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Mirrors > Home > ILE Home > Th. List > nn0opthlem1d | GIF version |
Description: A rather pretty lemma for nn0opth2 10311. (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 8845 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
3 | 2 | a1i 9 | . . . 4 ⊢ (𝜑 → 1 ∈ ℕ0) |
4 | 1, 3 | nn0addcld 8886 | . . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℕ0) |
5 | nn0opthlem1d.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℕ0) | |
6 | 4, 5 | nn0le2msqd 10306 | . 2 ⊢ (𝜑 → ((𝐴 + 1) ≤ 𝐶 ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶))) |
7 | nn0ltp1le 8968 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
8 | 1, 5, 7 | syl2anc 406 | . 2 ⊢ (𝜑 → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) |
9 | 1, 1 | nn0mulcld 8887 | . . . . 5 ⊢ (𝜑 → (𝐴 · 𝐴) ∈ ℕ0) |
10 | 2nn0 8846 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
11 | 10 | a1i 9 | . . . . . 6 ⊢ (𝜑 → 2 ∈ ℕ0) |
12 | 11, 1 | nn0mulcld 8887 | . . . . 5 ⊢ (𝜑 → (2 · 𝐴) ∈ ℕ0) |
13 | 9, 12 | nn0addcld 8886 | . . . 4 ⊢ (𝜑 → ((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0) |
14 | 5, 5 | nn0mulcld 8887 | . . . 4 ⊢ (𝜑 → (𝐶 · 𝐶) ∈ ℕ0) |
15 | nn0ltp1le 8968 | . . . 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 406 | . . 3 ⊢ (𝜑 → (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶))) |
17 | 1 | nn0cnd 8884 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
18 | 1cnd 7654 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
19 | binom2 10244 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2))) | |
20 | 17, 18, 19 | syl2anc 406 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2))) |
21 | 17, 18 | addcld 7657 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℂ) |
22 | 21 | sqvald 10262 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1))) |
23 | 17 | sqvald 10262 | . . . . . . . 8 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴)) |
24 | 23 | oveq1d 5721 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑2) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · (𝐴 · 1)))) |
25 | 18 | sqvald 10262 | . . . . . . 7 ⊢ (𝜑 → (1↑2) = (1 · 1)) |
26 | 24, 25 | oveq12d 5724 | . . . . . 6 ⊢ (𝜑 → (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1))) |
27 | 20, 22, 26 | 3eqtr3d 2140 | . . . . 5 ⊢ (𝜑 → ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1))) |
28 | 17 | mulid1d 7655 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 · 1) = 𝐴) |
29 | 28 | oveq2d 5722 | . . . . . . 7 ⊢ (𝜑 → (2 · (𝐴 · 1)) = (2 · 𝐴)) |
30 | 29 | oveq2d 5722 | . . . . . 6 ⊢ (𝜑 → ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · 𝐴))) |
31 | 18 | mulid1d 7655 | . . . . . 6 ⊢ (𝜑 → (1 · 1) = 1) |
32 | 30, 31 | oveq12d 5724 | . . . . 5 ⊢ (𝜑 → (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1)) |
33 | 27, 32 | eqtrd 2132 | . . . 4 ⊢ (𝜑 → ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1)) |
34 | 33 | breq1d 3885 | . . 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 1299 ∈ wcel 1448 class class class wbr 3875 (class class class)co 5706 ℂcc 7498 1c1 7501 + caddc 7503 · cmul 7505 < clt 7672 ≤ cle 7673 2c2 8629 ℕ0cn0 8829 ↑cexp 10133 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-n0 8830 df-z 8907 df-uz 9177 df-seqfrec 10060 df-exp 10134 |
This theorem is referenced by: nn0opthlem2d 10308 |
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