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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 7759 | . . . 4 | |
2 | reapval 8331 | . . . 4 #ℝ | |
3 | 1, 2 | mpan2 421 | . . 3 #ℝ |
4 | lt0neg1 8223 | . . . . . . . . . 10 | |
5 | renegcl 8016 | . . . . . . . . . . 11 | |
6 | ltxrlt 7823 | . . . . . . . . . . 11 | |
7 | 1, 5, 6 | sylancr 410 | . . . . . . . . . 10 |
8 | 4, 7 | bitrd 187 | . . . . . . . . 9 |
9 | 8 | pm5.32i 449 | . . . . . . . 8 |
10 | ax-precex 7723 | . . . . . . . . . 10 | |
11 | simpr 109 | . . . . . . . . . . 11 | |
12 | 11 | reximi 2527 | . . . . . . . . . 10 |
13 | 10, 12 | syl 14 | . . . . . . . . 9 |
14 | 5, 13 | sylan 281 | . . . . . . . 8 |
15 | 9, 14 | sylbi 120 | . . . . . . 7 |
16 | recn 7746 | . . . . . . . . . . . . 13 | |
17 | 16 | negnegd 8057 | . . . . . . . . . . . 12 |
18 | 17 | oveq2d 5783 | . . . . . . . . . . 11 |
19 | 18 | eqeq1d 2146 | . . . . . . . . . 10 |
20 | 19 | pm5.32i 449 | . . . . . . . . 9 |
21 | renegcl 8016 | . . . . . . . . . 10 | |
22 | negeq 7948 | . . . . . . . . . . . . 13 | |
23 | 22 | oveq2d 5783 | . . . . . . . . . . . 12 |
24 | 23 | eqeq1d 2146 | . . . . . . . . . . 11 |
25 | 24 | rspcev 2784 | . . . . . . . . . 10 |
26 | 21, 25 | sylan 281 | . . . . . . . . 9 |
27 | 20, 26 | sylbir 134 | . . . . . . . 8 |
28 | 27 | adantl 275 | . . . . . . 7 |
29 | 15, 28 | rexlimddv 2552 | . . . . . 6 |
30 | recn 7746 | . . . . . . . . . 10 | |
31 | recn 7746 | . . . . . . . . . 10 | |
32 | mul2neg 8153 | . . . . . . . . . 10 | |
33 | 30, 31, 32 | syl2an 287 | . . . . . . . . 9 |
34 | 33 | eqeq1d 2146 | . . . . . . . 8 |
35 | 34 | rexbidva 2432 | . . . . . . 7 |
36 | 35 | adantr 274 | . . . . . 6 |
37 | 29, 36 | mpbid 146 | . . . . 5 |
38 | 37 | ex 114 | . . . 4 |
39 | ltxrlt 7823 | . . . . . . . 8 | |
40 | 1, 39 | mpan 420 | . . . . . . 7 |
41 | 40 | pm5.32i 449 | . . . . . 6 |
42 | ax-precex 7723 | . . . . . . 7 | |
43 | simpr 109 | . . . . . . . 8 | |
44 | 43 | reximi 2527 | . . . . . . 7 |
45 | 42, 44 | syl 14 | . . . . . 6 |
46 | 41, 45 | sylbi 120 | . . . . 5 |
47 | 46 | ex 114 | . . . 4 |
48 | 38, 47 | jaod 706 | . . 3 |
49 | 3, 48 | sylbid 149 | . 2 #ℝ |
50 | 49 | imp 123 | 1 #ℝ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 wrex 2415 class class class wbr 3924 (class class class)co 5767 cc 7611 cr 7612 cc0 7613 c1 7614 cltrr 7617 cmul 7618 clt 7793 cneg 7927 #ℝ creap 8329 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 df-reap 8330 |
This theorem is referenced by: rimul 8340 recexap 8407 rerecclap 8483 |
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