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Mirrors > Home > ILE Home > Th. List > nnlesq | GIF version |
Description: A positive integer is less than or equal to its square. For general integers, see zzlesq 10729. (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 8962 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | 1 | mulridd 8009 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 · 1) = 𝑁) |
3 | nnge1 8977 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
4 | 1red 8007 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ) | |
5 | nnre 8961 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | nngt0 8979 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
7 | lemul2 8849 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (1 ≤ 𝑁 ↔ (𝑁 · 1) ≤ (𝑁 · 𝑁))) | |
8 | 4, 5, 5, 6, 7 | syl112anc 1253 | . . . 4 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 ↔ (𝑁 · 1) ≤ (𝑁 · 𝑁))) |
9 | 3, 8 | mpbid 147 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 · 1) ≤ (𝑁 · 𝑁)) |
10 | 2, 9 | eqbrtrrd 4045 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 · 𝑁)) |
11 | sqval 10618 | . . 3 ⊢ (𝑁 ∈ ℂ → (𝑁↑2) = (𝑁 · 𝑁)) | |
12 | 1, 11 | syl 14 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) = (𝑁 · 𝑁)) |
13 | 10, 12 | breqtrrd 4049 | 1 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁↑2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 (class class class)co 5900 ℂcc 7844 ℝcr 7845 0cc0 7846 1c1 7847 · cmul 7851 < clt 8027 ≤ cle 8028 ℕcn 8954 2c2 9005 ↑cexp 10559 |
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 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-po 4317 df-iso 4318 df-iord 4387 df-on 4389 df-ilim 4390 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-recs 6334 df-frec 6420 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-reap 8567 df-ap 8574 df-div 8665 df-inn 8955 df-2 9013 df-n0 9212 df-z 9289 df-uz 9564 df-seqfrec 10485 df-exp 10560 |
This theorem is referenced by: zzlesq 10729 |
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