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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 9290 | . 2 ⊢ 𝑀 ≤ (2 · 𝑀) |
3 | nn0lele2x.2 | . . . 4 ⊢ 𝑁 ∈ ℕ0 | |
4 | 3 | nn0rei 9251 | . . 3 ⊢ 𝑁 ∈ ℝ |
5 | 1 | nn0rei 9251 | . . 3 ⊢ 𝑀 ∈ ℝ |
6 | 2re 9052 | . . . 4 ⊢ 2 ∈ ℝ | |
7 | 6, 5 | remulcli 8033 | . . 3 ⊢ (2 · 𝑀) ∈ ℝ |
8 | 4, 5, 7 | letri 8127 | . 2 ⊢ ((𝑁 ≤ 𝑀 ∧ 𝑀 ≤ (2 · 𝑀)) → 𝑁 ≤ (2 · 𝑀)) |
9 | 2, 8 | mpan2 425 | 1 ⊢ (𝑁 ≤ 𝑀 → 𝑁 ≤ (2 · 𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2164 class class class wbr 4029 (class class class)co 5918 · cmul 7877 ≤ cle 8055 2c2 9033 ℕ0cn0 9240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-ltadd 7988 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-xp 4665 df-cnv 4667 df-iota 5215 df-fv 5262 df-ov 5921 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-inn 8983 df-2 9041 df-n0 9241 |
This theorem is referenced by: (None) |
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