Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > expnass | GIF version |
Description: A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.) |
Ref | Expression |
---|---|
expnass | ⊢ ((3↑3)↑3) < (3↑(3↑3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3cn 8795 | . . 3 ⊢ 3 ∈ ℂ | |
2 | 3nn0 8995 | . . 3 ⊢ 3 ∈ ℕ0 | |
3 | expmul 10338 | . . 3 ⊢ ((3 ∈ ℂ ∧ 3 ∈ ℕ0 ∧ 3 ∈ ℕ0) → (3↑(3 · 3)) = ((3↑3)↑3)) | |
4 | 1, 2, 2, 3 | mp3an 1315 | . 2 ⊢ (3↑(3 · 3)) = ((3↑3)↑3) |
5 | 3re 8794 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 2, 2 | nn0mulcli 9015 | . . . 4 ⊢ (3 · 3) ∈ ℕ0 |
7 | 6 | nn0zi 9076 | . . 3 ⊢ (3 · 3) ∈ ℤ |
8 | 2, 2 | nn0expcli 10319 | . . . 4 ⊢ (3↑3) ∈ ℕ0 |
9 | 8 | nn0zi 9076 | . . 3 ⊢ (3↑3) ∈ ℤ |
10 | 1lt3 8891 | . . . 4 ⊢ 1 < 3 | |
11 | 1 | sqvali 10372 | . . . . 5 ⊢ (3↑2) = (3 · 3) |
12 | 2z 9082 | . . . . . 6 ⊢ 2 ∈ ℤ | |
13 | 3z 9083 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | 2lt3 8890 | . . . . . . 7 ⊢ 2 < 3 | |
15 | ltexp2a 10345 | . . . . . . 7 ⊢ (((3 ∈ ℝ ∧ 2 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (1 < 3 ∧ 2 < 3)) → (3↑2) < (3↑3)) | |
16 | 10, 14, 15 | mpanr12 435 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℤ ∧ 3 ∈ ℤ) → (3↑2) < (3↑3)) |
17 | 5, 12, 13, 16 | mp3an 1315 | . . . . 5 ⊢ (3↑2) < (3↑3) |
18 | 11, 17 | eqbrtrri 3951 | . . . 4 ⊢ (3 · 3) < (3↑3) |
19 | ltexp2a 10345 | . . . 4 ⊢ (((3 ∈ ℝ ∧ (3 · 3) ∈ ℤ ∧ (3↑3) ∈ ℤ) ∧ (1 < 3 ∧ (3 · 3) < (3↑3))) → (3↑(3 · 3)) < (3↑(3↑3))) | |
20 | 10, 18, 19 | mpanr12 435 | . . 3 ⊢ ((3 ∈ ℝ ∧ (3 · 3) ∈ ℤ ∧ (3↑3) ∈ ℤ) → (3↑(3 · 3)) < (3↑(3↑3))) |
21 | 5, 7, 9, 20 | mp3an 1315 | . 2 ⊢ (3↑(3 · 3)) < (3↑(3↑3)) |
22 | 4, 21 | eqbrtrri 3951 | 1 ⊢ ((3↑3)↑3) < (3↑(3↑3)) |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 962 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℂcc 7618 ℝcr 7619 1c1 7621 · cmul 7625 < clt 7800 2c2 8771 3c3 8772 ℕ0cn0 8977 ℤcz 9054 ↑cexp 10292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-seqfrec 10219 df-exp 10293 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |