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Mirrors > Home > ILE Home > Th. List > resqrexlemnmsq | GIF version |
Description: Lemma for resqrex 10830. 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 10811 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ+) |
5 | resqrexlemnmsq.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 4, 5 | ffvelrnd 5564 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ+) |
7 | 6 | rpred 9513 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ) |
8 | 7 | resqcld 10481 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝑁)↑2) ∈ ℝ) |
9 | 8 | recnd 7818 | . . 3 ⊢ (𝜑 → ((𝐹‘𝑁)↑2) ∈ ℂ) |
10 | resqrexlemnmsq.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
11 | 4, 10 | ffvelrnd 5564 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ+) |
12 | 11 | rpred 9513 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ) |
13 | 12 | resqcld 10481 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝑀)↑2) ∈ ℝ) |
14 | 13 | recnd 7818 | . . 3 ⊢ (𝜑 → ((𝐹‘𝑀)↑2) ∈ ℂ) |
15 | 2 | recnd 7818 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
16 | 9, 14, 15 | nnncan2d 8132 | . 2 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) = (((𝐹‘𝑁)↑2) − ((𝐹‘𝑀)↑2))) |
17 | 8, 2 | resubcld 8167 | . . . 4 ⊢ (𝜑 → (((𝐹‘𝑁)↑2) − 𝐴) ∈ ℝ) |
18 | 13, 2 | resubcld 8167 | . . . 4 ⊢ (𝜑 → (((𝐹‘𝑀)↑2) − 𝐴) ∈ ℝ) |
19 | 17, 18 | resubcld 8167 | . . 3 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) ∈ ℝ) |
20 | 1nn 8755 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
21 | 20 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ) |
22 | 4, 21 | ffvelrnd 5564 | . . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ+) |
23 | 2z 9106 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
24 | 23 | a1i 9 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℤ) |
25 | 22, 24 | rpexpcld 10479 | . . . . 5 ⊢ (𝜑 → ((𝐹‘1)↑2) ∈ ℝ+) |
26 | 4nn 8907 | . . . . . . . 8 ⊢ 4 ∈ ℕ | |
27 | 26 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → 4 ∈ ℕ) |
28 | 27 | nnrpd 9511 | . . . . . 6 ⊢ (𝜑 → 4 ∈ ℝ+) |
29 | 5 | nnzd 9196 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
30 | 1zzd 9105 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ) | |
31 | 29, 30 | zsubcld 9202 | . . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
32 | 28, 31 | rpexpcld 10479 | . . . . 5 ⊢ (𝜑 → (4↑(𝑁 − 1)) ∈ ℝ+) |
33 | 25, 32 | rpdivcld 9531 | . . . 4 ⊢ (𝜑 → (((𝐹‘1)↑2) / (4↑(𝑁 − 1))) ∈ ℝ+) |
34 | 33 | rpred 9513 | . . 3 ⊢ (𝜑 → (((𝐹‘1)↑2) / (4↑(𝑁 − 1))) ∈ ℝ) |
35 | 1, 2, 3 | resqrexlemover 10814 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → 𝐴 < ((𝐹‘𝑀)↑2)) |
36 | 10, 35 | mpdan 418 | . . . . 5 ⊢ (𝜑 → 𝐴 < ((𝐹‘𝑀)↑2)) |
37 | difrp 9509 | . . . . . 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 9546 | . . 3 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) < (((𝐹‘𝑁)↑2) − 𝐴)) |
41 | 1, 2, 3 | resqrexlemcalc3 10820 | . . . 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 8209 | . 2 ⊢ (𝜑 → ((((𝐹‘𝑁)↑2) − 𝐴) − (((𝐹‘𝑀)↑2) − 𝐴)) < (((𝐹‘1)↑2) / (4↑(𝑁 − 1)))) |
44 | 16, 43 | eqbrtrrd 3960 | 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 1332 ∈ wcel 1481 {csn 3532 class class class wbr 3937 × cxp 4545 ‘cfv 5131 (class class class)co 5782 ∈ cmpo 5784 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 < clt 7824 ≤ cle 7825 − cmin 7957 / cdiv 8456 ℕcn 8744 2c2 8795 4c4 8797 ℤcz 9078 ℝ+crp 9470 seqcseq 10249 ↑cexp 10323 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-rp 9471 df-seqfrec 10250 df-exp 10324 |
This theorem is referenced by: resqrexlemnm 10822 |
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