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Mirrors > Home > ILE Home > Th. List > resqrexlemdec | GIF version |
Description: Lemma for resqrex 10039. 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 10022 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) = (((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) / 2)) |
5 | 2 | adantr 270 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℝ) |
6 | 1, 2, 3 | resqrexlemf 10020 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ+) |
7 | 6 | ffvelrnda 5328 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ℝ+) |
8 | 5, 7 | rerpdivcld 8875 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐴 / (𝐹‘𝑁)) ∈ ℝ) |
9 | 7 | rpred 8843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ℝ) |
10 | 1, 2, 3 | resqrexlemover 10023 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 < ((𝐹‘𝑁)↑2)) |
11 | 7 | rpcnd 8845 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘𝑁) ∈ ℂ) |
12 | 11 | sqvald 9688 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝑁)↑2) = ((𝐹‘𝑁) · (𝐹‘𝑁))) |
13 | 10, 12 | breqtrd 3811 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 < ((𝐹‘𝑁) · (𝐹‘𝑁))) |
14 | 5, 9, 7 | ltdivmuld 8895 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐴 / (𝐹‘𝑁)) < (𝐹‘𝑁) ↔ 𝐴 < ((𝐹‘𝑁) · (𝐹‘𝑁)))) |
15 | 13, 14 | mpbird 165 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐴 / (𝐹‘𝑁)) < (𝐹‘𝑁)) |
16 | 8, 9, 9, 15 | ltadd2dd 7582 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) < ((𝐹‘𝑁) + (𝐹‘𝑁))) |
17 | 11 | 2timesd 8329 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (2 · (𝐹‘𝑁)) = ((𝐹‘𝑁) + (𝐹‘𝑁))) |
18 | 16, 17 | breqtrrd 3813 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) < (2 · (𝐹‘𝑁))) |
19 | 9, 8 | readdcld 7199 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) ∈ ℝ) |
20 | 2rp 8809 | . . . . 5 ⊢ 2 ∈ ℝ+ | |
21 | 20 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 2 ∈ ℝ+) |
22 | 19, 9, 21 | ltdivmuld 8895 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) / 2) < (𝐹‘𝑁) ↔ ((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) < (2 · (𝐹‘𝑁)))) |
23 | 18, 22 | mpbird 165 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘𝑁) + (𝐴 / (𝐹‘𝑁))) / 2) < (𝐹‘𝑁)) |
24 | 4, 23 | eqbrtrd 3807 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐹‘(𝑁 + 1)) < (𝐹‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 {csn 3400 class class class wbr 3787 × cxp 4363 ‘cfv 4926 (class class class)co 5537 ↦ cmpt2 5539 ℝcr 7031 0cc0 7032 1c1 7033 + caddc 7035 · cmul 7037 < clt 7204 ≤ cle 7205 / cdiv 7816 ℕcn 8095 2c2 8145 ℝ+crp 8804 seqcseq 9510 ↑cexp 9561 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3895 ax-sep 3898 ax-nul 3906 ax-pow 3950 ax-pr 3966 ax-un 4190 ax-setind 4282 ax-iinf 4331 ax-cnex 7118 ax-resscn 7119 ax-1cn 7120 ax-1re 7121 ax-icn 7122 ax-addcl 7123 ax-addrcl 7124 ax-mulcl 7125 ax-mulrcl 7126 ax-addcom 7127 ax-mulcom 7128 ax-addass 7129 ax-mulass 7130 ax-distr 7131 ax-i2m1 7132 ax-0lt1 7133 ax-1rid 7134 ax-0id 7135 ax-rnegex 7136 ax-precex 7137 ax-cnre 7138 ax-pre-ltirr 7139 ax-pre-ltwlin 7140 ax-pre-lttrn 7141 ax-pre-apti 7142 ax-pre-ltadd 7143 ax-pre-mulgt0 7144 ax-pre-mulext 7145 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rmo 2357 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3253 df-if 3354 df-pw 3386 df-sn 3406 df-pr 3407 df-op 3409 df-uni 3604 df-int 3639 df-iun 3682 df-br 3788 df-opab 3842 df-mpt 3843 df-tr 3878 df-id 4050 df-po 4053 df-iso 4054 df-iord 4123 df-on 4125 df-ilim 4126 df-suc 4128 df-iom 4334 df-xp 4371 df-rel 4372 df-cnv 4373 df-co 4374 df-dm 4375 df-rn 4376 df-res 4377 df-ima 4378 df-iota 4891 df-fun 4928 df-fn 4929 df-f 4930 df-f1 4931 df-fo 4932 df-f1o 4933 df-fv 4934 df-riota 5493 df-ov 5540 df-oprab 5541 df-mpt2 5542 df-1st 5792 df-2nd 5793 df-recs 5948 df-frec 6034 df-pnf 7206 df-mnf 7207 df-xr 7208 df-ltxr 7209 df-le 7210 df-sub 7337 df-neg 7338 df-reap 7731 df-ap 7738 df-div 7817 df-inn 8096 df-2 8154 df-3 8155 df-4 8156 df-n0 8345 df-z 8422 df-uz 8690 df-rp 8805 df-iseq 9511 df-iexp 9562 |
This theorem is referenced by: resqrexlemdecn 10025 |
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