . . . . 8
24		oveq1 6024	. . . . . . . . 9
25		oveq2 6025	. . . . . . . . . 10
26	25	oveq2d 6033	. . . . . . . . 9
27	24, 26, 10	ovmpog 6155	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1273	. . . . . . 7
29	16, 28	eqtrd 2264	. . . . . 6
30	29, 23	eqeltrd 2308	. . . . 5
31	30	rgen 2585	. . . 4
32		ffnfv 5805	. . . 4 $/\$
33	12, 31, 32	mpbir2an 950	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6327	. . . . . . . 8
36		xp2nd 6328	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8781	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5639	. . . . . . . . 9
41		fveq2 5639	. . . . . . . . . 10
42		fveq2 5639	. . . . . . . . . . 11
43	42	oveq2d 6033	. . . . . . . . . 10
44	41, 43	oveq12d 6035	. . . . . . . . 9
45	40, 44	eqeq12d 2246	. . . . . . . 8
46	45, 29	vtoclga 2870	. . . . . . 7
47	29, 46	eqeqan12d 2247	. . . . . 6 $/\$
48		1st2nd2 6337	. . . . . . . 8
49	13, 48	eqeqan12d 2247	. . . . . . 7 $/\$
50		vex 2805	. . . . . . . . 9
51		1stexg 6329	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6330	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4329	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2586	. . 3
60		dff13 5908	. . 3 $/\$
61	33, 59, 60	mpbir2an 950	. 2
62		cnre 8174	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8207	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8207	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8199	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8198	. . . . . . . . 9 $/\$
70		oveq1 6024	. . . . . . . . . 10
71		oveq2 6025	. . . . . . . . . . 11
72	71	oveq2d 6033	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6155	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1273	. . . . . . . 8 $/\$
75	74	eqeq2d 2243	. . . . . . 7 $/\$
76	75	2rexbiia 2548	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5639	. . . . . . . 8
79		df-ov 6020	. . . . . . . 8
80	78, 79	eqtr4di 2282	. . . . . . 7
81	80	eqeq2d 2243	. . . . . 6
82	81	rexxp 4874	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2585	. . 3
85		dffo3 5794	. . 3 $/\$
86	33, 84, 85	mpbir2an 950	. 2
87		df-f1o 5333	. 2 $/\$
88	61, 86, 87	mpbir2an 950	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1397

wcel 2202

wral 2510

wrex 2511

cvv 2802

cop 3672

cxp 4723

wfn 5321

wf 5322

wf1 5323

wfo 5324

wf1o 5325

cfv 5326 (class class class)co 6017

cmpo 6019

c1st 6300

c2nd 6301

cc 8029

cr 8030

ci 8033

caddc 8034

cmul 8036

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-pnf 8215 df-mnf 8216 df-ltxr 8218 df-sub 8351 df-neg 8352 df-reap 8754

This theorem is referenced by: cnrecnv 11470

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