![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nn0opthlem1 | Structured version Visualization version GIF version |
Description: A rather pretty lemma for nn0opthi 13357. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
nn0opthlem1.1 | ⊢ 𝐴 ∈ ℕ0 |
nn0opthlem1.2 | ⊢ 𝐶 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0opthlem1 | ⊢ (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0opthlem1.1 | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
2 | 1nn0 11643 | . . . 4 ⊢ 1 ∈ ℕ0 | |
3 | 1, 2 | nn0addcli 11664 | . . 3 ⊢ (𝐴 + 1) ∈ ℕ0 |
4 | nn0opthlem1.2 | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
5 | 3, 4 | nn0le2msqi 13354 | . 2 ⊢ ((𝐴 + 1) ≤ 𝐶 ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶)) |
6 | nn0ltp1le 11770 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
7 | 1, 4, 6 | mp2an 683 | . 2 ⊢ (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶) |
8 | 1, 1 | nn0mulcli 11665 | . . . . 5 ⊢ (𝐴 · 𝐴) ∈ ℕ0 |
9 | 2nn0 11644 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 9, 1 | nn0mulcli 11665 | . . . . 5 ⊢ (2 · 𝐴) ∈ ℕ0 |
11 | 8, 10 | nn0addcli 11664 | . . . 4 ⊢ ((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0 |
12 | 4, 4 | nn0mulcli 11665 | . . . 4 ⊢ (𝐶 · 𝐶) ∈ ℕ0 |
13 | nn0ltp1le 11770 | . . . 4 ⊢ ((((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0 ∧ (𝐶 · 𝐶) ∈ ℕ0) → (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶))) | |
14 | 11, 12, 13 | mp2an 683 | . . 3 ⊢ (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶)) |
15 | 1 | nn0cni 11638 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
16 | ax-1cn 10317 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
17 | 15, 16 | binom2i 13275 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) |
18 | 15, 16 | addcli 10370 | . . . . . . 7 ⊢ (𝐴 + 1) ∈ ℂ |
19 | 18 | sqvali 13244 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1)) |
20 | 15 | sqvali 13244 | . . . . . . . 8 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
21 | 20 | oveq1i 6920 | . . . . . . 7 ⊢ ((𝐴↑2) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) |
22 | 16 | sqvali 13244 | . . . . . . 7 ⊢ (1↑2) = (1 · 1) |
23 | 21, 22 | oveq12i 6922 | . . . . . 6 ⊢ (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
24 | 17, 19, 23 | 3eqtr3i 2857 | . . . . 5 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
25 | 15 | mulid1i 10368 | . . . . . . . 8 ⊢ (𝐴 · 1) = 𝐴 |
26 | 25 | oveq2i 6921 | . . . . . . 7 ⊢ (2 · (𝐴 · 1)) = (2 · 𝐴) |
27 | 26 | oveq2i 6921 | . . . . . 6 ⊢ ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · 𝐴)) |
28 | 16 | mulid1i 10368 | . . . . . 6 ⊢ (1 · 1) = 1 |
29 | 27, 28 | oveq12i 6922 | . . . . 5 ⊢ (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
30 | 24, 29 | eqtri 2849 | . . . 4 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
31 | 30 | breq1i 4882 | . . 3 ⊢ (((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶)) |
32 | 14, 31 | bitr4i 270 | . 2 ⊢ (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶)) |
33 | 5, 7, 32 | 3bitr4i 295 | 1 ⊢ (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2164 class class class wbr 4875 (class class class)co 6910 1c1 10260 + caddc 10262 · cmul 10264 < clt 10398 ≤ cle 10399 2c2 11413 ℕ0cn0 11625 ↑cexp 13161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-seq 13103 df-exp 13162 |
This theorem is referenced by: nn0opthlem2 13356 |
Copyright terms: Public domain | W3C validator |