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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 11797 | . . 3 ⊢ 3 ∈ ℂ | |
2 | 3nn0 11994 | . . 3 ⊢ 3 ∈ ℕ0 | |
3 | expmul 13566 | . . 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 11796 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 2, 2 | nn0mulcli 12014 | . . . 4 ⊢ (3 · 3) ∈ ℕ0 |
7 | 6 | nn0zi 12088 | . . 3 ⊢ (3 · 3) ∈ ℤ |
8 | 2, 2 | nn0expcli 13547 | . . . 4 ⊢ (3↑3) ∈ ℕ0 |
9 | 8 | nn0zi 12088 | . . 3 ⊢ (3↑3) ∈ ℤ |
10 | 1lt3 11889 | . . . 4 ⊢ 1 < 3 | |
11 | 1 | sqvali 13635 | . . . . 5 ⊢ (3↑2) = (3 · 3) |
12 | 2z 12095 | . . . . . 6 ⊢ 2 ∈ ℤ | |
13 | 3z 12096 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | 2lt3 11888 | . . . . . . 7 ⊢ 2 < 3 | |
15 | ltexp2a 13622 | . . . . . . 7 ⊢ (((3 ∈ ℝ ∧ 2 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (1 < 3 ∧ 2 < 3)) → (3↑2) < (3↑3)) | |
16 | 10, 14, 15 | mpanr12 705 | . . . . . 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 5053 | . . . 4 ⊢ (3 · 3) < (3↑3) |
19 | ltexp2a 13622 | . . . 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 705 | . . 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 5053 | 1 ⊢ ((3↑3)↑3) < (3↑(3↑3)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 (class class class)co 7170 ℂcc 10613 ℝcr 10614 1c1 10616 · cmul 10620 < clt 10753 2c2 11771 3c3 11772 ℕ0cn0 11976 ℤcz 12062 ↑cexp 13521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-seq 13461 df-exp 13522 |
This theorem is referenced by: (None) |
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