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Theorem cnref1o 10030

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 8175), 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 8344	. . . . . . 7 $/\$
3		ax-icn 8264	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8344	. . . . . . . 8 $/\$
7	4, 6	mulcld 8336	. . . . . . 7 $/\$
8	2, 7	addcld 8335	. . . . . 6 $/\$
9	8	rgen2a 2604	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6428	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6399	. . . . . . . . 9
14	13	fveq2d 5694	. . . . . . . 8
15		df-ov 6078	. . . . . . . 8
16	14, 15	eqtr4di 2289	. . . . . . 7
17		xp1st 6389	. . . . . . . 8
18		xp2nd 6390	. . . . . . . 8
19	17	recnd 8344	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8344	. . . . . . . . . 10
22	20, 21	mulcld 8336	. . . . . . . . 9
23	19, 22	addcld 8335	. . . . . . . 8
24		oveq1 6082	. . . . . . . . 9
25		oveq2 6083	. . . . . . . . . 10
26	25	oveq2d 6091	. . . . . . . . 9
27	24, 26, 10	ovmpog 6213	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1278	. . . . . . 7
29	16, 28	eqtrd 2271	. . . . . 6
30	29, 23	eqeltrd 2315	. . . . 5
31	30	rgen 2603	. . . 4
32		ffnfv 5857	. . . 4 $/\$
33	12, 31, 32	mpbir2an 955	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6389	. . . . . . . 8
36		xp2nd 6390	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8920	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5690	. . . . . . . . 9
41		fveq2 5690	. . . . . . . . . 10
42		fveq2 5690	. . . . . . . . . . 11
43	42	oveq2d 6091	. . . . . . . . . 10
44	41, 43	oveq12d 6093	. . . . . . . . 9
45	40, 44	eqeq12d 2253	. . . . . . . 8
46	45, 29	vtoclga 2889	. . . . . . 7
47	29, 46	eqeqan12d 2254	. . . . . 6 $/\$
48		1st2nd2 6399	. . . . . . . 8
49	13, 48	eqeqan12d 2254	. . . . . . 7 $/\$
50		vex 2824	. . . . . . . . 9
51		1stexg 6391	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6392	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4372	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2604	. . 3
60		dff13 5964	. . 3 $/\$
61	33, 59, 60	mpbir2an 955	. 2
62		cnre 8312	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8344	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8344	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8336	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8335	. . . . . . . . 9 $/\$
70		oveq1 6082	. . . . . . . . . 10
71		oveq2 6083	. . . . . . . . . . 11
72	71	oveq2d 6091	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6213	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1278	. . . . . . . 8 $/\$
75	74	eqeq2d 2250	. . . . . . 7 $/\$
76	75	2rexbiia 2566	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5690	. . . . . . . 8
79		df-ov 6078	. . . . . . . 8
80	78, 79	eqtr4di 2289	. . . . . . 7
81	80	eqeq2d 2250	. . . . . 6
82	81	rexxp 4919	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2603	. . 3
85		dffo3 5846	. . 3 $/\$
86	33, 84, 85	mpbir2an 955	. 2
87		df-f1o 5379	. 2 $/\$
88	61, 86, 87	mpbir2an 955	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1402

wcel 2209

wral 2528

wrex 2529

cvv 2821

cop 3708

cxp 4767

wfn 5367

wf 5368

wf1 5369

wfo 5370

wf1o 5371

cfv 5372 (class class class)co 6075

cmpo 6077

c1st 6362

c2nd 6363

cc 8167

cr 8168

ci 8171

caddc 8172

cmul 8174

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-pnf 8352 df-mnf 8353 df-ltxr 8355 df-sub 8489 df-neg 8490 df-reap 8893

This theorem is referenced by: cnrecnv 11654

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