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Mirrors > Home > MPE Home > Th. List > nn0opthlem1 | Structured version Visualization version GIF version |
Description: A rather pretty lemma for nn0opthi 12919. (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 11185 | . . . 4 ⊢ 1 ∈ ℕ0 | |
3 | 1, 2 | nn0addcli 11207 | . . 3 ⊢ (𝐴 + 1) ∈ ℕ0 |
4 | nn0opthlem1.2 | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
5 | 3, 4 | nn0le2msqi 12916 | . 2 ⊢ ((𝐴 + 1) ≤ 𝐶 ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶)) |
6 | nn0ltp1le 11312 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
7 | 1, 4, 6 | mp2an 704 | . 2 ⊢ (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶) |
8 | 1, 1 | nn0mulcli 11208 | . . . . 5 ⊢ (𝐴 · 𝐴) ∈ ℕ0 |
9 | 2nn0 11186 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 9, 1 | nn0mulcli 11208 | . . . . 5 ⊢ (2 · 𝐴) ∈ ℕ0 |
11 | 8, 10 | nn0addcli 11207 | . . . 4 ⊢ ((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0 |
12 | 4, 4 | nn0mulcli 11208 | . . . 4 ⊢ (𝐶 · 𝐶) ∈ ℕ0 |
13 | nn0ltp1le 11312 | . . . 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 704 | . . 3 ⊢ (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶)) |
15 | 1 | nn0cni 11181 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
16 | ax-1cn 9873 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
17 | 15, 16 | binom2i 12836 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) |
18 | 15, 16 | addcli 9923 | . . . . . . 7 ⊢ (𝐴 + 1) ∈ ℂ |
19 | 18 | sqvali 12805 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1)) |
20 | 15 | sqvali 12805 | . . . . . . . 8 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
21 | 20 | oveq1i 6559 | . . . . . . 7 ⊢ ((𝐴↑2) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) |
22 | 16 | sqvali 12805 | . . . . . . 7 ⊢ (1↑2) = (1 · 1) |
23 | 21, 22 | oveq12i 6561 | . . . . . 6 ⊢ (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
24 | 17, 19, 23 | 3eqtr3i 2640 | . . . . 5 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
25 | 15 | mulid1i 9921 | . . . . . . . 8 ⊢ (𝐴 · 1) = 𝐴 |
26 | 25 | oveq2i 6560 | . . . . . . 7 ⊢ (2 · (𝐴 · 1)) = (2 · 𝐴) |
27 | 26 | oveq2i 6560 | . . . . . 6 ⊢ ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · 𝐴)) |
28 | 16 | mulid1i 9921 | . . . . . 6 ⊢ (1 · 1) = 1 |
29 | 27, 28 | oveq12i 6561 | . . . . 5 ⊢ (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
30 | 24, 29 | eqtri 2632 | . . . 4 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
31 | 30 | breq1i 4590 | . . 3 ⊢ (((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶)) |
32 | 14, 31 | bitr4i 266 | . 2 ⊢ (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶)) |
33 | 5, 7, 32 | 3bitr4i 291 | 1 ⊢ (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 2c2 10947 ℕ0cn0 11169 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: nn0opthlem2 12918 |
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