Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > qdencn | Unicode version |
Description: The set of complex numbers whose real and imaginary parts are rational is dense in the complex plane. This is a two dimensional analogue to qdenre 10929 (and also would hold for with the usual metric; this is not about complex numbers in particular). (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
qdencn.q |
Ref | Expression |
---|---|
qdencn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . 4 | |
2 | 1 | recld 10665 | . . 3 |
3 | simpr 109 | . . . 4 | |
4 | 3 | rphalfcld 9451 | . . 3 |
5 | qdenre 10929 | . . 3 | |
6 | 2, 4, 5 | syl2anc 408 | . 2 |
7 | simpll 503 | . . . . 5 | |
8 | 7 | imcld 10666 | . . . 4 |
9 | 4 | adantr 274 | . . . 4 |
10 | qdenre 10929 | . . . 4 | |
11 | 8, 9, 10 | syl2anc 408 | . . 3 |
12 | qcn 9382 | . . . . . . . 8 | |
13 | 12 | ad2antrl 481 | . . . . . . 7 |
14 | 13 | adantr 274 | . . . . . 6 |
15 | ax-icn 7683 | . . . . . . . 8 | |
16 | 15 | a1i 9 | . . . . . . 7 |
17 | qcn 9382 | . . . . . . . 8 | |
18 | 17 | ad2antrl 481 | . . . . . . 7 |
19 | 16, 18 | mulcld 7754 | . . . . . 6 |
20 | 14, 19 | addcld 7753 | . . . . 5 |
21 | qre 9373 | . . . . . . . . . 10 | |
22 | 21 | ad2antrl 481 | . . . . . . . . 9 |
23 | 22 | adantr 274 | . . . . . . . 8 |
24 | qre 9373 | . . . . . . . . 9 | |
25 | 24 | ad2antrl 481 | . . . . . . . 8 |
26 | 23, 25 | crred 10703 | . . . . . . 7 |
27 | simplrl 509 | . . . . . . 7 | |
28 | 26, 27 | eqeltrd 2194 | . . . . . 6 |
29 | 23, 25 | crimd 10704 | . . . . . . 7 |
30 | simprl 505 | . . . . . . 7 | |
31 | 29, 30 | eqeltrd 2194 | . . . . . 6 |
32 | 28, 31 | jca 304 | . . . . 5 |
33 | fveq2 5389 | . . . . . . . 8 | |
34 | 33 | eleq1d 2186 | . . . . . . 7 |
35 | fveq2 5389 | . . . . . . . 8 | |
36 | 35 | eleq1d 2186 | . . . . . . 7 |
37 | 34, 36 | anbi12d 464 | . . . . . 6 |
38 | qdencn.q | . . . . . 6 | |
39 | 37, 38 | elrab2 2816 | . . . . 5 |
40 | 20, 32, 39 | sylanbrc 413 | . . . 4 |
41 | 7 | adantr 274 | . . . . . . 7 |
42 | 20, 41 | subcld 8041 | . . . . . 6 |
43 | 42 | abscld 10908 | . . . . 5 |
44 | 2 | ad2antrr 479 | . . . . . . . . 9 |
45 | 44 | recnd 7762 | . . . . . . . 8 |
46 | 14, 45 | subcld 8041 | . . . . . . 7 |
47 | 46 | abscld 10908 | . . . . . 6 |
48 | 8 | adantr 274 | . . . . . . . . 9 |
49 | 48 | recnd 7762 | . . . . . . . 8 |
50 | 18, 49 | subcld 8041 | . . . . . . 7 |
51 | 50 | abscld 10908 | . . . . . 6 |
52 | 47, 51 | readdcld 7763 | . . . . 5 |
53 | 3 | ad2antrr 479 | . . . . . 6 |
54 | 53 | rpred 9438 | . . . . 5 |
55 | 1 | replimd 10668 | . . . . . . . . . . 11 |
56 | 55 | oveq2d 5758 | . . . . . . . . . 10 |
57 | 56 | ad2antrr 479 | . . . . . . . . 9 |
58 | 16, 49 | mulcld 7754 | . . . . . . . . . 10 |
59 | 14, 19, 45, 58 | addsub4d 8088 | . . . . . . . . 9 |
60 | 57, 59 | eqtrd 2150 | . . . . . . . 8 |
61 | 60 | fveq2d 5393 | . . . . . . 7 |
62 | 19, 58 | subcld 8041 | . . . . . . . 8 |
63 | 46, 62 | abstrid 10923 | . . . . . . 7 |
64 | 61, 63 | eqbrtrd 3920 | . . . . . 6 |
65 | 16, 50 | absmuld 10921 | . . . . . . . 8 |
66 | 16, 18, 49 | subdid 8144 | . . . . . . . . 9 |
67 | 66 | fveq2d 5393 | . . . . . . . 8 |
68 | absi 10786 | . . . . . . . . . 10 | |
69 | 68 | oveq1i 5752 | . . . . . . . . 9 |
70 | 51 | recnd 7762 | . . . . . . . . . 10 |
71 | 70 | mulid2d 7752 | . . . . . . . . 9 |
72 | 69, 71 | syl5eq 2162 | . . . . . . . 8 |
73 | 65, 67, 72 | 3eqtr3d 2158 | . . . . . . 7 |
74 | 73 | oveq2d 5758 | . . . . . 6 |
75 | 64, 74 | breqtrd 3924 | . . . . 5 |
76 | simplrr 510 | . . . . . 6 | |
77 | simprr 506 | . . . . . 6 | |
78 | 47, 51, 54, 76, 77 | lt2halvesd 8925 | . . . . 5 |
79 | 43, 52, 54, 75, 78 | lelttrd 7855 | . . . 4 |
80 | oveq1 5749 | . . . . . . 7 | |
81 | 80 | fveq2d 5393 | . . . . . 6 |
82 | 81 | breq1d 3909 | . . . . 5 |
83 | 82 | rspcev 2763 | . . . 4 |
84 | 40, 79, 83 | syl2anc 408 | . . 3 |
85 | 11, 84 | rexlimddv 2531 | . 2 |
86 | 6, 85 | rexlimddv 2531 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wrex 2394 crab 2397 class class class wbr 3899 cfv 5093 (class class class)co 5742 cc 7586 cr 7587 c1 7589 ci 7590 caddc 7591 cmul 7593 clt 7768 cle 7769 cmin 7901 cdiv 8399 c2 8735 cq 9367 crp 9397 cre 10567 cim 10568 cabs 10724 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 df-div 8400 df-inn 8685 df-2 8743 df-3 8744 df-4 8745 df-n0 8936 df-z 9013 df-uz 9283 df-q 9368 df-rp 9398 df-seqfrec 10174 df-exp 10248 df-cj 10569 df-re 10570 df-im 10571 df-rsqrt 10725 df-abs 10726 |
This theorem is referenced by: (None) |
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