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Theorem cnref1o 10001

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 8149), 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 8318	. . . . . . 7 $/\$
3		ax-icn 8238	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8318	. . . . . . . 8 $/\$
7	4, 6	mulcld 8310	. . . . . . 7 $/\$
8	2, 7	addcld 8309	. . . . . 6 $/\$
9	8	rgen2a 2598	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6411	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6382	. . . . . . . . 9
14	13	fveq2d 5679	. . . . . . . 8
15		df-ov 6061	. . . . . . . 8
16	14, 15	eqtr4di 2285	. . . . . . 7
17		xp1st 6372	. . . . . . . 8
18		xp2nd 6373	. . . . . . . 8
19	17	recnd 8318	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8318	. . . . . . . . . 10
22	20, 21	mulcld 8310	. . . . . . . . 9
23	19, 22	addcld 8309	. . . . . . . 8
24		oveq1 6065	. . . . . . . . 9
25		oveq2 6066	. . . . . . . . . 10
26	25	oveq2d 6074	. . . . . . . . 9
27	24, 26, 10	ovmpog 6196	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1274	. . . . . . 7
29	16, 28	eqtrd 2267	. . . . . 6
30	29, 23	eqeltrd 2311	. . . . 5
31	30	rgen 2597	. . . 4
32		ffnfv 5840	. . . 4 $/\$
33	12, 31, 32	mpbir2an 951	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6372	. . . . . . . 8
36		xp2nd 6373	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8893	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5675	. . . . . . . . 9
41		fveq2 5675	. . . . . . . . . 10
42		fveq2 5675	. . . . . . . . . . 11
43	42	oveq2d 6074	. . . . . . . . . 10
44	41, 43	oveq12d 6076	. . . . . . . . 9
45	40, 44	eqeq12d 2249	. . . . . . . 8
46	45, 29	vtoclga 2883	. . . . . . 7
47	29, 46	eqeqan12d 2250	. . . . . 6 $/\$
48		1st2nd2 6382	. . . . . . . 8
49	13, 48	eqeqan12d 2250	. . . . . . 7 $/\$
50		vex 2818	. . . . . . . . 9
51		1stexg 6374	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6375	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4358	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2598	. . 3
60		dff13 5947	. . 3 $/\$
61	33, 59, 60	mpbir2an 951	. 2
62		cnre 8286	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8318	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8318	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8310	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8309	. . . . . . . . 9 $/\$
70		oveq1 6065	. . . . . . . . . 10
71		oveq2 6066	. . . . . . . . . . 11
72	71	oveq2d 6074	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6196	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1274	. . . . . . . 8 $/\$
75	74	eqeq2d 2246	. . . . . . 7 $/\$
76	75	2rexbiia 2560	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5675	. . . . . . . 8
79		df-ov 6061	. . . . . . . 8
80	78, 79	eqtr4di 2285	. . . . . . 7
81	80	eqeq2d 2246	. . . . . 6
82	81	rexxp 4904	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2597	. . 3
85		dffo3 5829	. . 3 $/\$
86	33, 84, 85	mpbir2an 951	. 2
87		df-f1o 5364	. 2 $/\$
88	61, 86, 87	mpbir2an 951	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1398

wcel 2205

wral 2522

wrex 2523

cvv 2815

cop 3697

cxp 4752

wfn 5352

wf 5353

wf1 5354

wfo 5355

wf1o 5356

cfv 5357 (class class class)co 6058

cmpo 6060

c1st 6345

c2nd 6346

cc 8141

cr 8142

ci 8145

caddc 8146

cmul 8148

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-pnf 8326 df-mnf 8327 df-ltxr 8329 df-sub 8462 df-neg 8463 df-reap 8866

This theorem is referenced by: cnrecnv 11620

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