Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7087. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axcnre |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 6987 | . 2 | |
2 | eqeq1 2087 | . . 3 | |
3 | 2 | 2rexbidv 2391 | . 2 |
4 | opelreal 6996 | . . . . . 6 | |
5 | opelreal 6996 | . . . . . 6 | |
6 | 4, 5 | anbi12i 447 | . . . . 5 |
7 | 6 | biimpri 131 | . . . 4 |
8 | df-i 6990 | . . . . . . . . 9 | |
9 | 8 | oveq1i 5542 | . . . . . . . 8 |
10 | 0r 6927 | . . . . . . . . . 10 | |
11 | 1sr 6928 | . . . . . . . . . . 11 | |
12 | mulcnsr 7003 | . . . . . . . . . . 11 | |
13 | 10, 11, 12 | mpanl12 426 | . . . . . . . . . 10 |
14 | 10, 13 | mpan2 415 | . . . . . . . . 9 |
15 | mulcomsrg 6934 | . . . . . . . . . . . . . 14 | |
16 | 10, 15 | mpan 414 | . . . . . . . . . . . . 13 |
17 | 00sr 6946 | . . . . . . . . . . . . 13 | |
18 | 16, 17 | eqtrd 2113 | . . . . . . . . . . . 12 |
19 | 18 | oveq1d 5547 | . . . . . . . . . . 11 |
20 | 00sr 6946 | . . . . . . . . . . . . . . . 16 | |
21 | 11, 20 | ax-mp 7 | . . . . . . . . . . . . . . 15 |
22 | 21 | oveq2i 5543 | . . . . . . . . . . . . . 14 |
23 | m1r 6929 | . . . . . . . . . . . . . . 15 | |
24 | 00sr 6946 | . . . . . . . . . . . . . . 15 | |
25 | 23, 24 | ax-mp 7 | . . . . . . . . . . . . . 14 |
26 | 22, 25 | eqtri 2101 | . . . . . . . . . . . . 13 |
27 | 26 | oveq2i 5543 | . . . . . . . . . . . 12 |
28 | 0idsr 6944 | . . . . . . . . . . . . 13 | |
29 | 10, 28 | ax-mp 7 | . . . . . . . . . . . 12 |
30 | 27, 29 | eqtri 2101 | . . . . . . . . . . 11 |
31 | 19, 30 | syl6eq 2129 | . . . . . . . . . 10 |
32 | mulcomsrg 6934 | . . . . . . . . . . . . . 14 | |
33 | 11, 32 | mpan 414 | . . . . . . . . . . . . 13 |
34 | 1idsr 6945 | . . . . . . . . . . . . 13 | |
35 | 33, 34 | eqtrd 2113 | . . . . . . . . . . . 12 |
36 | 35 | oveq1d 5547 | . . . . . . . . . . 11 |
37 | 00sr 6946 | . . . . . . . . . . . . . 14 | |
38 | 10, 37 | ax-mp 7 | . . . . . . . . . . . . 13 |
39 | 38 | oveq2i 5543 | . . . . . . . . . . . 12 |
40 | 0idsr 6944 | . . . . . . . . . . . 12 | |
41 | 39, 40 | syl5eq 2125 | . . . . . . . . . . 11 |
42 | 36, 41 | eqtrd 2113 | . . . . . . . . . 10 |
43 | 31, 42 | opeq12d 3578 | . . . . . . . . 9 |
44 | 14, 43 | eqtrd 2113 | . . . . . . . 8 |
45 | 9, 44 | syl5eq 2125 | . . . . . . 7 |
46 | 45 | oveq2d 5548 | . . . . . 6 |
47 | 46 | adantl 271 | . . . . 5 |
48 | addcnsr 7002 | . . . . . . 7 | |
49 | 10, 48 | mpanl2 425 | . . . . . 6 |
50 | 10, 49 | mpanr1 427 | . . . . 5 |
51 | 0idsr 6944 | . . . . . 6 | |
52 | addcomsrg 6932 | . . . . . . . 8 | |
53 | 10, 52 | mpan 414 | . . . . . . 7 |
54 | 53, 40 | eqtrd 2113 | . . . . . 6 |
55 | opeq12 3572 | . . . . . 6 | |
56 | 51, 54, 55 | syl2an 283 | . . . . 5 |
57 | 47, 50, 56 | 3eqtrrd 2118 | . . . 4 |
58 | vex 2604 | . . . . . 6 | |
59 | opexg 3983 | . . . . . 6 | |
60 | 58, 10, 59 | mp2an 416 | . . . . 5 |
61 | vex 2604 | . . . . . 6 | |
62 | opexg 3983 | . . . . . 6 | |
63 | 61, 10, 62 | mp2an 416 | . . . . 5 |
64 | eleq1 2141 | . . . . . . 7 | |
65 | eleq1 2141 | . . . . . . 7 | |
66 | 64, 65 | bi2anan9 570 | . . . . . 6 |
67 | oveq1 5539 | . . . . . . . 8 | |
68 | oveq2 5540 | . . . . . . . . 9 | |
69 | 68 | oveq2d 5548 | . . . . . . . 8 |
70 | 67, 69 | sylan9eq 2133 | . . . . . . 7 |
71 | 70 | eqeq2d 2092 | . . . . . 6 |
72 | 66, 71 | anbi12d 456 | . . . . 5 |
73 | 60, 63, 72 | spc2ev 2693 | . . . 4 |
74 | 7, 57, 73 | syl2anc 403 | . . 3 |
75 | r2ex 2386 | . . 3 | |
76 | 74, 75 | sylibr 132 | . 2 |
77 | 1, 3, 76 | optocl 4434 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wex 1421 wcel 1433 wrex 2349 cvv 2601 cop 3401 (class class class)co 5532 cnr 6487 c0r 6488 c1r 6489 cm1r 6490 cplr 6491 cmr 6492 cc 6979 cr 6980 ci 6983 caddc 6984 cmul 6986 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-imp 6659 df-enr 6903 df-nr 6904 df-plr 6905 df-mr 6906 df-0r 6908 df-1r 6909 df-m1r 6910 df-c 6987 df-i 6990 df-r 6991 df-add 6992 df-mul 6993 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |