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Mirrors > Home > ILE Home > Th. List > cvg1nlemcxze | GIF version |
Description: Lemma for cvg1n 9585. Rearranging an expression related to the rate of convergence. (Contributed by Jim Kingdon, 6-Aug-2021.) |
Ref | Expression |
---|---|
cvg1nlemcxze.c | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
cvg1nlemcxze.x | ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
cvg1nlemcxze.z | ⊢ (𝜑 → 𝑍 ∈ ℕ) |
cvg1nlemcxze.e | ⊢ (𝜑 → 𝐸 ∈ ℕ) |
cvg1nlemcxze.a | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
cvg1nlemcxze.1 | ⊢ (𝜑 → ((((𝐶 · 2) / 𝑋) / 𝑍) + 𝐴) < 𝐸) |
Ref | Expression |
---|---|
cvg1nlemcxze | ⊢ (𝜑 → (𝐶 / (𝐸 · 𝑍)) < (𝑋 / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvg1nlemcxze.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
2 | 1 | rpcnd 8624 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
3 | 2cnd 7988 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ) | |
4 | cvg1nlemcxze.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ+) | |
5 | 4 | rpcnd 8624 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
6 | 4 | rpap0d 8628 | . . . . . . 7 ⊢ (𝜑 → 𝑋 # 0) |
7 | 2, 3, 5, 6 | div23apd 7802 | . . . . . 6 ⊢ (𝜑 → ((𝐶 · 2) / 𝑋) = ((𝐶 / 𝑋) · 2)) |
8 | 2rp 8588 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ+ | |
9 | 8 | a1i 9 | . . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℝ+) |
10 | 1, 9 | rpmulcld 8639 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 · 2) ∈ ℝ+) |
11 | 10, 4 | rpdivcld 8640 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐶 · 2) / 𝑋) ∈ ℝ+) |
12 | cvg1nlemcxze.z | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ℕ) | |
13 | 12 | nnrpd 8621 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ ℝ+) |
14 | 11, 13 | rpdivcld 8640 | . . . . . . . . 9 ⊢ (𝜑 → (((𝐶 · 2) / 𝑋) / 𝑍) ∈ ℝ+) |
15 | 14 | rpred 8622 | . . . . . . . 8 ⊢ (𝜑 → (((𝐶 · 2) / 𝑋) / 𝑍) ∈ ℝ) |
16 | cvg1nlemcxze.a | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
17 | 16 | nnred 7927 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
18 | 15, 17 | readdcld 7055 | . . . . . . . 8 ⊢ (𝜑 → ((((𝐶 · 2) / 𝑋) / 𝑍) + 𝐴) ∈ ℝ) |
19 | cvg1nlemcxze.e | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℕ) | |
20 | 19 | nnred 7927 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
21 | 16 | nnrpd 8621 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
22 | 15, 21 | ltaddrpd 8656 | . . . . . . . 8 ⊢ (𝜑 → (((𝐶 · 2) / 𝑋) / 𝑍) < ((((𝐶 · 2) / 𝑋) / 𝑍) + 𝐴)) |
23 | cvg1nlemcxze.1 | . . . . . . . 8 ⊢ (𝜑 → ((((𝐶 · 2) / 𝑋) / 𝑍) + 𝐴) < 𝐸) | |
24 | 15, 18, 20, 22, 23 | lttrd 7140 | . . . . . . 7 ⊢ (𝜑 → (((𝐶 · 2) / 𝑋) / 𝑍) < 𝐸) |
25 | 11 | rpred 8622 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶 · 2) / 𝑋) ∈ ℝ) |
26 | 25, 20, 13 | ltdivmul2d 8675 | . . . . . . 7 ⊢ (𝜑 → ((((𝐶 · 2) / 𝑋) / 𝑍) < 𝐸 ↔ ((𝐶 · 2) / 𝑋) < (𝐸 · 𝑍))) |
27 | 24, 26 | mpbid 135 | . . . . . 6 ⊢ (𝜑 → ((𝐶 · 2) / 𝑋) < (𝐸 · 𝑍)) |
28 | 7, 27 | eqbrtrrd 3786 | . . . . 5 ⊢ (𝜑 → ((𝐶 / 𝑋) · 2) < (𝐸 · 𝑍)) |
29 | 1 | rpred 8622 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
30 | 29, 4 | rerpdivcld 8654 | . . . . . 6 ⊢ (𝜑 → (𝐶 / 𝑋) ∈ ℝ) |
31 | 19, 12 | nnmulcld 7962 | . . . . . . 7 ⊢ (𝜑 → (𝐸 · 𝑍) ∈ ℕ) |
32 | 31 | nnred 7927 | . . . . . 6 ⊢ (𝜑 → (𝐸 · 𝑍) ∈ ℝ) |
33 | 30, 32, 9 | ltmuldivd 8670 | . . . . 5 ⊢ (𝜑 → (((𝐶 / 𝑋) · 2) < (𝐸 · 𝑍) ↔ (𝐶 / 𝑋) < ((𝐸 · 𝑍) / 2))) |
34 | 28, 33 | mpbid 135 | . . . 4 ⊢ (𝜑 → (𝐶 / 𝑋) < ((𝐸 · 𝑍) / 2)) |
35 | 29, 9, 32, 4 | lt2mul2divd 8685 | . . . 4 ⊢ (𝜑 → ((𝐶 · 2) < ((𝐸 · 𝑍) · 𝑋) ↔ (𝐶 / 𝑋) < ((𝐸 · 𝑍) / 2))) |
36 | 34, 35 | mpbird 156 | . . 3 ⊢ (𝜑 → (𝐶 · 2) < ((𝐸 · 𝑍) · 𝑋)) |
37 | 31 | nncnd 7928 | . . . 4 ⊢ (𝜑 → (𝐸 · 𝑍) ∈ ℂ) |
38 | 37, 5 | mulcomd 7048 | . . 3 ⊢ (𝜑 → ((𝐸 · 𝑍) · 𝑋) = (𝑋 · (𝐸 · 𝑍))) |
39 | 36, 38 | breqtrd 3788 | . 2 ⊢ (𝜑 → (𝐶 · 2) < (𝑋 · (𝐸 · 𝑍))) |
40 | 4 | rpred 8622 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
41 | 31 | nnrpd 8621 | . . 3 ⊢ (𝜑 → (𝐸 · 𝑍) ∈ ℝ+) |
42 | 29, 9, 40, 41 | lt2mul2divd 8685 | . 2 ⊢ (𝜑 → ((𝐶 · 2) < (𝑋 · (𝐸 · 𝑍)) ↔ (𝐶 / (𝐸 · 𝑍)) < (𝑋 / 2))) |
43 | 39, 42 | mpbid 135 | 1 ⊢ (𝜑 → (𝐶 / (𝐸 · 𝑍)) < (𝑋 / 2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 class class class wbr 3764 (class class class)co 5512 + caddc 6892 · cmul 6894 < clt 7060 / cdiv 7651 ℕcn 7914 2c2 7964 ℝ+crp 8583 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-2 7973 df-rp 8584 |
This theorem is referenced by: cvg1nlemres 9584 |
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