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Mirrors > Home > ILE Home > Th. List > resqrexlemnmsq | GIF version |
Description: Lemma for resqrex 10798. 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 10779 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ+) |
5 | resqrexlemnmsq.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 4, 5 | ffvelrnd 5556 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ+) |
7 | 6 | rpred 9483 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ) |
8 | 7 | resqcld 10450 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝑁)↑2) ∈ ℝ) |
9 | 8 | recnd 7794 | . . 3 ⊢ (𝜑 → ((𝐹‘𝑁)↑2) ∈ ℂ) |
10 | resqrexlemnmsq.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
11 | 4, 10 | ffvelrnd 5556 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ+) |
12 | 11 | rpred 9483 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ) |
13 | 12 | resqcld 10450 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝑀)↑2) ∈ ℝ) |
14 | 13 | recnd 7794 | . . 3 ⊢ (𝜑 → ((𝐹‘𝑀)↑2) ∈ ℂ) |
15 | 2 | recnd 7794 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
16 | 9, 14, 15 | nnncan2d 8108 | . 2 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) = (((𝐹‘𝑁)↑2) − ((𝐹‘𝑀)↑2))) |
17 | 8, 2 | resubcld 8143 | . . . 4 ⊢ (𝜑 → (((𝐹‘𝑁)↑2) − 𝐴) ∈ ℝ) |
18 | 13, 2 | resubcld 8143 | . . . 4 ⊢ (𝜑 → (((𝐹‘𝑀)↑2) − 𝐴) ∈ ℝ) |
19 | 17, 18 | resubcld 8143 | . . 3 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) ∈ ℝ) |
20 | 1nn 8731 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
21 | 20 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ) |
22 | 4, 21 | ffvelrnd 5556 | . . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ+) |
23 | 2z 9082 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
24 | 23 | a1i 9 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
25 | 22, 24 | rpexpcld 10448 | . . . . 5 ⊢ (𝜑 → ((𝐹‘1)↑2) ∈ ℝ+) |
26 | 4nn 8883 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
27 | 26 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 4 ∈ ℕ) |
28 | 27 | nnrpd 9482 | . . . . . 6 ⊢ (𝜑 → 4 ∈ ℝ+) |
29 | 5 | nnzd 9172 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
30 | 1zzd 9081 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ) | |
31 | 29, 30 | zsubcld 9178 | . . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
32 | 28, 31 | rpexpcld 10448 | . . . . 5 ⊢ (𝜑 → (4↑(𝑁 − 1)) ∈ ℝ+) |
33 | 25, 32 | rpdivcld 9501 | . . . 4 ⊢ (𝜑 → (((𝐹‘1)↑2) / (4↑(𝑁 − 1))) ∈ ℝ+) |
34 | 33 | rpred 9483 | . . 3 ⊢ (𝜑 → (((𝐹‘1)↑2) / (4↑(𝑁 − 1))) ∈ ℝ) |
35 | 1, 2, 3 | resqrexlemover 10782 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → 𝐴 < ((𝐹‘𝑀)↑2)) |
36 | 10, 35 | mpdan 417 | . . . . 5 ⊢ (𝜑 → 𝐴 < ((𝐹‘𝑀)↑2)) |
37 | difrp 9480 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ ((𝐹‘𝑀)↑2) ∈ ℝ) → (𝐴 < ((𝐹‘𝑀)↑2) ↔ (((𝐹‘𝑀)↑2) − 𝐴) ∈ ℝ+)) | |
38 | 2, 13, 37 | syl2anc 408 | . . . . 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 9516 | . . 3 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) < (((𝐹‘𝑁)↑2) − 𝐴)) |
41 | 1, 2, 3 | resqrexlemcalc3 10788 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (((𝐹‘𝑁)↑2) − 𝐴) ≤ (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) |
42 | 5, 41 | mpdan 417 | . . 3 ⊢ (𝜑 → (((𝐹‘𝑁)↑2) − 𝐴) ≤ (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) |
43 | 19, 17, 34, 40, 42 | ltletrd 8185 | . 2 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) < (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) |
44 | 16, 43 | eqbrtrrd 3952 | 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 1331 ∈ wcel 1480 {csn 3527 class class class wbr 3929 × cxp 4537 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 < clt 7800 ≤ cle 7801 − cmin 7933 / cdiv 8432 ℕcn 8720 2c2 8771 4c4 8773 ℤcz 9054 ℝ+crp 9441 seqcseq 10218 ↑cexp 10292 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-seqfrec 10219 df-exp 10293 |
This theorem is referenced by: resqrexlemnm 10790 |
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