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Mirrors > Home > MPE Home > Th. List > nnlesq | Structured version Visualization version 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 11634 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | 1 | mulid1d 10646 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 · 1) = 𝑁) |
3 | nnge1 11653 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
4 | 1red 10630 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ) | |
5 | nnre 11633 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | nngt0 11656 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
7 | lemul2 11481 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (1 ≤ 𝑁 ↔ (𝑁 · 1) ≤ (𝑁 · 𝑁))) | |
8 | 4, 5, 5, 6, 7 | syl112anc 1366 | . . . 4 ⊢ (𝑁 ∈ ℕ → (1 ≤ 𝑁 ↔ (𝑁 · 1) ≤ (𝑁 · 𝑁))) |
9 | 3, 8 | mpbid 233 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 · 1) ≤ (𝑁 · 𝑁)) |
10 | 2, 9 | eqbrtrrd 5081 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 · 𝑁)) |
11 | sqval 13469 | . . 3 ⊢ (𝑁 ∈ ℂ → (𝑁↑2) = (𝑁 · 𝑁)) | |
12 | 1, 11 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁↑2) = (𝑁 · 𝑁)) |
13 | 10, 12 | breqtrrd 5085 | 1 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 · cmul 10530 < clt 10663 ≤ cle 10664 ℕcn 11626 2c2 11680 ↑cexp 13417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13358 df-exp 13418 |
This theorem is referenced by: prmreclem1 16240 |
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