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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 12364 | . . 3 ⊢ 9 ∈ ℕ | |
| 2 | 8nn 12361 | . . 3 ⊢ 8 ∈ ℕ | |
| 3 | 4nn0 12545 | . . 3 ⊢ 4 ∈ ℕ0 | |
| 4 | 0z 12624 | . . 3 ⊢ 0 ∈ ℤ | |
| 5 | 1nn0 12542 | . . 3 ⊢ 1 ∈ ℕ0 | |
| 6 | 2nn0 12543 | . . . 4 ⊢ 2 ∈ ℕ0 | |
| 7 | 7nn 12358 | . . . . . 6 ⊢ 7 ∈ ℕ | |
| 8 | 7 | nnzi 12641 | . . . . 5 ⊢ 7 ∈ ℤ |
| 9 | 8nn0 12549 | . . . . 5 ⊢ 8 ∈ ℕ0 | |
| 10 | 8cn 12363 | . . . . . . 7 ⊢ 8 ∈ ℂ | |
| 11 | exp1 14108 | . . . . . . 7 ⊢ (8 ∈ ℂ → (8↑1) = 8) | |
| 12 | 10, 11 | ax-mp 5 | . . . . . 6 ⊢ (8↑1) = 8 |
| 13 | 12 | oveq1i 7441 | . . . . 5 ⊢ ((8↑1) mod 9) = (8 mod 9) |
| 14 | 2t1e2 12429 | . . . . 5 ⊢ (2 · 1) = 2 | |
| 15 | 6nn0 12547 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
| 16 | 3nn0 12544 | . . . . . . 7 ⊢ 3 ∈ ℕ0 | |
| 17 | 3p1e4 12411 | . . . . . . 7 ⊢ (3 + 1) = 4 | |
| 18 | eqid 2737 | . . . . . . 7 ⊢ ;63 = ;63 | |
| 19 | 15, 16, 17, 18 | decsuc 12764 | . . . . . 6 ⊢ (;63 + 1) = ;64 |
| 20 | 9cn 12366 | . . . . . . . 8 ⊢ 9 ∈ ℂ | |
| 21 | 7cn 12360 | . . . . . . . 8 ⊢ 7 ∈ ℂ | |
| 22 | 9t7e63 12860 | . . . . . . . 8 ⊢ (9 · 7) = ;63 | |
| 23 | 20, 21, 22 | mulcomli 11270 | . . . . . . 7 ⊢ (7 · 9) = ;63 |
| 24 | 23 | oveq1i 7441 | . . . . . 6 ⊢ ((7 · 9) + 1) = (;63 + 1) |
| 25 | 8t8e64 12854 | . . . . . 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 17107 | . . . 4 ⊢ ((8↑2) mod 9) = (1 mod 9) |
| 28 | 2t2e4 12430 | . . . 4 ⊢ (2 · 2) = 4 | |
| 29 | 0p1e1 12388 | . . . . 5 ⊢ (0 + 1) = 1 | |
| 30 | 20 | mul02i 11450 | . . . . . 6 ⊢ (0 · 9) = 0 |
| 31 | 30 | oveq1i 7441 | . . . . 5 ⊢ ((0 · 9) + 1) = (0 + 1) |
| 32 | 1t1e1 12428 | . . . . 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 17107 | . . 3 ⊢ ((8↑4) mod 9) = (1 mod 9) |
| 35 | 4cn 12351 | . . . 4 ⊢ 4 ∈ ℂ | |
| 36 | 2cn 12341 | . . . 4 ⊢ 2 ∈ ℂ | |
| 37 | 4t2e8 12434 | . . . 4 ⊢ (4 · 2) = 8 | |
| 38 | 35, 36, 37 | mulcomli 11270 | . . 3 ⊢ (2 · 4) = 8 |
| 39 | 1, 2, 3, 4, 5, 5, 34, 38, 33 | mod2xi 17107 | . 2 ⊢ ((8↑8) mod 9) = (1 mod 9) |
| 40 | 1re 11261 | . . 3 ⊢ 1 ∈ ℝ | |
| 41 | nnrp 13046 | . . . 4 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
| 42 | 1, 41 | ax-mp 5 | . . 3 ⊢ 9 ∈ ℝ+ |
| 43 | 0le1 11786 | . . 3 ⊢ 0 ≤ 1 | |
| 44 | 1lt9 12472 | . . 3 ⊢ 1 < 9 | |
| 45 | modid 13936 | . . 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 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 ℕcn 12266 2c2 12321 3c3 12322 4c4 12323 6c6 12325 7c7 12326 8c8 12327 9c9 12328 ;cdc 12733 ℝ+crp 13034 mod cmo 13909 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: 9fppr8 47724 nfermltl8rev 47729 |
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