Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > caucvgrelemcau | Unicode version |
Description: Lemma for caucvgre 10721. Converting the Cauchy condition. (Contributed by Jim Kingdon, 20-Jul-2021.) |
Ref | Expression |
---|---|
caucvgre.f | |
caucvgre.cau |
Ref | Expression |
---|---|
caucvgrelemcau |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 504 | . . . . . . 7 | |
2 | 1 | nnred 8701 | . . . . . 6 |
3 | simpr 109 | . . . . . . 7 | |
4 | 3 | nnred 8701 | . . . . . 6 |
5 | ltle 7819 | . . . . . 6 | |
6 | 2, 4, 5 | syl2anc 408 | . . . . 5 |
7 | eluznn 9362 | . . . . . . . . . . . 12 | |
8 | 7 | ex 114 | . . . . . . . . . . 11 |
9 | nnz 9041 | . . . . . . . . . . . . 13 | |
10 | eluz1 9298 | . . . . . . . . . . . . 13 | |
11 | 9, 10 | syl 14 | . . . . . . . . . . . 12 |
12 | simpr 109 | . . . . . . . . . . . 12 | |
13 | 11, 12 | syl6bi 162 | . . . . . . . . . . 11 |
14 | 8, 13 | jcad 305 | . . . . . . . . . 10 |
15 | nnz 9041 | . . . . . . . . . . . 12 | |
16 | 15 | anim1i 338 | . . . . . . . . . . 11 |
17 | 16, 11 | syl5ibr 155 | . . . . . . . . . 10 |
18 | 14, 17 | impbid 128 | . . . . . . . . 9 |
19 | 18 | adantl 275 | . . . . . . . 8 |
20 | 19 | biimpar 295 | . . . . . . 7 |
21 | caucvgre.cau | . . . . . . . . 9 | |
22 | 21 | r19.21bi 2497 | . . . . . . . 8 |
23 | 22 | r19.21bi 2497 | . . . . . . 7 |
24 | 20, 23 | syldan 280 | . . . . . 6 |
25 | 24 | expr 372 | . . . . 5 |
26 | 6, 25 | syld 45 | . . . 4 |
27 | ltxrlt 7798 | . . . . 5 | |
28 | 2, 4, 27 | syl2anc 408 | . . . 4 |
29 | caucvgre.f | . . . . . . . . 9 | |
30 | 29 | ad2antrr 479 | . . . . . . . 8 |
31 | 30, 1 | ffvelrnd 5524 | . . . . . . 7 |
32 | 30, 3 | ffvelrnd 5524 | . . . . . . . 8 |
33 | 1 | nnrecred 8735 | . . . . . . . 8 |
34 | 32, 33 | readdcld 7763 | . . . . . . 7 |
35 | ltxrlt 7798 | . . . . . . 7 | |
36 | 31, 34, 35 | syl2anc 408 | . . . . . 6 |
37 | nnap0 8717 | . . . . . . . . . 10 # | |
38 | 1, 37 | syl 14 | . . . . . . . . 9 # |
39 | caucvgrelemrec 10719 | . . . . . . . . 9 # | |
40 | 2, 38, 39 | syl2anc 408 | . . . . . . . 8 |
41 | 40 | oveq2d 5758 | . . . . . . 7 |
42 | 41 | breq2d 3911 | . . . . . 6 |
43 | 36, 42 | bitr4d 190 | . . . . 5 |
44 | 31, 33 | readdcld 7763 | . . . . . . 7 |
45 | ltxrlt 7798 | . . . . . . 7 | |
46 | 32, 44, 45 | syl2anc 408 | . . . . . 6 |
47 | 40 | oveq2d 5758 | . . . . . . 7 |
48 | 47 | breq2d 3911 | . . . . . 6 |
49 | 46, 48 | bitr4d 190 | . . . . 5 |
50 | 43, 49 | anbi12d 464 | . . . 4 |
51 | 26, 28, 50 | 3imtr3d 201 | . . 3 |
52 | 51 | ralrimiva 2482 | . 2 |
53 | 52 | ralrimiva 2482 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 class class class wbr 3899 wf 5089 cfv 5093 crio 5697 (class class class)co 5742 cr 7587 cc0 7588 c1 7589 caddc 7591 cltrr 7592 cmul 7593 clt 7768 cle 7769 # cap 8311 cdiv 8400 cn 8688 cz 9022 cuz 9294 |
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-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 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 |
This theorem depends on definitions: df-bi 116 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-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 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-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-z 9023 df-uz 9295 |
This theorem is referenced by: caucvgre 10721 |
Copyright terms: Public domain | W3C validator |