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Mirrors > Home > MPE Home > Th. List > nn0opthlem2 | Structured version Visualization version GIF version |
Description: Lemma for nn0opthi 14306. (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 12561 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℕ0 |
4 | nn0opth.3 | . . . 4 ⊢ 𝐶 ∈ ℕ0 | |
5 | 3, 4 | nn0opthlem1 14304 | . . 3 ⊢ ((𝐴 + 𝐵) < 𝐶 ↔ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵))) < (𝐶 · 𝐶)) |
6 | 2 | nn0rei 12535 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
7 | 6, 1 | nn0addge2i 12573 | . . . . 5 ⊢ 𝐵 ≤ (𝐴 + 𝐵) |
8 | 3, 2 | nn0lele2xi 12580 | . . . . . 6 ⊢ (𝐵 ≤ (𝐴 + 𝐵) → 𝐵 ≤ (2 · (𝐴 + 𝐵))) |
9 | 2re 12338 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
10 | 3 | nn0rei 12535 | . . . . . . . 8 ⊢ (𝐴 + 𝐵) ∈ ℝ |
11 | 9, 10 | remulcli 11275 | . . . . . . 7 ⊢ (2 · (𝐴 + 𝐵)) ∈ ℝ |
12 | 10, 10 | remulcli 11275 | . . . . . . 7 ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) ∈ ℝ |
13 | 6, 11, 12 | leadd2i 11817 | . . . . . 6 ⊢ (𝐵 ≤ (2 · (𝐴 + 𝐵)) ↔ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) ≤ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵)))) |
14 | 8, 13 | sylib 218 | . . . . 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 11274 | . . . . 5 ⊢ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) ∈ ℝ |
17 | 12, 11 | readdcli 11274 | . . . . 5 ⊢ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵))) ∈ ℝ |
18 | 4 | nn0rei 12535 | . . . . . 6 ⊢ 𝐶 ∈ ℝ |
19 | 18, 18 | remulcli 11275 | . . . . 5 ⊢ (𝐶 · 𝐶) ∈ ℝ |
20 | 16, 17, 19 | lelttri 11386 | . . . 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 690 | . . 3 ⊢ ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + (2 · (𝐴 + 𝐵))) < (𝐶 · 𝐶) → (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < (𝐶 · 𝐶)) |
22 | 5, 21 | sylbi 217 | . 2 ⊢ ((𝐴 + 𝐵) < 𝐶 → (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < (𝐶 · 𝐶)) |
23 | nn0opth.4 | . . . 4 ⊢ 𝐷 ∈ ℕ0 | |
24 | 19, 23 | nn0addge1i 12572 | . . 3 ⊢ (𝐶 · 𝐶) ≤ ((𝐶 · 𝐶) + 𝐷) |
25 | 23 | nn0rei 12535 | . . . . 5 ⊢ 𝐷 ∈ ℝ |
26 | 19, 25 | readdcli 11274 | . . . 4 ⊢ ((𝐶 · 𝐶) + 𝐷) ∈ ℝ |
27 | 16, 19, 26 | ltletri 11387 | . . 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 691 | . 2 ⊢ ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < (𝐶 · 𝐶) → (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) < ((𝐶 · 𝐶) + 𝐷)) |
29 | 16, 26 | ltnei 11383 | . 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 2106 ≠ wne 2938 class class class wbr 5148 (class class class)co 7431 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 2c2 12319 ℕ0cn0 12524 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-exp 14100 |
This theorem is referenced by: nn0opthi 14306 |
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