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Mirrors > Home > MPE Home > Th. List > nn0opthlem1 | Structured version Visualization version GIF version |
Description: A rather pretty lemma for nn0opthi 13306. (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 11594 | . . . 4 ⊢ 1 ∈ ℕ0 | |
3 | 1, 2 | nn0addcli 11615 | . . 3 ⊢ (𝐴 + 1) ∈ ℕ0 |
4 | nn0opthlem1.2 | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
5 | 3, 4 | nn0le2msqi 13303 | . 2 ⊢ ((𝐴 + 1) ≤ 𝐶 ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶)) |
6 | nn0ltp1le 11721 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
7 | 1, 4, 6 | mp2an 684 | . 2 ⊢ (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶) |
8 | 1, 1 | nn0mulcli 11616 | . . . . 5 ⊢ (𝐴 · 𝐴) ∈ ℕ0 |
9 | 2nn0 11595 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 9, 1 | nn0mulcli 11616 | . . . . 5 ⊢ (2 · 𝐴) ∈ ℕ0 |
11 | 8, 10 | nn0addcli 11615 | . . . 4 ⊢ ((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0 |
12 | 4, 4 | nn0mulcli 11616 | . . . 4 ⊢ (𝐶 · 𝐶) ∈ ℕ0 |
13 | nn0ltp1le 11721 | . . . 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 684 | . . 3 ⊢ (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶)) |
15 | 1 | nn0cni 11589 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
16 | ax-1cn 10280 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
17 | 15, 16 | binom2i 13224 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) |
18 | 15, 16 | addcli 10333 | . . . . . . 7 ⊢ (𝐴 + 1) ∈ ℂ |
19 | 18 | sqvali 13193 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1)) |
20 | 15 | sqvali 13193 | . . . . . . . 8 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
21 | 20 | oveq1i 6886 | . . . . . . 7 ⊢ ((𝐴↑2) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) |
22 | 16 | sqvali 13193 | . . . . . . 7 ⊢ (1↑2) = (1 · 1) |
23 | 21, 22 | oveq12i 6888 | . . . . . 6 ⊢ (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
24 | 17, 19, 23 | 3eqtr3i 2827 | . . . . 5 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
25 | 15 | mulid1i 10331 | . . . . . . . 8 ⊢ (𝐴 · 1) = 𝐴 |
26 | 25 | oveq2i 6887 | . . . . . . 7 ⊢ (2 · (𝐴 · 1)) = (2 · 𝐴) |
27 | 26 | oveq2i 6887 | . . . . . 6 ⊢ ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · 𝐴)) |
28 | 16 | mulid1i 10331 | . . . . . 6 ⊢ (1 · 1) = 1 |
29 | 27, 28 | oveq12i 6888 | . . . . 5 ⊢ (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
30 | 24, 29 | eqtri 2819 | . . . 4 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
31 | 30 | breq1i 4848 | . . 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 2157 class class class wbr 4841 (class class class)co 6876 1c1 10223 + caddc 10225 · cmul 10227 < clt 10361 ≤ cle 10362 2c2 11364 ℕ0cn0 11576 ↑cexp 13110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-n0 11577 df-z 11663 df-uz 11927 df-seq 13052 df-exp 13111 |
This theorem is referenced by: nn0opthlem2 13305 |
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