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Mirrors > Home > ILE Home > Th. List > cvg1nlemcxze | Unicode version |
Description: Lemma for cvg1n 10726. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.) |
Ref | Expression |
---|---|
cvg1nlemcxze.c | |
cvg1nlemcxze.x | |
cvg1nlemcxze.z | |
cvg1nlemcxze.e | |
cvg1nlemcxze.a | |
cvg1nlemcxze.1 |
Ref | Expression |
---|---|
cvg1nlemcxze |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvg1nlemcxze.c | . . . . . . . 8 | |
2 | 1 | rpcnd 9453 | . . . . . . 7 |
3 | 2cnd 8761 | . . . . . . 7 | |
4 | cvg1nlemcxze.x | . . . . . . . 8 | |
5 | 4 | rpcnd 9453 | . . . . . . 7 |
6 | 4 | rpap0d 9457 | . . . . . . 7 # |
7 | 2, 3, 5, 6 | div23apd 8556 | . . . . . 6 |
8 | 2rp 9414 | . . . . . . . . . . . . 13 | |
9 | 8 | a1i 9 | . . . . . . . . . . . 12 |
10 | 1, 9 | rpmulcld 9468 | . . . . . . . . . . 11 |
11 | 10, 4 | rpdivcld 9469 | . . . . . . . . . 10 |
12 | cvg1nlemcxze.z | . . . . . . . . . . 11 | |
13 | 12 | nnrpd 9450 | . . . . . . . . . 10 |
14 | 11, 13 | rpdivcld 9469 | . . . . . . . . 9 |
15 | 14 | rpred 9451 | . . . . . . . 8 |
16 | cvg1nlemcxze.a | . . . . . . . . . 10 | |
17 | 16 | nnred 8701 | . . . . . . . . 9 |
18 | 15, 17 | readdcld 7763 | . . . . . . . 8 |
19 | cvg1nlemcxze.e | . . . . . . . . 9 | |
20 | 19 | nnred 8701 | . . . . . . . 8 |
21 | 16 | nnrpd 9450 | . . . . . . . . 9 |
22 | 15, 21 | ltaddrpd 9485 | . . . . . . . 8 |
23 | cvg1nlemcxze.1 | . . . . . . . 8 | |
24 | 15, 18, 20, 22, 23 | lttrd 7856 | . . . . . . 7 |
25 | 11 | rpred 9451 | . . . . . . . 8 |
26 | 25, 20, 13 | ltdivmul2d 9504 | . . . . . . 7 |
27 | 24, 26 | mpbid 146 | . . . . . 6 |
28 | 7, 27 | eqbrtrrd 3922 | . . . . 5 |
29 | 1 | rpred 9451 | . . . . . . 7 |
30 | 29, 4 | rerpdivcld 9483 | . . . . . 6 |
31 | 19, 12 | nnmulcld 8737 | . . . . . . 7 |
32 | 31 | nnred 8701 | . . . . . 6 |
33 | 30, 32, 9 | ltmuldivd 9499 | . . . . 5 |
34 | 28, 33 | mpbid 146 | . . . 4 |
35 | 29, 9, 32, 4 | lt2mul2divd 9520 | . . . 4 |
36 | 34, 35 | mpbird 166 | . . 3 |
37 | 31 | nncnd 8702 | . . . 4 |
38 | 37, 5 | mulcomd 7755 | . . 3 |
39 | 36, 38 | breqtrd 3924 | . 2 |
40 | 4 | rpred 9451 | . . 3 |
41 | 31 | nnrpd 9450 | . . 3 |
42 | 29, 9, 40, 41 | lt2mul2divd 9520 | . 2 |
43 | 39, 42 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1465 class class class wbr 3899 (class class class)co 5742 caddc 7591 cmul 7593 clt 7768 cdiv 8400 cn 8688 c2 8739 crp 9409 |
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-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-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 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-2 8747 df-rp 9410 |
This theorem is referenced by: cvg1nlemres 10725 |
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