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Mirrors > Home > ILE Home > Th. List > cnref1o | Unicode version |
Description: There is a natural one-to-one mapping from to , where we map to . In our construction of the complex numbers, this is in fact our definition of (see df-c 7750), 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.) |
Ref | Expression |
---|---|
cnref1o.1 |
Ref | Expression |
---|---|
cnref1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . . . . 8 | |
2 | 1 | recnd 7918 | . . . . . . 7 |
3 | ax-icn 7839 | . . . . . . . . 9 | |
4 | 3 | a1i 9 | . . . . . . . 8 |
5 | simpr 109 | . . . . . . . . 9 | |
6 | 5 | recnd 7918 | . . . . . . . 8 |
7 | 4, 6 | mulcld 7910 | . . . . . . 7 |
8 | 2, 7 | addcld 7909 | . . . . . 6 |
9 | 8 | rgen2a 2518 | . . . . 5 |
10 | cnref1o.1 | . . . . . 6 | |
11 | 10 | fnmpo 6162 | . . . . 5 |
12 | 9, 11 | ax-mp 5 | . . . 4 |
13 | 1st2nd2 6135 | . . . . . . . . 9 | |
14 | 13 | fveq2d 5484 | . . . . . . . 8 |
15 | df-ov 5839 | . . . . . . . 8 | |
16 | 14, 15 | eqtr4di 2215 | . . . . . . 7 |
17 | xp1st 6125 | . . . . . . . 8 | |
18 | xp2nd 6126 | . . . . . . . 8 | |
19 | 17 | recnd 7918 | . . . . . . . . 9 |
20 | 3 | a1i 9 | . . . . . . . . . 10 |
21 | 18 | recnd 7918 | . . . . . . . . . 10 |
22 | 20, 21 | mulcld 7910 | . . . . . . . . 9 |
23 | 19, 22 | addcld 7909 | . . . . . . . 8 |
24 | oveq1 5843 | . . . . . . . . 9 | |
25 | oveq2 5844 | . . . . . . . . . 10 | |
26 | 25 | oveq2d 5852 | . . . . . . . . 9 |
27 | 24, 26, 10 | ovmpog 5967 | . . . . . . . 8 |
28 | 17, 18, 23, 27 | syl3anc 1227 | . . . . . . 7 |
29 | 16, 28 | eqtrd 2197 | . . . . . 6 |
30 | 29, 23 | eqeltrd 2241 | . . . . 5 |
31 | 30 | rgen 2517 | . . . 4 |
32 | ffnfv 5637 | . . . 4 | |
33 | 12, 31, 32 | mpbir2an 931 | . . 3 |
34 | 17, 18 | jca 304 | . . . . . . 7 |
35 | xp1st 6125 | . . . . . . . 8 | |
36 | xp2nd 6126 | . . . . . . . 8 | |
37 | 35, 36 | jca 304 | . . . . . . 7 |
38 | cru 8491 | . . . . . . 7 | |
39 | 34, 37, 38 | syl2an 287 | . . . . . 6 |
40 | fveq2 5480 | . . . . . . . . 9 | |
41 | fveq2 5480 | . . . . . . . . . 10 | |
42 | fveq2 5480 | . . . . . . . . . . 11 | |
43 | 42 | oveq2d 5852 | . . . . . . . . . 10 |
44 | 41, 43 | oveq12d 5854 | . . . . . . . . 9 |
45 | 40, 44 | eqeq12d 2179 | . . . . . . . 8 |
46 | 45, 29 | vtoclga 2787 | . . . . . . 7 |
47 | 29, 46 | eqeqan12d 2180 | . . . . . 6 |
48 | 1st2nd2 6135 | . . . . . . . 8 | |
49 | 13, 48 | eqeqan12d 2180 | . . . . . . 7 |
50 | vex 2724 | . . . . . . . . 9 | |
51 | 1stexg 6127 | . . . . . . . . 9 | |
52 | 50, 51 | ax-mp 5 | . . . . . . . 8 |
53 | 2ndexg 6128 | . . . . . . . . 9 | |
54 | 50, 53 | ax-mp 5 | . . . . . . . 8 |
55 | 52, 54 | opth 4209 | . . . . . . 7 |
56 | 49, 55 | bitrdi 195 | . . . . . 6 |
57 | 39, 47, 56 | 3bitr4d 219 | . . . . 5 |
58 | 57 | biimpd 143 | . . . 4 |
59 | 58 | rgen2a 2518 | . . 3 |
60 | dff13 5730 | . . 3 | |
61 | 33, 59, 60 | mpbir2an 931 | . 2 |
62 | cnre 7886 | . . . . . 6 | |
63 | simpl 108 | . . . . . . . . 9 | |
64 | simpr 109 | . . . . . . . . 9 | |
65 | 63 | recnd 7918 | . . . . . . . . . 10 |
66 | 3 | a1i 9 | . . . . . . . . . . 11 |
67 | 64 | recnd 7918 | . . . . . . . . . . 11 |
68 | 66, 67 | mulcld 7910 | . . . . . . . . . 10 |
69 | 65, 68 | addcld 7909 | . . . . . . . . 9 |
70 | oveq1 5843 | . . . . . . . . . 10 | |
71 | oveq2 5844 | . . . . . . . . . . 11 | |
72 | 71 | oveq2d 5852 | . . . . . . . . . 10 |
73 | 70, 72, 10 | ovmpog 5967 | . . . . . . . . 9 |
74 | 63, 64, 69, 73 | syl3anc 1227 | . . . . . . . 8 |
75 | 74 | eqeq2d 2176 | . . . . . . 7 |
76 | 75 | 2rexbiia 2480 | . . . . . 6 |
77 | 62, 76 | sylibr 133 | . . . . 5 |
78 | fveq2 5480 | . . . . . . . 8 | |
79 | df-ov 5839 | . . . . . . . 8 | |
80 | 78, 79 | eqtr4di 2215 | . . . . . . 7 |
81 | 80 | eqeq2d 2176 | . . . . . 6 |
82 | 81 | rexxp 4742 | . . . . 5 |
83 | 77, 82 | sylibr 133 | . . . 4 |
84 | 83 | rgen 2517 | . . 3 |
85 | dffo3 5626 | . . 3 | |
86 | 33, 84, 85 | mpbir2an 931 | . 2 |
87 | df-f1o 5189 | . 2 | |
88 | 61, 86, 87 | mpbir2an 931 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 wrex 2443 cvv 2721 cop 3573 cxp 4596 wfn 5177 wf 5178 wf1 5179 wfo 5180 wf1o 5181 cfv 5182 (class class class)co 5836 cmpo 5838 c1st 6098 c2nd 6099 cc 7742 cr 7743 ci 7746 caddc 7747 cmul 7749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-sub 8062 df-neg 8063 df-reap 8464 |
This theorem is referenced by: cnrecnv 10838 |
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