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Mirrors > Home > ILE Home > Th. List > recexre | Unicode version |
Description: Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.) |
Ref | Expression |
---|---|
recexre | #ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7119 | . . . 4 | |
2 | reapval 7676 | . . . 4 #ℝ | |
3 | 1, 2 | mpan2 415 | . . 3 #ℝ |
4 | lt0neg1 7572 | . . . . . . . . . 10 | |
5 | renegcl 7369 | . . . . . . . . . . 11 | |
6 | ltxrlt 7178 | . . . . . . . . . . 11 | |
7 | 1, 5, 6 | sylancr 405 | . . . . . . . . . 10 |
8 | 4, 7 | bitrd 186 | . . . . . . . . 9 |
9 | 8 | pm5.32i 441 | . . . . . . . 8 |
10 | ax-precex 7086 | . . . . . . . . . 10 | |
11 | simpr 108 | . . . . . . . . . . 11 | |
12 | 11 | reximi 2458 | . . . . . . . . . 10 |
13 | 10, 12 | syl 14 | . . . . . . . . 9 |
14 | 5, 13 | sylan 277 | . . . . . . . 8 |
15 | 9, 14 | sylbi 119 | . . . . . . 7 |
16 | recn 7106 | . . . . . . . . . . . . 13 | |
17 | 16 | negnegd 7410 | . . . . . . . . . . . 12 |
18 | 17 | oveq2d 5548 | . . . . . . . . . . 11 |
19 | 18 | eqeq1d 2089 | . . . . . . . . . 10 |
20 | 19 | pm5.32i 441 | . . . . . . . . 9 |
21 | renegcl 7369 | . . . . . . . . . 10 | |
22 | negeq 7301 | . . . . . . . . . . . . 13 | |
23 | 22 | oveq2d 5548 | . . . . . . . . . . . 12 |
24 | 23 | eqeq1d 2089 | . . . . . . . . . . 11 |
25 | 24 | rspcev 2701 | . . . . . . . . . 10 |
26 | 21, 25 | sylan 277 | . . . . . . . . 9 |
27 | 20, 26 | sylbir 133 | . . . . . . . 8 |
28 | 27 | adantl 271 | . . . . . . 7 |
29 | 15, 28 | rexlimddv 2481 | . . . . . 6 |
30 | recn 7106 | . . . . . . . . . 10 | |
31 | recn 7106 | . . . . . . . . . 10 | |
32 | mul2neg 7502 | . . . . . . . . . 10 | |
33 | 30, 31, 32 | syl2an 283 | . . . . . . . . 9 |
34 | 33 | eqeq1d 2089 | . . . . . . . 8 |
35 | 34 | rexbidva 2365 | . . . . . . 7 |
36 | 35 | adantr 270 | . . . . . 6 |
37 | 29, 36 | mpbid 145 | . . . . 5 |
38 | 37 | ex 113 | . . . 4 |
39 | ltxrlt 7178 | . . . . . . . 8 | |
40 | 1, 39 | mpan 414 | . . . . . . 7 |
41 | 40 | pm5.32i 441 | . . . . . 6 |
42 | ax-precex 7086 | . . . . . . 7 | |
43 | simpr 108 | . . . . . . . 8 | |
44 | 43 | reximi 2458 | . . . . . . 7 |
45 | 42, 44 | syl 14 | . . . . . 6 |
46 | 41, 45 | sylbi 119 | . . . . 5 |
47 | 46 | ex 113 | . . . 4 |
48 | 38, 47 | jaod 669 | . . 3 |
49 | 3, 48 | sylbid 148 | . 2 #ℝ |
50 | 49 | imp 122 | 1 #ℝ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 wrex 2349 class class class wbr 3785 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 c1 6982 cltrr 6985 cmul 6986 clt 7153 cneg 7280 #ℝ creap 7674 |
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-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 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-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675 |
This theorem is referenced by: rimul 7685 recexap 7743 rerecclap 7818 |
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