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Mirrors > Home > ILE Home > Th. List > resqrexlemdec | GIF version |
Description: Lemma for resqrex 9624. The sequence is decreasing. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.) |
Ref | Expression |
---|---|
resqrexlemex.seq | ⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+) |
resqrexlemex.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
resqrexlemex.agt0 | ⊢ (𝜑 → 0 ≤ 𝐴) |
Ref | Expression |
---|---|
resqrexlemdec | ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) < (𝐹‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqrexlemex.seq | . . 3 ⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+) | |
2 | resqrexlemex.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | resqrexlemex.agt0 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) | |
4 | 1, 2, 3 | resqrexlemfp1 9607 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) = (((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) / 2)) |
5 | 2 | adantr 261 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℝ) |
6 | 1, 2, 3 | resqrexlemf 9605 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ+) |
7 | 6 | ffvelrnda 5302 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ℝ+) |
8 | 5, 7 | rerpdivcld 8654 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐴 / (𝐹‘𝑁)) ∈ ℝ) |
9 | 7 | rpred 8622 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ℝ) |
10 | 1, 2, 3 | resqrexlemover 9608 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 < ((𝐹‘𝑁)↑2)) |
11 | 7 | rpcnd 8624 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ℂ) |
12 | 11 | sqvald 9378 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝑁)↑2) = ((𝐹‘𝑁) · (𝐹‘𝑁))) |
13 | 10, 12 | breqtrd 3788 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 < ((𝐹‘𝑁) · (𝐹‘𝑁))) |
14 | 5, 9, 7 | ltdivmuld 8674 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐴 / (𝐹‘𝑁)) < (𝐹‘𝑁) ↔ 𝐴 < ((𝐹‘𝑁) · (𝐹‘𝑁)))) |
15 | 13, 14 | mpbird 156 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐴 / (𝐹‘𝑁)) < (𝐹‘𝑁)) |
16 | 8, 9, 9, 15 | ltadd2dd 7419 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) < ((𝐹‘𝑁) + (𝐹‘𝑁))) |
17 | 11 | 2timesd 8167 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (2 · (𝐹‘𝑁)) = ((𝐹‘𝑁) + (𝐹‘𝑁))) |
18 | 16, 17 | breqtrrd 3790 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) < (2 · (𝐹‘𝑁))) |
19 | 9, 8 | readdcld 7055 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) ∈ ℝ) |
20 | 2rp 8588 | . . . . 5 ⊢ 2 ∈ ℝ+ | |
21 | 20 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 2 ∈ ℝ+) |
22 | 19, 9, 21 | ltdivmuld 8674 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) / 2) < (𝐹‘𝑁) ↔ ((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) < (2 · (𝐹‘𝑁)))) |
23 | 18, 22 | mpbird 156 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) / 2) < (𝐹‘𝑁)) |
24 | 4, 23 | eqbrtrd 3784 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) < (𝐹‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {csn 3375 class class class wbr 3764 × cxp 4343 ‘cfv 4902 (class class class)co 5512 ↦ cmpt2 5514 ℝcr 6888 0cc0 6889 1c1 6890 + caddc 6892 · cmul 6894 < clt 7060 ≤ cle 7061 / cdiv 7651 ℕcn 7914 2c2 7964 ℝ+crp 8583 seqcseq 9211 ↑cexp 9254 |
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-if 3332 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-frec 5978 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-3 7974 df-4 7975 df-n0 8182 df-z 8246 df-uz 8474 df-rp 8584 df-iseq 9212 df-iexp 9255 |
This theorem is referenced by: resqrexlemdecn 9610 |
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