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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 11721 | . . 3 ⊢ 3 ∈ ℂ | |
2 | 3nn0 11918 | . . 3 ⊢ 3 ∈ ℕ0 | |
3 | expmul 13477 | . . 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 11720 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 2, 2 | nn0mulcli 11938 | . . . 4 ⊢ (3 · 3) ∈ ℕ0 |
7 | 6 | nn0zi 12010 | . . 3 ⊢ (3 · 3) ∈ ℤ |
8 | 2, 2 | nn0expcli 13458 | . . . 4 ⊢ (3↑3) ∈ ℕ0 |
9 | 8 | nn0zi 12010 | . . 3 ⊢ (3↑3) ∈ ℤ |
10 | 1lt3 11813 | . . . 4 ⊢ 1 < 3 | |
11 | 1 | sqvali 13546 | . . . . 5 ⊢ (3↑2) = (3 · 3) |
12 | 2z 12017 | . . . . . 6 ⊢ 2 ∈ ℤ | |
13 | 3z 12018 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | 2lt3 11812 | . . . . . . 7 ⊢ 2 < 3 | |
15 | ltexp2a 13533 | . . . . . . 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 5091 | . . . 4 ⊢ (3 · 3) < (3↑3) |
19 | ltexp2a 13533 | . . . 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 5091 | 1 ⊢ ((3↑3)↑3) < (3↑(3↑3)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℂcc 10537 ℝcr 10538 1c1 10540 · cmul 10544 < clt 10677 2c2 11695 3c3 11696 ℕ0cn0 11900 ℤcz 11984 ↑cexp 13432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 |
This theorem is referenced by: (None) |
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