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Mirrors > Home > ILE Home > Th. List > nn0opthlem1d | GIF version |
Description: A rather pretty lemma for nn0opth2 10463. (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 8986 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
3 | 2 | a1i 9 | . . . 4 ⊢ (𝜑 → 1 ∈ ℕ0) |
4 | 1, 3 | nn0addcld 9027 | . . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ0) |
5 | nn0opthlem1d.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℕ0) | |
6 | 4, 5 | nn0le2msqd 10458 | . 2 ⊢ (𝜑 → ((𝐴 + 1) ≤ 𝐶 ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶))) |
7 | nn0ltp1le 9109 | . . 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 9028 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐴) ∈ ℕ0) |
10 | 2nn0 8987 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
11 | 10 | a1i 9 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ0) |
12 | 11, 1 | nn0mulcld 9028 | . . . . 5 ⊢ (𝜑 → (2 · 𝐴) ∈ ℕ0) |
13 | 9, 12 | nn0addcld 9027 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0) |
14 | 5, 5 | nn0mulcld 9028 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝐶) ∈ ℕ0) |
15 | nn0ltp1le 9109 | . . . 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 9025 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
18 | 1cnd 7775 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
19 | binom2 10396 | . . . . . . 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 7778 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℂ) |
22 | 21 | sqvald 10414 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1))) |
23 | 17 | sqvald 10414 | . . . . . . . 8 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴)) |
24 | 23 | oveq1d 5782 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑2) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · (𝐴 · 1)))) |
25 | 18 | sqvald 10414 | . . . . . . 7 ⊢ (𝜑 → (1↑2) = (1 · 1)) |
26 | 24, 25 | oveq12d 5785 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1))) |
27 | 20, 22, 26 | 3eqtr3d 2178 | . . . . 5 ⊢ (𝜑 → ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1))) |
28 | 17 | mulid1d 7776 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 · 1) = 𝐴) |
29 | 28 | oveq2d 5783 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐴 · 1)) = (2 · 𝐴)) |
30 | 29 | oveq2d 5783 | . . . . . 6 ⊢ (𝜑 → ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · 𝐴))) |
31 | 18 | mulid1d 7776 | . . . . . 6 ⊢ (𝜑 → (1 · 1) = 1) |
32 | 30, 31 | oveq12d 5785 | . . . . 5 ⊢ (𝜑 → (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1)) |
33 | 27, 32 | eqtrd 2170 | . . . 4 ⊢ (𝜑 → ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1)) |
34 | 33 | breq1d 3934 | . . 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 3924 (class class class)co 5767 ℂcc 7611 1c1 7614 + caddc 7616 · cmul 7618 < clt 7793 ≤ cle 7794 2c2 8764 ℕ0cn0 8970 ↑cexp 10285 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-n0 8971 df-z 9048 df-uz 9320 df-seqfrec 10212 df-exp 10286 |
This theorem is referenced by: nn0opthlem2d 10460 |
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