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Theorem cnref1o 9639

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 7808), 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 7976	. . . . . . 7 $/\$
3		ax-icn 7897	. . . . . . . . 9
4	3	a1i 9	. . . . . . . 8 $/\$
5		simpr 110	. . . . . . . . 9 $/\$
6	5	recnd 7976	. . . . . . . 8 $/\$
7	4, 6	mulcld 7968	. . . . . . 7 $/\$
8	2, 7	addcld 7967	. . . . . 6 $/\$
9	8	rgen2a 2531	. . . . 5
10		cnref1o.1	. . . . . 6
11	10	fnmpo 6197	. . . . 5
12	9, 11	ax-mp 5	. . . 4
13		1st2nd2 6170	. . . . . . . . 9
14	13	fveq2d 5515	. . . . . . . 8
15		df-ov 5872	. . . . . . . 8
16	14, 15	eqtr4di 2228	. . . . . . 7
17		xp1st 6160	. . . . . . . 8
18		xp2nd 6161	. . . . . . . 8
19	17	recnd 7976	. . . . . . . . 9
20	3	a1i 9	. . . . . . . . . 10
21	18	recnd 7976	. . . . . . . . . 10
22	20, 21	mulcld 7968	. . . . . . . . 9
23	19, 22	addcld 7967	. . . . . . . 8
24		oveq1 5876	. . . . . . . . 9
25		oveq2 5877	. . . . . . . . . 10
26	25	oveq2d 5885	. . . . . . . . 9
27	24, 26, 10	ovmpog 6003	. . . . . . . 8 $/\$ $/\$
28	17, 18, 23, 27	syl3anc 1238	. . . . . . 7
29	16, 28	eqtrd 2210	. . . . . 6
30	29, 23	eqeltrd 2254	. . . . 5
31	30	rgen 2530	. . . 4
32		ffnfv 5670	. . . 4 $/\$
33	12, 31, 32	mpbir2an 942	. . 3
34	17, 18	jca 306	. . . . . . 7 $/\$
35		xp1st 6160	. . . . . . . 8
36		xp2nd 6161	. . . . . . . 8
37	35, 36	jca 306	. . . . . . 7 $/\$
38		cru 8549	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
39	34, 37, 38	syl2an 289	. . . . . 6 $/\$ $/\$
40		fveq2 5511	. . . . . . . . 9
41		fveq2 5511	. . . . . . . . . 10
42		fveq2 5511	. . . . . . . . . . 11
43	42	oveq2d 5885	. . . . . . . . . 10
44	41, 43	oveq12d 5887	. . . . . . . . 9
45	40, 44	eqeq12d 2192	. . . . . . . 8
46	45, 29	vtoclga 2803	. . . . . . 7
47	29, 46	eqeqan12d 2193	. . . . . 6 $/\$
48		1st2nd2 6170	. . . . . . . 8
49	13, 48	eqeqan12d 2193	. . . . . . 7 $/\$
50		vex 2740	. . . . . . . . 9
51		1stexg 6162	. . . . . . . . 9
52	50, 51	ax-mp 5	. . . . . . . 8
53		2ndexg 6163	. . . . . . . . 9
54	50, 53	ax-mp 5	. . . . . . . 8
55	52, 54	opth 4234	. . . . . . 7 $/\$
56	49, 55	bitrdi 196	. . . . . 6 $/\$ $/\$
57	39, 47, 56	3bitr4d 220	. . . . 5 $/\$
58	57	biimpd 144	. . . 4 $/\$
59	58	rgen2a 2531	. . 3
60		dff13 5763	. . 3 $/\$
61	33, 59, 60	mpbir2an 942	. 2
62		cnre 7944	. . . . . 6
63		simpl 109	. . . . . . . . 9 $/\$
64		simpr 110	. . . . . . . . 9 $/\$
65	63	recnd 7976	. . . . . . . . . 10 $/\$
66	3	a1i 9	. . . . . . . . . . 11 $/\$
67	64	recnd 7976	. . . . . . . . . . 11 $/\$
68	66, 67	mulcld 7968	. . . . . . . . . 10 $/\$
69	65, 68	addcld 7967	. . . . . . . . 9 $/\$
70		oveq1 5876	. . . . . . . . . 10
71		oveq2 5877	. . . . . . . . . . 11
72	71	oveq2d 5885	. . . . . . . . . 10
73	70, 72, 10	ovmpog 6003	. . . . . . . . 9 $/\$ $/\$
74	63, 64, 69, 73	syl3anc 1238	. . . . . . . 8 $/\$
75	74	eqeq2d 2189	. . . . . . 7 $/\$
76	75	2rexbiia 2493	. . . . . 6
77	62, 76	sylibr 134	. . . . 5
78		fveq2 5511	. . . . . . . 8
79		df-ov 5872	. . . . . . . 8
80	78, 79	eqtr4di 2228	. . . . . . 7
81	80	eqeq2d 2189	. . . . . 6
82	81	rexxp 4767	. . . . 5
83	77, 82	sylibr 134	. . . 4
84	83	rgen 2530	. . 3
85		dffo3 5659	. . 3 $/\$
86	33, 84, 85	mpbir2an 942	. 2
87		df-f1o 5219	. 2 $/\$
88	61, 86, 87	mpbir2an 942	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148

wral 2455

wrex 2456

cvv 2737

cop 3594

cxp 4621

wfn 5207

wf 5208

wf1 5209

wfo 5210

wf1o 5211

cfv 5212 (class class class)co 5869

cmpo 5871

c1st 6133

c2nd 6134

cc 7800

cr 7801

ci 7804

caddc 7805

cmul 7807

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-pnf 7984 df-mnf 7985 df-ltxr 7987 df-sub 8120 df-neg 8121 df-reap 8522

This theorem is referenced by: cnrecnv 10903

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