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Mirrors > Home > MPE Home > Th. List > expnass | Structured version Visualization version 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 12331 | . . 3 ⊢ 3 ∈ ℂ | |
2 | 3nn0 12528 | . . 3 ⊢ 3 ∈ ℕ0 | |
3 | expmul 14112 | . . 3 ⊢ ((3 ∈ ℂ ∧ 3 ∈ ℕ0 ∧ 3 ∈ ℕ0) → (3↑(3 · 3)) = ((3↑3)↑3)) | |
4 | 1, 2, 2, 3 | mp3an 1457 | . 2 ⊢ (3↑(3 · 3)) = ((3↑3)↑3) |
5 | 3re 12330 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 2, 2 | nn0mulcli 12548 | . . . 4 ⊢ (3 · 3) ∈ ℕ0 |
7 | 6 | nn0zi 12625 | . . 3 ⊢ (3 · 3) ∈ ℤ |
8 | 2, 2 | nn0expcli 14093 | . . . 4 ⊢ (3↑3) ∈ ℕ0 |
9 | 8 | nn0zi 12625 | . . 3 ⊢ (3↑3) ∈ ℤ |
10 | 1lt3 12423 | . . . 4 ⊢ 1 < 3 | |
11 | 1 | sqvali 14183 | . . . . 5 ⊢ (3↑2) = (3 · 3) |
12 | 2z 12632 | . . . . . 6 ⊢ 2 ∈ ℤ | |
13 | 3z 12633 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | 2lt3 12422 | . . . . . . 7 ⊢ 2 < 3 | |
15 | ltexp2a 14170 | . . . . . . 7 ⊢ (((3 ∈ ℝ ∧ 2 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (1 < 3 ∧ 2 < 3)) → (3↑2) < (3↑3)) | |
16 | 10, 14, 15 | mpanr12 703 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℤ ∧ 3 ∈ ℤ) → (3↑2) < (3↑3)) |
17 | 5, 12, 13, 16 | mp3an 1457 | . . . . 5 ⊢ (3↑2) < (3↑3) |
18 | 11, 17 | eqbrtrri 5175 | . . . 4 ⊢ (3 · 3) < (3↑3) |
19 | ltexp2a 14170 | . . . 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 703 | . . 3 ⊢ ((3 ∈ ℝ ∧ (3 · 3) ∈ ℤ ∧ (3↑3) ∈ ℤ) → (3↑(3 · 3)) < (3↑(3↑3))) |
21 | 5, 7, 9, 20 | mp3an 1457 | . 2 ⊢ (3↑(3 · 3)) < (3↑(3↑3)) |
22 | 4, 21 | eqbrtrri 5175 | 1 ⊢ ((3↑3)↑3) < (3↑(3↑3)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7426 ℂcc 11144 ℝcr 11145 1c1 11147 · cmul 11151 < clt 11286 2c2 12305 3c3 12306 ℕ0cn0 12510 ℤcz 12596 ↑cexp 14066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-seq 14007 df-exp 14067 |
This theorem is referenced by: (None) |
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