Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > exple2lt6 | Structured version Visualization version GIF version |
Description: A nonnegative integer to the power of itself is less than 6 if it is less than or equal to 2. (Contributed by AV, 16-Mar-2019.) |
Ref | Expression |
---|---|
exple2lt6 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁↑𝑁) < 6) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0le2is012 12034 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) | |
2 | id 22 | . . . . 5 ⊢ (𝑁 = 0 → 𝑁 = 0) | |
3 | 2, 2 | oveq12d 7163 | . . . 4 ⊢ (𝑁 = 0 → (𝑁↑𝑁) = (0↑0)) |
4 | 0exp0e1 13422 | . . . . 5 ⊢ (0↑0) = 1 | |
5 | 1lt6 11810 | . . . . 5 ⊢ 1 < 6 | |
6 | 4, 5 | eqbrtri 5078 | . . . 4 ⊢ (0↑0) < 6 |
7 | 3, 6 | eqbrtrdi 5096 | . . 3 ⊢ (𝑁 = 0 → (𝑁↑𝑁) < 6) |
8 | id 22 | . . . . 5 ⊢ (𝑁 = 1 → 𝑁 = 1) | |
9 | 8, 8 | oveq12d 7163 | . . . 4 ⊢ (𝑁 = 1 → (𝑁↑𝑁) = (1↑1)) |
10 | ax-1cn 10583 | . . . . . 6 ⊢ 1 ∈ ℂ | |
11 | exp1 13423 | . . . . . 6 ⊢ (1 ∈ ℂ → (1↑1) = 1) | |
12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ (1↑1) = 1 |
13 | 12, 5 | eqbrtri 5078 | . . . 4 ⊢ (1↑1) < 6 |
14 | 9, 13 | eqbrtrdi 5096 | . . 3 ⊢ (𝑁 = 1 → (𝑁↑𝑁) < 6) |
15 | id 22 | . . . . 5 ⊢ (𝑁 = 2 → 𝑁 = 2) | |
16 | 15, 15 | oveq12d 7163 | . . . 4 ⊢ (𝑁 = 2 → (𝑁↑𝑁) = (2↑2)) |
17 | sq2 13548 | . . . . 5 ⊢ (2↑2) = 4 | |
18 | 4lt6 11807 | . . . . 5 ⊢ 4 < 6 | |
19 | 17, 18 | eqbrtri 5078 | . . . 4 ⊢ (2↑2) < 6 |
20 | 16, 19 | eqbrtrdi 5096 | . . 3 ⊢ (𝑁 = 2 → (𝑁↑𝑁) < 6) |
21 | 7, 14, 20 | 3jaoi 1419 | . 2 ⊢ ((𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2) → (𝑁↑𝑁) < 6) |
22 | 1, 21 | syl 17 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁↑𝑁) < 6) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 < clt 10663 ≤ cle 10664 2c2 11680 4c4 11682 6c6 11684 ℕ0cn0 11885 ↑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-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13358 df-exp 13418 |
This theorem is referenced by: pgrple2abl 44341 |
Copyright terms: Public domain | W3C validator |