Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > trirec0 | Unicode version |
Description: Every real number having
a reciprocal or equaling zero is equivalent to
real number trichotomy.
This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 13756). (Contributed by Jim Kingdon, 10-Jun-2024.) |
Ref | Expression |
---|---|
trirec0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 519 | . . . . . 6 | |
2 | simpr 109 | . . . . . . 7 | |
3 | 1, 2 | lt0ap0d 8538 | . . . . . 6 # |
4 | rerecclap 8617 | . . . . . . 7 # | |
5 | recn 7877 | . . . . . . . 8 | |
6 | recidap 8573 | . . . . . . . 8 # | |
7 | 5, 6 | sylan 281 | . . . . . . 7 # |
8 | oveq2 5844 | . . . . . . . . 9 | |
9 | 8 | eqeq1d 2173 | . . . . . . . 8 |
10 | 9 | rspcev 2825 | . . . . . . 7 |
11 | 4, 7, 10 | syl2anc 409 | . . . . . 6 # |
12 | 1, 3, 11 | syl2anc 409 | . . . . 5 |
13 | 12 | orcd 723 | . . . 4 |
14 | simpr 109 | . . . . 5 | |
15 | 14 | olcd 724 | . . . 4 |
16 | simpll 519 | . . . . . 6 | |
17 | simpr 109 | . . . . . . 7 | |
18 | 16, 17 | gt0ap0d 8518 | . . . . . 6 # |
19 | 16, 18, 11 | syl2anc 409 | . . . . 5 |
20 | 19 | orcd 723 | . . . 4 |
21 | 0re 7890 | . . . . . 6 | |
22 | breq2 3980 | . . . . . . . 8 | |
23 | eqeq2 2174 | . . . . . . . 8 | |
24 | breq1 3979 | . . . . . . . 8 | |
25 | 22, 23, 24 | 3orbi123d 1300 | . . . . . . 7 |
26 | 25 | rspcv 2821 | . . . . . 6 |
27 | 21, 26 | ax-mp 5 | . . . . 5 |
28 | 27 | adantl 275 | . . . 4 |
29 | 13, 15, 20, 28 | mpjao3dan 1296 | . . 3 |
30 | 29 | ralimiaa 2526 | . 2 |
31 | oveq1 5843 | . . . . . . 7 | |
32 | 31 | eqeq1d 2173 | . . . . . 6 |
33 | 32 | rexbidv 2465 | . . . . 5 |
34 | eqeq1 2171 | . . . . 5 | |
35 | 33, 34 | orbi12d 783 | . . . 4 |
36 | 35 | cbvralv 2689 | . . 3 |
37 | nfcv 2306 | . . . . . . . . 9 | |
38 | nfre1 2507 | . . . . . . . . . 10 | |
39 | nfv 1515 | . . . . . . . . . 10 | |
40 | 38, 39 | nfor 1561 | . . . . . . . . 9 |
41 | 37, 40 | nfralya 2504 | . . . . . . . 8 |
42 | nfv 1515 | . . . . . . . 8 | |
43 | 41, 42 | nfan 1552 | . . . . . . 7 |
44 | nfv 1515 | . . . . . . 7 | |
45 | simpr 109 | . . . . . . . . . . 11 | |
46 | simprr 522 | . . . . . . . . . . . . . 14 | |
47 | 46 | ad2antrr 480 | . . . . . . . . . . . . 13 |
48 | 47 | adantr 274 | . . . . . . . . . . . 12 |
49 | simprl 521 | . . . . . . . . . . . . . 14 | |
50 | 49 | ad2antrr 480 | . . . . . . . . . . . . 13 |
51 | 50 | adantr 274 | . . . . . . . . . . . 12 |
52 | 48, 51 | sublt0d 8459 | . . . . . . . . . . 11 |
53 | 45, 52 | mpbid 146 | . . . . . . . . . 10 |
54 | 53 | 3mix3d 1163 | . . . . . . . . 9 |
55 | simpr 109 | . . . . . . . . . . 11 | |
56 | 50 | adantr 274 | . . . . . . . . . . . 12 |
57 | 47 | adantr 274 | . . . . . . . . . . . 12 |
58 | 56, 57 | posdifd 8421 | . . . . . . . . . . 11 |
59 | 55, 58 | mpbird 166 | . . . . . . . . . 10 |
60 | 59 | 3mix1d 1161 | . . . . . . . . 9 |
61 | 47 | recnd 7918 | . . . . . . . . . . . 12 |
62 | 50 | recnd 7918 | . . . . . . . . . . . 12 |
63 | 61, 62 | subcld 8200 | . . . . . . . . . . 11 |
64 | simplr 520 | . . . . . . . . . . . 12 | |
65 | 64 | recnd 7918 | . . . . . . . . . . 11 |
66 | simpr 109 | . . . . . . . . . . . 12 | |
67 | 1ap0 8479 | . . . . . . . . . . . 12 # | |
68 | 66, 67 | eqbrtrdi 4015 | . . . . . . . . . . 11 # |
69 | 63, 65, 68 | mulap0bad 8547 | . . . . . . . . . 10 # |
70 | 46, 49 | resubcld 8270 | . . . . . . . . . . . 12 |
71 | 70 | ad2antrr 480 | . . . . . . . . . . 11 |
72 | reaplt 8477 | . . . . . . . . . . 11 # | |
73 | 71, 21, 72 | sylancl 410 | . . . . . . . . . 10 # |
74 | 69, 73 | mpbid 146 | . . . . . . . . 9 |
75 | 54, 60, 74 | mpjaodan 788 | . . . . . . . 8 |
76 | 75 | exp31 362 | . . . . . . 7 |
77 | 43, 44, 76 | rexlimd 2578 | . . . . . 6 |
78 | 77 | imp 123 | . . . . 5 |
79 | 46 | recnd 7918 | . . . . . . . . 9 |
80 | 79 | adantr 274 | . . . . . . . 8 |
81 | 49 | recnd 7918 | . . . . . . . . 9 |
82 | 81 | adantr 274 | . . . . . . . 8 |
83 | simpr 109 | . . . . . . . 8 | |
84 | 80, 82, 83 | subeq0d 8208 | . . . . . . 7 |
85 | 84 | equcomd 1694 | . . . . . 6 |
86 | 85 | 3mix2d 1162 | . . . . 5 |
87 | oveq1 5843 | . . . . . . . . 9 | |
88 | 87 | eqeq1d 2173 | . . . . . . . 8 |
89 | 88 | rexbidv 2465 | . . . . . . 7 |
90 | eqeq1 2171 | . . . . . . 7 | |
91 | 89, 90 | orbi12d 783 | . . . . . 6 |
92 | simpl 108 | . . . . . 6 | |
93 | 91, 92, 70 | rspcdva 2830 | . . . . 5 |
94 | 78, 86, 93 | mpjaodan 788 | . . . 4 |
95 | 94 | ralrimivva 2546 | . . 3 |
96 | 36, 95 | sylbi 120 | . 2 |
97 | 30, 96 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 w3o 966 wceq 1342 wcel 2135 wral 2442 wrex 2443 class class class wbr 3976 (class class class)co 5836 cc 7742 cr 7743 cc0 7744 c1 7745 cmul 7749 clt 7924 cmin 8060 # cap 8470 cdiv 8559 |
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-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3or 968 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-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 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-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 |
This theorem is referenced by: trirec0xor 13758 |
Copyright terms: Public domain | W3C validator |