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Mirrors > Home > MPE Home > Th. List > nn0opthlem2 | Structured version Visualization version GIF version |
Description: Lemma for nn0opthi 13629. (Contributed by Raph Levien, 10-Dec-2002.) (Revised by Scott Fenton, 8-Sep-2010.) |
Ref | Expression |
---|---|
nn0opth.1 | ⊢ 𝐴 ∈ ℕ0 |
nn0opth.2 | ⊢ 𝐵 ∈ ℕ0 |
nn0opth.3 | ⊢ 𝐶 ∈ ℕ0 |
nn0opth.4 | ⊢ 𝐷 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0opthlem2 | ⊢ ((𝐴 + 𝐵) < 𝐶 → ((𝐶 · 𝐶) + 𝐷) ≠ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0opth.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
2 | nn0opth.2 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
3 | 1, 2 | nn0addcli 11933 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℕ0 |
4 | nn0opth.3 | . . . 4 ⊢ 𝐶 ∈ ℕ0 | |
5 | 3, 4 | nn0opthlem1 13627 | . . 3 ⊢ ((𝐴 + 𝐵) < 𝐶 ↔ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵))) < (𝐶 · 𝐶)) |
6 | 2 | nn0rei 11907 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
7 | 6, 1 | nn0addge2i 11945 | . . . . 5 ⊢ 𝐵 ≤ (𝐴 + 𝐵) |
8 | 3, 2 | nn0lele2xi 11951 | . . . . . 6 ⊢ (𝐵 ≤ (𝐴 + 𝐵) → 𝐵 ≤ (2 · (𝐴 + 𝐵))) |
9 | 2re 11710 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
10 | 3 | nn0rei 11907 | . . . . . . . 8 ⊢ (𝐴 + 𝐵) ∈ ℝ |
11 | 9, 10 | remulcli 10656 | . . . . . . 7 ⊢ (2 · (𝐴 + 𝐵)) ∈ ℝ |
12 | 10, 10 | remulcli 10656 | . . . . . . 7 ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) ∈ ℝ |
13 | 6, 11, 12 | leadd2i 11195 | . . . . . 6 ⊢ (𝐵 ≤ (2 · (𝐴 + 𝐵)) ↔ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) ≤ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵)))) |
14 | 8, 13 | sylib 220 | . . . . 5 ⊢ (𝐵 ≤ (𝐴 + 𝐵) → (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) ≤ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵)))) |
15 | 7, 14 | ax-mp 5 | . . . 4 ⊢ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) ≤ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵))) |
16 | 12, 6 | readdcli 10655 | . . . . 5 ⊢ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) ∈ ℝ |
17 | 12, 11 | readdcli 10655 | . . . . 5 ⊢ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵))) ∈ ℝ |
18 | 4 | nn0rei 11907 | . . . . . 6 ⊢ 𝐶 ∈ ℝ |
19 | 18, 18 | remulcli 10656 | . . . . 5 ⊢ (𝐶 · 𝐶) ∈ ℝ |
20 | 16, 17, 19 | lelttri 10766 | . . . 4 ⊢ (((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) ≤ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵))) ∧ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵))) < (𝐶 · 𝐶)) → (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < (𝐶 · 𝐶)) |
21 | 15, 20 | mpan 688 | . . 3 ⊢ ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵))) < (𝐶 · 𝐶) → (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < (𝐶 · 𝐶)) |
22 | 5, 21 | sylbi 219 | . 2 ⊢ ((𝐴 + 𝐵) < 𝐶 → (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < (𝐶 · 𝐶)) |
23 | nn0opth.4 | . . . 4 ⊢ 𝐷 ∈ ℕ0 | |
24 | 19, 23 | nn0addge1i 11944 | . . 3 ⊢ (𝐶 · 𝐶) ≤ ((𝐶 · 𝐶) + 𝐷) |
25 | 23 | nn0rei 11907 | . . . . 5 ⊢ 𝐷 ∈ ℝ |
26 | 19, 25 | readdcli 10655 | . . . 4 ⊢ ((𝐶 · 𝐶) + 𝐷) ∈ ℝ |
27 | 16, 19, 26 | ltletri 10767 | . . 3 ⊢ (((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < (𝐶 · 𝐶) ∧ (𝐶 · 𝐶) ≤ ((𝐶 · 𝐶) + 𝐷)) → (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < ((𝐶 · 𝐶) + 𝐷)) |
28 | 24, 27 | mpan2 689 | . 2 ⊢ ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < (𝐶 · 𝐶) → (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < ((𝐶 · 𝐶) + 𝐷)) |
29 | 16, 26 | ltnei 10763 | . 2 ⊢ ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < ((𝐶 · 𝐶) + 𝐷) → ((𝐶 · 𝐶) + 𝐷) ≠ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵)) |
30 | 22, 28, 29 | 3syl 18 | 1 ⊢ ((𝐴 + 𝐵) < 𝐶 → ((𝐶 · 𝐶) + 𝐷) ≠ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 (class class class)co 7155 + caddc 10539 · cmul 10541 < clt 10674 ≤ cle 10675 2c2 11691 ℕ0cn0 11896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-seq 13369 df-exp 13429 |
This theorem is referenced by: nn0opthi 13629 |
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