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Mirrors > Home > ILE Home > Th. List > resqrexlemnmsq | GIF version |
Description: Lemma for resqrex 10926. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-Jul-2021.) |
Ref | Expression |
---|---|
resqrexlemex.seq | ⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) |
resqrexlemex.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
resqrexlemex.agt0 | ⊢ (𝜑 → 0 ≤ 𝐴) |
resqrexlemnmsq.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
resqrexlemnmsq.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
resqrexlemnmsq.nm | ⊢ (𝜑 → 𝑁 ≤ 𝑀) |
Ref | Expression |
---|---|
resqrexlemnmsq | ⊢ (𝜑 → (((𝐹‘𝑁)↑2) − ((𝐹‘𝑀)↑2)) < (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqrexlemex.seq | . . . . . . . 8 ⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) | |
2 | resqrexlemex.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | resqrexlemex.agt0 | . . . . . . . 8 ⊢ (𝜑 → 0 ≤ 𝐴) | |
4 | 1, 2, 3 | resqrexlemf 10907 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ+) |
5 | resqrexlemnmsq.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 4, 5 | ffvelrnd 5603 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ+) |
7 | 6 | rpred 9603 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ) |
8 | 7 | resqcld 10577 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝑁)↑2) ∈ ℝ) |
9 | 8 | recnd 7906 | . . 3 ⊢ (𝜑 → ((𝐹‘𝑁)↑2) ∈ ℂ) |
10 | resqrexlemnmsq.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
11 | 4, 10 | ffvelrnd 5603 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ+) |
12 | 11 | rpred 9603 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ) |
13 | 12 | resqcld 10577 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝑀)↑2) ∈ ℝ) |
14 | 13 | recnd 7906 | . . 3 ⊢ (𝜑 → ((𝐹‘𝑀)↑2) ∈ ℂ) |
15 | 2 | recnd 7906 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
16 | 9, 14, 15 | nnncan2d 8221 | . 2 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) = (((𝐹‘𝑁)↑2) − ((𝐹‘𝑀)↑2))) |
17 | 8, 2 | resubcld 8256 | . . . 4 ⊢ (𝜑 → (((𝐹‘𝑁)↑2) − 𝐴) ∈ ℝ) |
18 | 13, 2 | resubcld 8256 | . . . 4 ⊢ (𝜑 → (((𝐹‘𝑀)↑2) − 𝐴) ∈ ℝ) |
19 | 17, 18 | resubcld 8256 | . . 3 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) ∈ ℝ) |
20 | 1nn 8844 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
21 | 20 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ) |
22 | 4, 21 | ffvelrnd 5603 | . . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ+) |
23 | 2z 9195 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
24 | 23 | a1i 9 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
25 | 22, 24 | rpexpcld 10575 | . . . . 5 ⊢ (𝜑 → ((𝐹‘1)↑2) ∈ ℝ+) |
26 | 4nn 8996 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
27 | 26 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 4 ∈ ℕ) |
28 | 27 | nnrpd 9601 | . . . . . 6 ⊢ (𝜑 → 4 ∈ ℝ+) |
29 | 5 | nnzd 9285 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
30 | 1zzd 9194 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ) | |
31 | 29, 30 | zsubcld 9291 | . . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
32 | 28, 31 | rpexpcld 10575 | . . . . 5 ⊢ (𝜑 → (4↑(𝑁 − 1)) ∈ ℝ+) |
33 | 25, 32 | rpdivcld 9621 | . . . 4 ⊢ (𝜑 → (((𝐹‘1)↑2) / (4↑(𝑁 − 1))) ∈ ℝ+) |
34 | 33 | rpred 9603 | . . 3 ⊢ (𝜑 → (((𝐹‘1)↑2) / (4↑(𝑁 − 1))) ∈ ℝ) |
35 | 1, 2, 3 | resqrexlemover 10910 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → 𝐴 < ((𝐹‘𝑀)↑2)) |
36 | 10, 35 | mpdan 418 | . . . . 5 ⊢ (𝜑 → 𝐴 < ((𝐹‘𝑀)↑2)) |
37 | difrp 9599 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐹‘𝑀)↑2) ∈ ℝ) → (𝐴 < ((𝐹‘𝑀)↑2) ↔ (((𝐹‘𝑀)↑2) − 𝐴) ∈ ℝ+)) | |
38 | 2, 13, 37 | syl2anc 409 | . . . . 5 ⊢ (𝜑 → (𝐴 < ((𝐹‘𝑀)↑2) ↔ (((𝐹‘𝑀)↑2) − 𝐴) ∈ ℝ+)) |
39 | 36, 38 | mpbid 146 | . . . 4 ⊢ (𝜑 → (((𝐹‘𝑀)↑2) − 𝐴) ∈ ℝ+) |
40 | 17, 39 | ltsubrpd 9636 | . . 3 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) < (((𝐹‘𝑁)↑2) − 𝐴)) |
41 | 1, 2, 3 | resqrexlemcalc3 10916 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘𝑁)↑2) − 𝐴) ≤ (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) |
42 | 5, 41 | mpdan 418 | . . 3 ⊢ (𝜑 → (((𝐹‘𝑁)↑2) − 𝐴) ≤ (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) |
43 | 19, 17, 34, 40, 42 | ltletrd 8298 | . 2 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) < (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) |
44 | 16, 43 | eqbrtrrd 3988 | 1 ⊢ (𝜑 → (((𝐹‘𝑁)↑2) − ((𝐹‘𝑀)↑2)) < (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 ∈ wcel 2128 {csn 3560 class class class wbr 3965 × cxp 4584 ‘cfv 5170 (class class class)co 5824 ∈ cmpo 5826 ℝcr 7731 0cc0 7732 1c1 7733 + caddc 7735 < clt 7912 ≤ cle 7913 − cmin 8046 / cdiv 8545 ℕcn 8833 2c2 8884 4c4 8886 ℤcz 9167 ℝ+crp 9560 seqcseq 10344 ↑cexp 10418 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-rp 9561 df-seqfrec 10345 df-exp 10419 |
This theorem is referenced by: resqrexlemnm 10918 |
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