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Theorem cnref1o 9725

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 7885), 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 8055	. . . . . . 7 $/\$
3		ax-icn 7974	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 8055	. . . . . . . 8 $/\$
7	4, 6	mulcld 8047	. . . . . . 7 $/\$
8	2, 7	addcld 8046	. . . . . 6 $/\$
9	8	rgen2a 2551	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6260	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6233	. . . . . . . . 9
14	13	fveq2d 5562	. . . . . . . 8
15		df-ov 5925	. . . . . . . 8
16	14, 15	eqtr4di 2247	. . . . . . 7
17		xp1st 6223	. . . . . . . 8
18		xp2nd 6224	. . . . . . . 8
19	17	recnd 8055	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 8055	. . . . . . . . . 10
22	20, 21	mulcld 8047	. . . . . . . . 9
23	19, 22	addcld 8046	. . . . . . . 8
24		oveq1 5929	. . . . . . . . 9
25		oveq2 5930	. . . . . . . . . 10
26	25	oveq2d 5938	. . . . . . . . 9
27	24, 26, 10	ovmpog 6057	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1249	. . . . . . 7
29	16, 28	eqtrd 2229	. . . . . 6
30	29, 23	eqeltrd 2273	. . . . 5
31	30	rgen 2550	. . . 4
32		ffnfv 5720	. . . 4 $/\$
33	12, 31, 32	mpbir2an 944	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6223	. . . . . . . 8
36		xp2nd 6224	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8629	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5558	. . . . . . . . 9
41		fveq2 5558	. . . . . . . . . 10
42		fveq2 5558	. . . . . . . . . . 11
43	42	oveq2d 5938	. . . . . . . . . 10
44	41, 43	oveq12d 5940	. . . . . . . . 9
45	40, 44	eqeq12d 2211	. . . . . . . 8
46	45, 29	vtoclga 2830	. . . . . . 7
47	29, 46	eqeqan12d 2212	. . . . . 6 $/\$
48		1st2nd2 6233	. . . . . . . 8
49	13, 48	eqeqan12d 2212	. . . . . . 7 $/\$
50		vex 2766	. . . . . . . . 9
51		1stexg 6225	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6226	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4270	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2551	. . 3
60		dff13 5815	. . 3 $/\$
61	33, 59, 60	mpbir2an 944	. 2
62		cnre 8022	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 8055	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 8055	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 8047	. . . . . . . . . 10 $/\$
69	65, 68	addcld 8046	. . . . . . . . 9 $/\$
70		oveq1 5929	. . . . . . . . . 10
71		oveq2 5930	. . . . . . . . . . 11
72	71	oveq2d 5938	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6057	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1249	. . . . . . . 8 $/\$
75	74	eqeq2d 2208	. . . . . . 7 $/\$
76	75	2rexbiia 2513	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5558	. . . . . . . 8
79		df-ov 5925	. . . . . . . 8
80	78, 79	eqtr4di 2247	. . . . . . 7
81	80	eqeq2d 2208	. . . . . 6
82	81	rexxp 4810	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2550	. . 3
85		dffo3 5709	. . 3 $/\$
86	33, 84, 85	mpbir2an 944	. 2
87		df-f1o 5265	. 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 2167

wral 2475

wrex 2476

cvv 2763

cop 3625

cxp 4661

wfn 5253

wf 5254

wf1 5255

wfo 5256

wf1o 5257

cfv 5258 (class class class)co 5922

cmpo 5924

c1st 6196

c2nd 6197

cc 7877

cr 7878

ci 7881

caddc 7882

cmul 7884

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-pnf 8063 df-mnf 8064 df-ltxr 8066 df-sub 8199 df-neg 8200 df-reap 8602

This theorem is referenced by: cnrecnv 11075

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