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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 12241 | . . 3 ⊢ 3 ∈ ℂ | |
2 | 3nn0 12438 | . . 3 ⊢ 3 ∈ ℕ0 | |
3 | expmul 14020 | . . 3 ⊢ ((3 ∈ ℂ ∧ 3 ∈ ℕ0 ∧ 3 ∈ ℕ0) → (3↑(3 · 3)) = ((3↑3)↑3)) | |
4 | 1, 2, 2, 3 | mp3an 1462 | . 2 ⊢ (3↑(3 · 3)) = ((3↑3)↑3) |
5 | 3re 12240 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 2, 2 | nn0mulcli 12458 | . . . 4 ⊢ (3 · 3) ∈ ℕ0 |
7 | 6 | nn0zi 12535 | . . 3 ⊢ (3 · 3) ∈ ℤ |
8 | 2, 2 | nn0expcli 14001 | . . . 4 ⊢ (3↑3) ∈ ℕ0 |
9 | 8 | nn0zi 12535 | . . 3 ⊢ (3↑3) ∈ ℤ |
10 | 1lt3 12333 | . . . 4 ⊢ 1 < 3 | |
11 | 1 | sqvali 14091 | . . . . 5 ⊢ (3↑2) = (3 · 3) |
12 | 2z 12542 | . . . . . 6 ⊢ 2 ∈ ℤ | |
13 | 3z 12543 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | 2lt3 12332 | . . . . . . 7 ⊢ 2 < 3 | |
15 | ltexp2a 14078 | . . . . . . 7 ⊢ (((3 ∈ ℝ ∧ 2 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (1 < 3 ∧ 2 < 3)) → (3↑2) < (3↑3)) | |
16 | 10, 14, 15 | mpanr12 704 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 2 ∈ ℤ ∧ 3 ∈ ℤ) → (3↑2) < (3↑3)) |
17 | 5, 12, 13, 16 | mp3an 1462 | . . . . 5 ⊢ (3↑2) < (3↑3) |
18 | 11, 17 | eqbrtrri 5133 | . . . 4 ⊢ (3 · 3) < (3↑3) |
19 | ltexp2a 14078 | . . . 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 704 | . . 3 ⊢ ((3 ∈ ℝ ∧ (3 · 3) ∈ ℤ ∧ (3↑3) ∈ ℤ) → (3↑(3 · 3)) < (3↑(3↑3))) |
21 | 5, 7, 9, 20 | mp3an 1462 | . 2 ⊢ (3↑(3 · 3)) < (3↑(3↑3)) |
22 | 4, 21 | eqbrtrri 5133 | 1 ⊢ ((3↑3)↑3) < (3↑(3↑3)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5110 (class class class)co 7362 ℂcc 11056 ℝcr 11057 1c1 11059 · cmul 11063 < clt 11196 2c2 12215 3c3 12216 ℕ0cn0 12420 ℤcz 12506 ↑cexp 13974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-seq 13914 df-exp 13975 |
This theorem is referenced by: (None) |
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