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Mirrors > Home > ILE Home > Th. List > nn0lele2xi | GIF version |
Description: 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
nn0lele2x.1 | ⊢ 𝑀 ∈ ℕ0 |
nn0lele2x.2 | ⊢ 𝑁 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0lele2xi | ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ (2 · 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0lele2x.1 | . . 3 ⊢ 𝑀 ∈ ℕ0 | |
2 | 1 | nn0le2xi 8338 | . 2 ⊢ 𝑀 ≤ (2 · 𝑀) |
3 | nn0lele2x.2 | . . . 4 ⊢ 𝑁 ∈ ℕ0 | |
4 | 3 | nn0rei 8299 | . . 3 ⊢ 𝑁 ∈ ℝ |
5 | 1 | nn0rei 8299 | . . 3 ⊢ 𝑀 ∈ ℝ |
6 | 2re 8109 | . . . 4 ⊢ 2 ∈ ℝ | |
7 | 6, 5 | remulcli 7133 | . . 3 ⊢ (2 · 𝑀) ∈ ℝ |
8 | 4, 5, 7 | letri 7218 | . 2 ⊢ ((𝑁 ≤ 𝑀 ∧ 𝑀 ≤ (2 · 𝑀)) → 𝑁 ≤ (2 · 𝑀)) |
9 | 2, 8 | mpan2 415 | 1 ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ (2 · 𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 · cmul 6986 ≤ cle 7154 2c2 8089 ℕ0cn0 8288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-iota 4887 df-fv 4930 df-ov 5535 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-inn 8040 df-2 8098 df-n0 8289 |
This theorem is referenced by: (None) |
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