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Mirrors > Home > MPE Home > Th. List > nn0opthlem1 | Structured version Visualization version GIF version |
Description: A rather pretty lemma for nn0opthi 13982. (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 12247 | . . . 4 ⊢ 1 ∈ ℕ0 | |
3 | 1, 2 | nn0addcli 12268 | . . 3 ⊢ (𝐴 + 1) ∈ ℕ0 |
4 | nn0opthlem1.2 | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
5 | 3, 4 | nn0le2msqi 13979 | . 2 ⊢ ((𝐴 + 1) ≤ 𝐶 ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶)) |
6 | nn0ltp1le 12376 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
7 | 1, 4, 6 | mp2an 689 | . 2 ⊢ (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶) |
8 | 1, 1 | nn0mulcli 12269 | . . . . 5 ⊢ (𝐴 · 𝐴) ∈ ℕ0 |
9 | 2nn0 12248 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 9, 1 | nn0mulcli 12269 | . . . . 5 ⊢ (2 · 𝐴) ∈ ℕ0 |
11 | 8, 10 | nn0addcli 12268 | . . . 4 ⊢ ((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0 |
12 | 4, 4 | nn0mulcli 12269 | . . . 4 ⊢ (𝐶 · 𝐶) ∈ ℕ0 |
13 | nn0ltp1le 12376 | . . . 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 689 | . . 3 ⊢ (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶)) |
15 | 1 | nn0cni 12243 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
16 | ax-1cn 10927 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
17 | 15, 16 | binom2i 13926 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) |
18 | 15, 16 | addcli 10979 | . . . . . . 7 ⊢ (𝐴 + 1) ∈ ℂ |
19 | 18 | sqvali 13895 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1)) |
20 | 15 | sqvali 13895 | . . . . . . . 8 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
21 | 20 | oveq1i 7287 | . . . . . . 7 ⊢ ((𝐴↑2) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) |
22 | 16 | sqvali 13895 | . . . . . . 7 ⊢ (1↑2) = (1 · 1) |
23 | 21, 22 | oveq12i 7289 | . . . . . 6 ⊢ (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
24 | 17, 19, 23 | 3eqtr3i 2774 | . . . . 5 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
25 | 15 | mulid1i 10977 | . . . . . . . 8 ⊢ (𝐴 · 1) = 𝐴 |
26 | 25 | oveq2i 7288 | . . . . . . 7 ⊢ (2 · (𝐴 · 1)) = (2 · 𝐴) |
27 | 26 | oveq2i 7288 | . . . . . 6 ⊢ ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · 𝐴)) |
28 | 16 | mulid1i 10977 | . . . . . 6 ⊢ (1 · 1) = 1 |
29 | 27, 28 | oveq12i 7289 | . . . . 5 ⊢ (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
30 | 24, 29 | eqtri 2766 | . . . 4 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
31 | 30 | breq1i 5083 | . . 3 ⊢ (((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶)) |
32 | 14, 31 | bitr4i 277 | . 2 ⊢ (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶)) |
33 | 5, 7, 32 | 3bitr4i 303 | 1 ⊢ (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2106 class class class wbr 5076 (class class class)co 7277 1c1 10870 + caddc 10872 · cmul 10874 < clt 11007 ≤ cle 11008 2c2 12026 ℕ0cn0 12231 ↑cexp 13780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-n0 12232 df-z 12318 df-uz 12581 df-seq 13720 df-exp 13781 |
This theorem is referenced by: nn0opthlem2 13981 |
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