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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 12340 | . . 3 ⊢ 9 ∈ ℕ | |
2 | 8nn 12337 | . . 3 ⊢ 8 ∈ ℕ | |
3 | 4nn0 12521 | . . 3 ⊢ 4 ∈ ℕ0 | |
4 | 0z 12599 | . . 3 ⊢ 0 ∈ ℤ | |
5 | 1nn0 12518 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 2nn0 12519 | . . . 4 ⊢ 2 ∈ ℕ0 | |
7 | 7nn 12334 | . . . . . 6 ⊢ 7 ∈ ℕ | |
8 | 7 | nnzi 12616 | . . . . 5 ⊢ 7 ∈ ℤ |
9 | 8nn0 12525 | . . . . 5 ⊢ 8 ∈ ℕ0 | |
10 | 8cn 12339 | . . . . . . 7 ⊢ 8 ∈ ℂ | |
11 | exp1 14064 | . . . . . . 7 ⊢ (8 ∈ ℂ → (8↑1) = 8) | |
12 | 10, 11 | ax-mp 5 | . . . . . 6 ⊢ (8↑1) = 8 |
13 | 12 | oveq1i 7426 | . . . . 5 ⊢ ((8↑1) mod 9) = (8 mod 9) |
14 | 2t1e2 12405 | . . . . 5 ⊢ (2 · 1) = 2 | |
15 | 6nn0 12523 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
16 | 3nn0 12520 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
17 | 3p1e4 12387 | . . . . . . 7 ⊢ (3 + 1) = 4 | |
18 | eqid 2725 | . . . . . . 7 ⊢ ;63 = ;63 | |
19 | 15, 16, 17, 18 | decsuc 12738 | . . . . . 6 ⊢ (;63 + 1) = ;64 |
20 | 9cn 12342 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
21 | 7cn 12336 | . . . . . . . 8 ⊢ 7 ∈ ℂ | |
22 | 9t7e63 12834 | . . . . . . . 8 ⊢ (9 · 7) = ;63 | |
23 | 20, 21, 22 | mulcomli 11253 | . . . . . . 7 ⊢ (7 · 9) = ;63 |
24 | 23 | oveq1i 7426 | . . . . . 6 ⊢ ((7 · 9) + 1) = (;63 + 1) |
25 | 8t8e64 12828 | . . . . . 6 ⊢ (8 · 8) = ;64 | |
26 | 19, 24, 25 | 3eqtr4i 2763 | . . . . 5 ⊢ ((7 · 9) + 1) = (8 · 8) |
27 | 1, 2, 5, 8, 9, 5, 13, 14, 26 | mod2xi 17037 | . . . 4 ⊢ ((8↑2) mod 9) = (1 mod 9) |
28 | 2t2e4 12406 | . . . 4 ⊢ (2 · 2) = 4 | |
29 | 0p1e1 12364 | . . . . 5 ⊢ (0 + 1) = 1 | |
30 | 20 | mul02i 11433 | . . . . . 6 ⊢ (0 · 9) = 0 |
31 | 30 | oveq1i 7426 | . . . . 5 ⊢ ((0 · 9) + 1) = (0 + 1) |
32 | 1t1e1 12404 | . . . . 5 ⊢ (1 · 1) = 1 | |
33 | 29, 31, 32 | 3eqtr4i 2763 | . . . 4 ⊢ ((0 · 9) + 1) = (1 · 1) |
34 | 1, 2, 6, 4, 5, 5, 27, 28, 33 | mod2xi 17037 | . . 3 ⊢ ((8↑4) mod 9) = (1 mod 9) |
35 | 4cn 12327 | . . . 4 ⊢ 4 ∈ ℂ | |
36 | 2cn 12317 | . . . 4 ⊢ 2 ∈ ℂ | |
37 | 4t2e8 12410 | . . . 4 ⊢ (4 · 2) = 8 | |
38 | 35, 36, 37 | mulcomli 11253 | . . 3 ⊢ (2 · 4) = 8 |
39 | 1, 2, 3, 4, 5, 5, 34, 38, 33 | mod2xi 17037 | . 2 ⊢ ((8↑8) mod 9) = (1 mod 9) |
40 | 1re 11244 | . . 3 ⊢ 1 ∈ ℝ | |
41 | nnrp 13017 | . . . 4 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
42 | 1, 41 | ax-mp 5 | . . 3 ⊢ 9 ∈ ℝ+ |
43 | 0le1 11767 | . . 3 ⊢ 0 ≤ 1 | |
44 | 1lt9 12448 | . . 3 ⊢ 1 < 9 | |
45 | modid 13893 | . . 3 ⊢ (((1 ∈ ℝ ∧ 9 ∈ ℝ+) ∧ (0 ≤ 1 ∧ 1 < 9)) → (1 mod 9) = 1) | |
46 | 40, 42, 43, 44, 45 | mp4an 691 | . 2 ⊢ (1 mod 9) = 1 |
47 | 39, 46 | eqtri 2753 | 1 ⊢ ((8↑8) mod 9) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 class class class wbr 5143 (class class class)co 7416 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 + caddc 11141 · cmul 11143 < clt 11278 ≤ cle 11279 ℕcn 12242 2c2 12297 3c3 12298 4c4 12299 6c6 12301 7c7 12302 8c8 12303 9c9 12304 ;cdc 12707 ℝ+crp 13006 mod cmo 13866 ↑cexp 14058 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 |
This theorem is referenced by: 9fppr8 47140 nfermltl8rev 47145 |
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