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Mirrors > Home > MPE Home > Th. List > Mathboxes > 8exp8mod9 | Structured version Visualization version GIF version |
Description: Eight to the eighth power modulo nine is one. (Contributed by AV, 2-Jun-2023.) |
Ref | Expression |
---|---|
8exp8mod9 | ⊢ ((8↑8) mod 9) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn 12185 | . . 3 ⊢ 9 ∈ ℕ | |
2 | 8nn 12182 | . . 3 ⊢ 8 ∈ ℕ | |
3 | 4nn0 12366 | . . 3 ⊢ 4 ∈ ℕ0 | |
4 | 0z 12444 | . . 3 ⊢ 0 ∈ ℤ | |
5 | 1nn0 12363 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 2nn0 12364 | . . . 4 ⊢ 2 ∈ ℕ0 | |
7 | 7nn 12179 | . . . . . 6 ⊢ 7 ∈ ℕ | |
8 | 7 | nnzi 12461 | . . . . 5 ⊢ 7 ∈ ℤ |
9 | 8nn0 12370 | . . . . 5 ⊢ 8 ∈ ℕ0 | |
10 | 8cn 12184 | . . . . . . 7 ⊢ 8 ∈ ℂ | |
11 | exp1 13903 | . . . . . . 7 ⊢ (8 ∈ ℂ → (8↑1) = 8) | |
12 | 10, 11 | ax-mp 5 | . . . . . 6 ⊢ (8↑1) = 8 |
13 | 12 | oveq1i 7360 | . . . . 5 ⊢ ((8↑1) mod 9) = (8 mod 9) |
14 | 2t1e2 12250 | . . . . 5 ⊢ (2 · 1) = 2 | |
15 | 6nn0 12368 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
16 | 3nn0 12365 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
17 | 3p1e4 12232 | . . . . . . 7 ⊢ (3 + 1) = 4 | |
18 | eqid 2738 | . . . . . . 7 ⊢ ;63 = ;63 | |
19 | 15, 16, 17, 18 | decsuc 12583 | . . . . . 6 ⊢ (;63 + 1) = ;64 |
20 | 9cn 12187 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
21 | 7cn 12181 | . . . . . . . 8 ⊢ 7 ∈ ℂ | |
22 | 9t7e63 12679 | . . . . . . . 8 ⊢ (9 · 7) = ;63 | |
23 | 20, 21, 22 | mulcomli 11098 | . . . . . . 7 ⊢ (7 · 9) = ;63 |
24 | 23 | oveq1i 7360 | . . . . . 6 ⊢ ((7 · 9) + 1) = (;63 + 1) |
25 | 8t8e64 12673 | . . . . . 6 ⊢ (8 · 8) = ;64 | |
26 | 19, 24, 25 | 3eqtr4i 2776 | . . . . 5 ⊢ ((7 · 9) + 1) = (8 · 8) |
27 | 1, 2, 5, 8, 9, 5, 13, 14, 26 | mod2xi 16877 | . . . 4 ⊢ ((8↑2) mod 9) = (1 mod 9) |
28 | 2t2e4 12251 | . . . 4 ⊢ (2 · 2) = 4 | |
29 | 0p1e1 12209 | . . . . 5 ⊢ (0 + 1) = 1 | |
30 | 20 | mul02i 11278 | . . . . . 6 ⊢ (0 · 9) = 0 |
31 | 30 | oveq1i 7360 | . . . . 5 ⊢ ((0 · 9) + 1) = (0 + 1) |
32 | 1t1e1 12249 | . . . . 5 ⊢ (1 · 1) = 1 | |
33 | 29, 31, 32 | 3eqtr4i 2776 | . . . 4 ⊢ ((0 · 9) + 1) = (1 · 1) |
34 | 1, 2, 6, 4, 5, 5, 27, 28, 33 | mod2xi 16877 | . . 3 ⊢ ((8↑4) mod 9) = (1 mod 9) |
35 | 4cn 12172 | . . . 4 ⊢ 4 ∈ ℂ | |
36 | 2cn 12162 | . . . 4 ⊢ 2 ∈ ℂ | |
37 | 4t2e8 12255 | . . . 4 ⊢ (4 · 2) = 8 | |
38 | 35, 36, 37 | mulcomli 11098 | . . 3 ⊢ (2 · 4) = 8 |
39 | 1, 2, 3, 4, 5, 5, 34, 38, 33 | mod2xi 16877 | . 2 ⊢ ((8↑8) mod 9) = (1 mod 9) |
40 | 1re 11089 | . . 3 ⊢ 1 ∈ ℝ | |
41 | nnrp 12856 | . . . 4 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
42 | 1, 41 | ax-mp 5 | . . 3 ⊢ 9 ∈ ℝ+ |
43 | 0le1 11612 | . . 3 ⊢ 0 ≤ 1 | |
44 | 1lt9 12293 | . . 3 ⊢ 1 < 9 | |
45 | modid 13731 | . . 3 ⊢ (((1 ∈ ℝ ∧ 9 ∈ ℝ+) ∧ (0 ≤ 1 ∧ 1 < 9)) → (1 mod 9) = 1) | |
46 | 40, 42, 43, 44, 45 | mp4an 692 | . 2 ⊢ (1 mod 9) = 1 |
47 | 39, 46 | eqtri 2766 | 1 ⊢ ((8↑8) mod 9) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 class class class wbr 5104 (class class class)co 7350 ℂcc 10983 ℝcr 10984 0cc0 10985 1c1 10986 + caddc 10988 · cmul 10990 < clt 11123 ≤ cle 11124 ℕcn 12087 2c2 12142 3c3 12143 4c4 12144 6c6 12146 7c7 12147 8c8 12148 9c9 12149 ;cdc 12552 ℝ+crp 12845 mod cmo 13704 ↑cexp 13897 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-sup 9312 df-inf 9313 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12553 df-uz 12698 df-rp 12846 df-fl 13627 df-mod 13705 df-seq 13837 df-exp 13898 |
This theorem is referenced by: 9fppr8 45720 nfermltl8rev 45725 |
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