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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 11583 | . . 3 ⊢ 9 ∈ ℕ | |
2 | 8nn 11580 | . . 3 ⊢ 8 ∈ ℕ | |
3 | 4nn0 11764 | . . 3 ⊢ 4 ∈ ℕ0 | |
4 | 0z 11840 | . . 3 ⊢ 0 ∈ ℤ | |
5 | 1nn0 11761 | . . 3 ⊢ 1 ∈ ℕ0 | |
6 | 2nn0 11762 | . . . 4 ⊢ 2 ∈ ℕ0 | |
7 | 7nn 11577 | . . . . . 6 ⊢ 7 ∈ ℕ | |
8 | 7 | nnzi 11855 | . . . . 5 ⊢ 7 ∈ ℤ |
9 | 8nn0 11768 | . . . . 5 ⊢ 8 ∈ ℕ0 | |
10 | 8cn 11582 | . . . . . . 7 ⊢ 8 ∈ ℂ | |
11 | exp1 13285 | . . . . . . 7 ⊢ (8 ∈ ℂ → (8↑1) = 8) | |
12 | 10, 11 | ax-mp 5 | . . . . . 6 ⊢ (8↑1) = 8 |
13 | 12 | oveq1i 7026 | . . . . 5 ⊢ ((8↑1) mod 9) = (8 mod 9) |
14 | 2t1e2 11648 | . . . . 5 ⊢ (2 · 1) = 2 | |
15 | 6nn0 11766 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
16 | 3nn0 11763 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
17 | 3p1e4 11630 | . . . . . . 7 ⊢ (3 + 1) = 4 | |
18 | eqid 2795 | . . . . . . 7 ⊢ ;63 = ;63 | |
19 | 15, 16, 17, 18 | decsuc 11978 | . . . . . 6 ⊢ (;63 + 1) = ;64 |
20 | 9cn 11585 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
21 | 7cn 11579 | . . . . . . . 8 ⊢ 7 ∈ ℂ | |
22 | 9t7e63 12075 | . . . . . . . 8 ⊢ (9 · 7) = ;63 | |
23 | 20, 21, 22 | mulcomli 10496 | . . . . . . 7 ⊢ (7 · 9) = ;63 |
24 | 23 | oveq1i 7026 | . . . . . 6 ⊢ ((7 · 9) + 1) = (;63 + 1) |
25 | 8t8e64 12069 | . . . . . 6 ⊢ (8 · 8) = ;64 | |
26 | 19, 24, 25 | 3eqtr4i 2829 | . . . . 5 ⊢ ((7 · 9) + 1) = (8 · 8) |
27 | 1, 2, 5, 8, 9, 5, 13, 14, 26 | mod2xi 16234 | . . . 4 ⊢ ((8↑2) mod 9) = (1 mod 9) |
28 | 2t2e4 11649 | . . . 4 ⊢ (2 · 2) = 4 | |
29 | 0p1e1 11607 | . . . . 5 ⊢ (0 + 1) = 1 | |
30 | 20 | mul02i 10676 | . . . . . 6 ⊢ (0 · 9) = 0 |
31 | 30 | oveq1i 7026 | . . . . 5 ⊢ ((0 · 9) + 1) = (0 + 1) |
32 | 1t1e1 11647 | . . . . 5 ⊢ (1 · 1) = 1 | |
33 | 29, 31, 32 | 3eqtr4i 2829 | . . . 4 ⊢ ((0 · 9) + 1) = (1 · 1) |
34 | 1, 2, 6, 4, 5, 5, 27, 28, 33 | mod2xi 16234 | . . 3 ⊢ ((8↑4) mod 9) = (1 mod 9) |
35 | 4cn 11570 | . . . 4 ⊢ 4 ∈ ℂ | |
36 | 2cn 11560 | . . . 4 ⊢ 2 ∈ ℂ | |
37 | 4t2e8 11653 | . . . 4 ⊢ (4 · 2) = 8 | |
38 | 35, 36, 37 | mulcomli 10496 | . . 3 ⊢ (2 · 4) = 8 |
39 | 1, 2, 3, 4, 5, 5, 34, 38, 33 | mod2xi 16234 | . 2 ⊢ ((8↑8) mod 9) = (1 mod 9) |
40 | 1re 10487 | . . 3 ⊢ 1 ∈ ℝ | |
41 | nnrp 12250 | . . . 4 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
42 | 1, 41 | ax-mp 5 | . . 3 ⊢ 9 ∈ ℝ+ |
43 | 0le1 11011 | . . 3 ⊢ 0 ≤ 1 | |
44 | 1lt9 11691 | . . 3 ⊢ 1 < 9 | |
45 | modid 13114 | . . 3 ⊢ (((1 ∈ ℝ ∧ 9 ∈ ℝ+) ∧ (0 ≤ 1 ∧ 1 < 9)) → (1 mod 9) = 1) | |
46 | 40, 42, 43, 44, 45 | mp4an 689 | . 2 ⊢ (1 mod 9) = 1 |
47 | 39, 46 | eqtri 2819 | 1 ⊢ ((8↑8) mod 9) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2081 class class class wbr 4962 (class class class)co 7016 ℂcc 10381 ℝcr 10382 0cc0 10383 1c1 10384 + caddc 10386 · cmul 10388 < clt 10521 ≤ cle 10522 ℕcn 11486 2c2 11540 3c3 11541 4c4 11542 6c6 11544 7c7 11545 8c8 11546 9c9 11547 ;cdc 11947 ℝ+crp 12239 mod cmo 13087 ↑cexp 13279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-rp 12240 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 |
This theorem is referenced by: 9fppr8 43404 nfermltl8rev 43409 |
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