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Mirrors > Home > ILE Home > Th. List > axcnre | Unicode version |
Description: A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 6995. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axcnre |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 6895 | . 2 | |
2 | eqeq1 2046 | . . 3 | |
3 | 2 | 2rexbidv 2349 | . 2 |
4 | opelreal 6904 | . . . . . 6 | |
5 | opelreal 6904 | . . . . . 6 | |
6 | 4, 5 | anbi12i 433 | . . . . 5 |
7 | 6 | biimpri 124 | . . . 4 |
8 | df-i 6898 | . . . . . . . . 9 | |
9 | 8 | oveq1i 5522 | . . . . . . . 8 |
10 | 0r 6835 | . . . . . . . . . 10 | |
11 | 1sr 6836 | . . . . . . . . . . 11 | |
12 | mulcnsr 6911 | . . . . . . . . . . 11 | |
13 | 10, 11, 12 | mpanl12 412 | . . . . . . . . . 10 |
14 | 10, 13 | mpan2 401 | . . . . . . . . 9 |
15 | mulcomsrg 6842 | . . . . . . . . . . . . . 14 | |
16 | 10, 15 | mpan 400 | . . . . . . . . . . . . 13 |
17 | 00sr 6854 | . . . . . . . . . . . . 13 | |
18 | 16, 17 | eqtrd 2072 | . . . . . . . . . . . 12 |
19 | 18 | oveq1d 5527 | . . . . . . . . . . 11 |
20 | 00sr 6854 | . . . . . . . . . . . . . . . 16 | |
21 | 11, 20 | ax-mp 7 | . . . . . . . . . . . . . . 15 |
22 | 21 | oveq2i 5523 | . . . . . . . . . . . . . 14 |
23 | m1r 6837 | . . . . . . . . . . . . . . 15 | |
24 | 00sr 6854 | . . . . . . . . . . . . . . 15 | |
25 | 23, 24 | ax-mp 7 | . . . . . . . . . . . . . 14 |
26 | 22, 25 | eqtri 2060 | . . . . . . . . . . . . 13 |
27 | 26 | oveq2i 5523 | . . . . . . . . . . . 12 |
28 | 0idsr 6852 | . . . . . . . . . . . . 13 | |
29 | 10, 28 | ax-mp 7 | . . . . . . . . . . . 12 |
30 | 27, 29 | eqtri 2060 | . . . . . . . . . . 11 |
31 | 19, 30 | syl6eq 2088 | . . . . . . . . . 10 |
32 | mulcomsrg 6842 | . . . . . . . . . . . . . 14 | |
33 | 11, 32 | mpan 400 | . . . . . . . . . . . . 13 |
34 | 1idsr 6853 | . . . . . . . . . . . . 13 | |
35 | 33, 34 | eqtrd 2072 | . . . . . . . . . . . 12 |
36 | 35 | oveq1d 5527 | . . . . . . . . . . 11 |
37 | 00sr 6854 | . . . . . . . . . . . . . 14 | |
38 | 10, 37 | ax-mp 7 | . . . . . . . . . . . . 13 |
39 | 38 | oveq2i 5523 | . . . . . . . . . . . 12 |
40 | 0idsr 6852 | . . . . . . . . . . . 12 | |
41 | 39, 40 | syl5eq 2084 | . . . . . . . . . . 11 |
42 | 36, 41 | eqtrd 2072 | . . . . . . . . . 10 |
43 | 31, 42 | opeq12d 3557 | . . . . . . . . 9 |
44 | 14, 43 | eqtrd 2072 | . . . . . . . 8 |
45 | 9, 44 | syl5eq 2084 | . . . . . . 7 |
46 | 45 | oveq2d 5528 | . . . . . 6 |
47 | 46 | adantl 262 | . . . . 5 |
48 | addcnsr 6910 | . . . . . . 7 | |
49 | 10, 48 | mpanl2 411 | . . . . . 6 |
50 | 10, 49 | mpanr1 413 | . . . . 5 |
51 | 0idsr 6852 | . . . . . 6 | |
52 | addcomsrg 6840 | . . . . . . . 8 | |
53 | 10, 52 | mpan 400 | . . . . . . 7 |
54 | 53, 40 | eqtrd 2072 | . . . . . 6 |
55 | opeq12 3551 | . . . . . 6 | |
56 | 51, 54, 55 | syl2an 273 | . . . . 5 |
57 | 47, 50, 56 | 3eqtrrd 2077 | . . . 4 |
58 | vex 2560 | . . . . . 6 | |
59 | opexg 3964 | . . . . . 6 | |
60 | 58, 10, 59 | mp2an 402 | . . . . 5 |
61 | vex 2560 | . . . . . 6 | |
62 | opexg 3964 | . . . . . 6 | |
63 | 61, 10, 62 | mp2an 402 | . . . . 5 |
64 | eleq1 2100 | . . . . . . 7 | |
65 | eleq1 2100 | . . . . . . 7 | |
66 | 64, 65 | bi2anan9 538 | . . . . . 6 |
67 | oveq1 5519 | . . . . . . . 8 | |
68 | oveq2 5520 | . . . . . . . . 9 | |
69 | 68 | oveq2d 5528 | . . . . . . . 8 |
70 | 67, 69 | sylan9eq 2092 | . . . . . . 7 |
71 | 70 | eqeq2d 2051 | . . . . . 6 |
72 | 66, 71 | anbi12d 442 | . . . . 5 |
73 | 60, 63, 72 | spc2ev 2648 | . . . 4 |
74 | 7, 57, 73 | syl2anc 391 | . . 3 |
75 | r2ex 2344 | . . 3 | |
76 | 74, 75 | sylibr 137 | . 2 |
77 | 1, 3, 76 | optocl 4416 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 wrex 2307 cvv 2557 cop 3378 (class class class)co 5512 cnr 6395 c0r 6396 c1r 6397 cm1r 6398 cplr 6399 cmr 6400 cc 6887 cr 6888 ci 6891 caddc 6892 cmul 6894 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-imp 6567 df-enr 6811 df-nr 6812 df-plr 6813 df-mr 6814 df-0r 6816 df-1r 6817 df-m1r 6818 df-c 6895 df-i 6898 df-r 6899 df-add 6900 df-mul 6901 |
This theorem is referenced by: (None) |
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