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Mirrors > Home > ILE Home > Th. List > nn0opthlem1d | GIF version |
Description: A rather pretty lemma for nn0opth2 10470. (Contributed by Jim Kingdon, 31-Oct-2021.) |
Ref | Expression |
---|---|
nn0opthlem1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
nn0opthlem1d.2 | ⊢ (𝜑 → 𝐶 ∈ ℕ0) |
Ref | Expression |
---|---|
nn0opthlem1d | ⊢ (𝜑 → (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0opthlem1d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
2 | 1nn0 8993 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
3 | 2 | a1i 9 | . . . 4 ⊢ (𝜑 → 1 ∈ ℕ0) |
4 | 1, 3 | nn0addcld 9034 | . . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ0) |
5 | nn0opthlem1d.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℕ0) | |
6 | 4, 5 | nn0le2msqd 10465 | . 2 ⊢ (𝜑 → ((𝐴 + 1) ≤ 𝐶 ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶))) |
7 | nn0ltp1le 9116 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
8 | 1, 5, 7 | syl2anc 408 | . 2 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) |
9 | 1, 1 | nn0mulcld 9035 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐴) ∈ ℕ0) |
10 | 2nn0 8994 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
11 | 10 | a1i 9 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ0) |
12 | 11, 1 | nn0mulcld 9035 | . . . . 5 ⊢ (𝜑 → (2 · 𝐴) ∈ ℕ0) |
13 | 9, 12 | nn0addcld 9034 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0) |
14 | 5, 5 | nn0mulcld 9035 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝐶) ∈ ℕ0) |
15 | nn0ltp1le 9116 | . . . 4 ⊢ ((((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0 ∧ (𝐶 · 𝐶) ∈ ℕ0) → (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶))) | |
16 | 13, 14, 15 | syl2anc 408 | . . 3 ⊢ (𝜑 → (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶))) |
17 | 1 | nn0cnd 9032 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
18 | 1cnd 7782 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
19 | binom2 10403 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2))) | |
20 | 17, 18, 19 | syl2anc 408 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2))) |
21 | 17, 18 | addcld 7785 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℂ) |
22 | 21 | sqvald 10421 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1))) |
23 | 17 | sqvald 10421 | . . . . . . . 8 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴)) |
24 | 23 | oveq1d 5789 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑2) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · (𝐴 · 1)))) |
25 | 18 | sqvald 10421 | . . . . . . 7 ⊢ (𝜑 → (1↑2) = (1 · 1)) |
26 | 24, 25 | oveq12d 5792 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1))) |
27 | 20, 22, 26 | 3eqtr3d 2180 | . . . . 5 ⊢ (𝜑 → ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1))) |
28 | 17 | mulid1d 7783 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 · 1) = 𝐴) |
29 | 28 | oveq2d 5790 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐴 · 1)) = (2 · 𝐴)) |
30 | 29 | oveq2d 5790 | . . . . . 6 ⊢ (𝜑 → ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · 𝐴))) |
31 | 18 | mulid1d 7783 | . . . . . 6 ⊢ (𝜑 → (1 · 1) = 1) |
32 | 30, 31 | oveq12d 5792 | . . . . 5 ⊢ (𝜑 → (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1)) |
33 | 27, 32 | eqtrd 2172 | . . . 4 ⊢ (𝜑 → ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1)) |
34 | 33 | breq1d 3939 | . . 3 ⊢ (𝜑 → (((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶))) |
35 | 16, 34 | bitr4d 190 | . 2 ⊢ (𝜑 → (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶))) |
36 | 6, 8, 35 | 3bitr4d 219 | 1 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℂcc 7618 1c1 7621 + caddc 7623 · cmul 7625 < clt 7800 ≤ cle 7801 2c2 8771 ℕ0cn0 8977 ↑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-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 df-exp 10293 |
This theorem is referenced by: nn0opthlem2d 10467 |
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