Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nnlesq | GIF version |
Description: A positive integer is less than or equal to its square. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
nnlesq | ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nncn 8047 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | 1 | mulid1d 7136 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 · 1) = 𝑁) |
3 | nnge1 8062 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
4 | 1red 7134 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ) | |
5 | nnre 8046 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | nngt0 8064 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
7 | lemul2 7935 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (1 ≤ 𝑁 ↔ (𝑁 · 1) ≤ (𝑁 · 𝑁))) | |
8 | 4, 5, 5, 6, 7 | syl112anc 1173 | . . . 4 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 ↔ (𝑁 · 1) ≤ (𝑁 · 𝑁))) |
9 | 3, 8 | mpbid 145 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 · 1) ≤ (𝑁 · 𝑁)) |
10 | 2, 9 | eqbrtrrd 3807 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 · 𝑁)) |
11 | sqval 9534 | . . 3 ⊢ (𝑁 ∈ ℂ → (𝑁↑2) = (𝑁 · 𝑁)) | |
12 | 1, 11 | syl 14 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) = (𝑁 · 𝑁)) |
13 | 10, 12 | breqtrrd 3811 | 1 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁↑2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℂcc 6979 ℝcr 6980 0cc0 6981 1c1 6982 · cmul 6986 < clt 7153 ≤ cle 7154 ℕcn 8039 2c2 8089 ↑cexp 9475 |
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-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 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-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 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-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-n0 8289 df-z 8352 df-uz 8620 df-iseq 9432 df-iexp 9476 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |