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Mirrors > Home > MPE Home > Th. List > nn0opthlem1 | Structured version Visualization version GIF version |
Description: A rather pretty lemma for nn0opthi 14247. (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 12504 | . . . 4 ⊢ 1 ∈ ℕ0 | |
3 | 1, 2 | nn0addcli 12525 | . . 3 ⊢ (𝐴 + 1) ∈ ℕ0 |
4 | nn0opthlem1.2 | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
5 | 3, 4 | nn0le2msqi 14244 | . 2 ⊢ ((𝐴 + 1) ≤ 𝐶 ↔ ((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶)) |
6 | nn0ltp1le 12636 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
7 | 1, 4, 6 | mp2an 691 | . 2 ⊢ (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶) |
8 | 1, 1 | nn0mulcli 12526 | . . . . 5 ⊢ (𝐴 · 𝐴) ∈ ℕ0 |
9 | 2nn0 12505 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
10 | 9, 1 | nn0mulcli 12526 | . . . . 5 ⊢ (2 · 𝐴) ∈ ℕ0 |
11 | 8, 10 | nn0addcli 12525 | . . . 4 ⊢ ((𝐴 · 𝐴) + (2 · 𝐴)) ∈ ℕ0 |
12 | 4, 4 | nn0mulcli 12526 | . . . 4 ⊢ (𝐶 · 𝐶) ∈ ℕ0 |
13 | nn0ltp1le 12636 | . . . 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 691 | . . 3 ⊢ (((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶)) |
15 | 1 | nn0cni 12500 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
16 | ax-1cn 11182 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
17 | 15, 16 | binom2i 14193 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) |
18 | 15, 16 | addcli 11236 | . . . . . . 7 ⊢ (𝐴 + 1) ∈ ℂ |
19 | 18 | sqvali 14161 | . . . . . 6 ⊢ ((𝐴 + 1)↑2) = ((𝐴 + 1) · (𝐴 + 1)) |
20 | 15 | sqvali 14161 | . . . . . . . 8 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
21 | 20 | oveq1i 7424 | . . . . . . 7 ⊢ ((𝐴↑2) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) |
22 | 16 | sqvali 14161 | . . . . . . 7 ⊢ (1↑2) = (1 · 1) |
23 | 21, 22 | oveq12i 7426 | . . . . . 6 ⊢ (((𝐴↑2) + (2 · (𝐴 · 1))) + (1↑2)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
24 | 17, 19, 23 | 3eqtr3i 2763 | . . . . 5 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) |
25 | 15 | mulridi 11234 | . . . . . . . 8 ⊢ (𝐴 · 1) = 𝐴 |
26 | 25 | oveq2i 7425 | . . . . . . 7 ⊢ (2 · (𝐴 · 1)) = (2 · 𝐴) |
27 | 26 | oveq2i 7425 | . . . . . 6 ⊢ ((𝐴 · 𝐴) + (2 · (𝐴 · 1))) = ((𝐴 · 𝐴) + (2 · 𝐴)) |
28 | 16 | mulridi 11234 | . . . . . 6 ⊢ (1 · 1) = 1 |
29 | 27, 28 | oveq12i 7426 | . . . . 5 ⊢ (((𝐴 · 𝐴) + (2 · (𝐴 · 1))) + (1 · 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
30 | 24, 29 | eqtri 2755 | . . . 4 ⊢ ((𝐴 + 1) · (𝐴 + 1)) = (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) |
31 | 30 | breq1i 5149 | . . 3 ⊢ (((𝐴 + 1) · (𝐴 + 1)) ≤ (𝐶 · 𝐶) ↔ (((𝐴 · 𝐴) + (2 · 𝐴)) + 1) ≤ (𝐶 · 𝐶)) |
32 | 14, 31 | bitr4i 278 | . 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 2099 class class class wbr 5142 (class class class)co 7414 1c1 11125 + caddc 11127 · cmul 11129 < clt 11264 ≤ cle 11265 2c2 12283 ℕ0cn0 12488 ↑cexp 14044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-z 12575 df-uz 12839 df-seq 13985 df-exp 14045 |
This theorem is referenced by: nn0opthlem2 14246 |
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