Mathbox for Alexander van der Vekens |
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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 11925 | . . 3 ⊢ 9 ∈ ℕ | |
2 | 8nn 11922 | . . 3 ⊢ 8 ∈ ℕ | |
3 | 4nn0 12106 | . . 3 ⊢ 4 ∈ ℕ0 | |
4 | 0z 12184 | . . 3 ⊢ 0 ∈ ℤ | |
5 | 1nn0 12103 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 2nn0 12104 | . . . 4 ⊢ 2 ∈ ℕ0 | |
7 | 7nn 11919 | . . . . . 6 ⊢ 7 ∈ ℕ | |
8 | 7 | nnzi 12198 | . . . . 5 ⊢ 7 ∈ ℤ |
9 | 8nn0 12110 | . . . . 5 ⊢ 8 ∈ ℕ0 | |
10 | 8cn 11924 | . . . . . . 7 ⊢ 8 ∈ ℂ | |
11 | exp1 13638 | . . . . . . 7 ⊢ (8 ∈ ℂ → (8↑1) = 8) | |
12 | 10, 11 | ax-mp 5 | . . . . . 6 ⊢ (8↑1) = 8 |
13 | 12 | oveq1i 7220 | . . . . 5 ⊢ ((8↑1) mod 9) = (8 mod 9) |
14 | 2t1e2 11990 | . . . . 5 ⊢ (2 · 1) = 2 | |
15 | 6nn0 12108 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
16 | 3nn0 12105 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
17 | 3p1e4 11972 | . . . . . . 7 ⊢ (3 + 1) = 4 | |
18 | eqid 2737 | . . . . . . 7 ⊢ ;63 = ;63 | |
19 | 15, 16, 17, 18 | decsuc 12321 | . . . . . 6 ⊢ (;63 + 1) = ;64 |
20 | 9cn 11927 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
21 | 7cn 11921 | . . . . . . . 8 ⊢ 7 ∈ ℂ | |
22 | 9t7e63 12417 | . . . . . . . 8 ⊢ (9 · 7) = ;63 | |
23 | 20, 21, 22 | mulcomli 10839 | . . . . . . 7 ⊢ (7 · 9) = ;63 |
24 | 23 | oveq1i 7220 | . . . . . 6 ⊢ ((7 · 9) + 1) = (;63 + 1) |
25 | 8t8e64 12411 | . . . . . 6 ⊢ (8 · 8) = ;64 | |
26 | 19, 24, 25 | 3eqtr4i 2775 | . . . . 5 ⊢ ((7 · 9) + 1) = (8 · 8) |
27 | 1, 2, 5, 8, 9, 5, 13, 14, 26 | mod2xi 16619 | . . . 4 ⊢ ((8↑2) mod 9) = (1 mod 9) |
28 | 2t2e4 11991 | . . . 4 ⊢ (2 · 2) = 4 | |
29 | 0p1e1 11949 | . . . . 5 ⊢ (0 + 1) = 1 | |
30 | 20 | mul02i 11018 | . . . . . 6 ⊢ (0 · 9) = 0 |
31 | 30 | oveq1i 7220 | . . . . 5 ⊢ ((0 · 9) + 1) = (0 + 1) |
32 | 1t1e1 11989 | . . . . 5 ⊢ (1 · 1) = 1 | |
33 | 29, 31, 32 | 3eqtr4i 2775 | . . . 4 ⊢ ((0 · 9) + 1) = (1 · 1) |
34 | 1, 2, 6, 4, 5, 5, 27, 28, 33 | mod2xi 16619 | . . 3 ⊢ ((8↑4) mod 9) = (1 mod 9) |
35 | 4cn 11912 | . . . 4 ⊢ 4 ∈ ℂ | |
36 | 2cn 11902 | . . . 4 ⊢ 2 ∈ ℂ | |
37 | 4t2e8 11995 | . . . 4 ⊢ (4 · 2) = 8 | |
38 | 35, 36, 37 | mulcomli 10839 | . . 3 ⊢ (2 · 4) = 8 |
39 | 1, 2, 3, 4, 5, 5, 34, 38, 33 | mod2xi 16619 | . 2 ⊢ ((8↑8) mod 9) = (1 mod 9) |
40 | 1re 10830 | . . 3 ⊢ 1 ∈ ℝ | |
41 | nnrp 12594 | . . . 4 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
42 | 1, 41 | ax-mp 5 | . . 3 ⊢ 9 ∈ ℝ+ |
43 | 0le1 11352 | . . 3 ⊢ 0 ≤ 1 | |
44 | 1lt9 12033 | . . 3 ⊢ 1 < 9 | |
45 | modid 13466 | . . 3 ⊢ (((1 ∈ ℝ ∧ 9 ∈ ℝ+) ∧ (0 ≤ 1 ∧ 1 < 9)) → (1 mod 9) = 1) | |
46 | 40, 42, 43, 44, 45 | mp4an 693 | . 2 ⊢ (1 mod 9) = 1 |
47 | 39, 46 | eqtri 2765 | 1 ⊢ ((8↑8) mod 9) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 class class class wbr 5050 (class class class)co 7210 ℂcc 10724 ℝcr 10725 0cc0 10726 1c1 10727 + caddc 10729 · cmul 10731 < clt 10864 ≤ cle 10865 ℕcn 11827 2c2 11882 3c3 11883 4c4 11884 6c6 11886 7c7 11887 8c8 11888 9c9 11889 ;cdc 12290 ℝ+crp 12583 mod cmo 13439 ↑cexp 13632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-sup 9055 df-inf 9056 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-7 11895 df-8 11896 df-9 11897 df-n0 12088 df-z 12174 df-dec 12291 df-uz 12436 df-rp 12584 df-fl 13364 df-mod 13440 df-seq 13572 df-exp 13633 |
This theorem is referenced by: 9fppr8 44860 nfermltl8rev 44865 |
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