2exp340mod341 - Mathbox for Alexander van der Vekens

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Theorem 2exp340mod341 48221

Description: Eight to the eighth power modulo nine is one. (Contributed by AV, 3-Jun-2023.)

**Assertion**
Ref	Expression
2exp340mod341	⊢ ((2↑;;340) mod ;;341) = 1

**Proof of Theorem 2exp340mod341**
Step	Hyp	Ref	Expression
1		3nn0 12446	. . . . 5 ⊢ 3 ∈ ℕ₀
2		4nn0 12447	. . . . 5 ⊢ 4 ∈ ℕ₀
3	1, 2	deccl 12650	. . . 4 ⊢ ;34 ∈ ℕ₀
4		1nn 12176	. . . 4 ⊢ 1 ∈ ℕ
5	3, 4	decnncl 12655	. . 3 ⊢ ;;341 ∈ ℕ
6		2nn 12245	. . 3 ⊢ 2 ∈ ℕ
7		1nn0 12444	. . . . 5 ⊢ 1 ∈ ℕ₀
8		7nn0 12450	. . . . 5 ⊢ 7 ∈ ℕ₀
9	7, 8	deccl 12650	. . . 4 ⊢ ;17 ∈ ℕ₀
10		0nn0 12443	. . . 4 ⊢ 0 ∈ ℕ₀
11	9, 10	deccl 12650	. . 3 ⊢ ;;170 ∈ ℕ₀
12		0z 12526	. . 3 ⊢ 0 ∈ ℤ
13		8nn0 12451	. . . . 5 ⊢ 8 ∈ ℕ₀
14		5nn0 12448	. . . . 5 ⊢ 5 ∈ ℕ₀
15	13, 14	deccl 12650	. . . 4 ⊢ ;85 ∈ ℕ₀
16		3z 12551	. . . 4 ⊢ 3 ∈ ℤ
17		2nn0 12445	. . . . 5 ⊢ 2 ∈ ℕ₀
18	1, 17	deccl 12650	. . . 4 ⊢ ;32 ∈ ℕ₀
19	13, 2	deccl 12650	. . . . 5 ⊢ ;84 ∈ ℕ₀
20		6nn0 12449	. . . . . 6 ⊢ 6 ∈ ℕ₀
21	7, 20	deccl 12650	. . . . 5 ⊢ ;16 ∈ ℕ₀
22	2, 17	deccl 12650	. . . . . 6 ⊢ ;42 ∈ ℕ₀
23	17, 7	deccl 12650	. . . . . . 7 ⊢ ;21 ∈ ℕ₀
24	17, 10	deccl 12650	. . . . . . . 8 ⊢ ;20 ∈ ℕ₀
25	7, 10	deccl 12650	. . . . . . . . 9 ⊢ ;10 ∈ ℕ₀
26		2exp5 17047	. . . . . . . . . . 11 ⊢ (2↑5) = ;32
27	26	oveq1i 7370	. . . . . . . . . 10 ⊢ ((2↑5) mod ;;341) = (;32 mod ;;341)
28		5cn 12260	. . . . . . . . . . 11 ⊢ 5 ∈ ℂ
29		2cn 12247	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
30		5t2e10 12735	. . . . . . . . . . 11 ⊢ (5 · 2) = ;10
31	28, 29, 30	mulcomli 11145	. . . . . . . . . 10 ⊢ (2 · 5) = ;10
32	25, 17	deccl 12650	. . . . . . . . . . . 12 ⊢ ;;102 ∈ ℕ₀
33		3p1e4 12312	. . . . . . . . . . . 12 ⊢ (3 + 1) = 4
34		eqid 2737	. . . . . . . . . . . 12 ⊢ ;;;1023 = ;;;1023
35	32, 1, 33, 34	decsuc 12666	. . . . . . . . . . 11 ⊢ (;;;1023 + 1) = ;;;1024
36	1, 3, 7	decmulnc 12702	. . . . . . . . . . . . 13 ⊢ (3 · ;;341) = ;(3 · ;34)(3 · 1)
37		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ ;34 = ;34
38		3t3e9 12334	. . . . . . . . . . . . . . . . 17 ⊢ (3 · 3) = 9
39	38	oveq1i 7370	. . . . . . . . . . . . . . . 16 ⊢ ((3 · 3) + 1) = (9 + 1)
40		9p1e10 12637	. . . . . . . . . . . . . . . 16 ⊢ (9 + 1) = ;10
41	39, 40	eqtri 2760	. . . . . . . . . . . . . . 15 ⊢ ((3 · 3) + 1) = ;10
42		4cn 12257	. . . . . . . . . . . . . . . 16 ⊢ 4 ∈ ℂ
43		3cn 12253	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℂ
44		4t3e12 12733	. . . . . . . . . . . . . . . 16 ⊢ (4 · 3) = ;12
45	42, 43, 44	mulcomli 11145	. . . . . . . . . . . . . . 15 ⊢ (3 · 4) = ;12
46	1, 1, 2, 37, 17, 7, 41, 45	decmul2c 12701	. . . . . . . . . . . . . 14 ⊢ (3 · ;34) = ;;102
47	43	mulridi 11140	. . . . . . . . . . . . . 14 ⊢ (3 · 1) = 3
48	46, 47	deceq12i 12644	. . . . . . . . . . . . 13 ⊢ ;(3 · ;34)(3 · 1) = ;;;1023
49	36, 48	eqtri 2760	. . . . . . . . . . . 12 ⊢ (3 · ;;341) = ;;;1023
50	49	oveq1i 7370	. . . . . . . . . . 11 ⊢ ((3 · ;;341) + 1) = (;;;1023 + 1)
51		eqid 2737	. . . . . . . . . . . 12 ⊢ ;32 = ;32
52	1, 1, 17	decmulnc 12702	. . . . . . . . . . . . . 14 ⊢ (3 · ;32) = ;(3 · 3)(3 · 2)
53	52	oveq1i 7370	. . . . . . . . . . . . 13 ⊢ ((3 · ;32) + 6) = (;(3 · 3)(3 · 2) + 6)
54		9nn0 12452	. . . . . . . . . . . . . 14 ⊢ 9 ∈ ℕ₀
55		3t2e6 12333	. . . . . . . . . . . . . . 15 ⊢ (3 · 2) = 6
56	38, 55	deceq12i 12644	. . . . . . . . . . . . . 14 ⊢ ;(3 · 3)(3 · 2) = ;96
57		6p6e12 12709	. . . . . . . . . . . . . 14 ⊢ (6 + 6) = ;12
58	54, 20, 20, 56, 40, 17, 57	decaddci 12696	. . . . . . . . . . . . 13 ⊢ (;(3 · 3)(3 · 2) + 6) = ;;102
59	53, 58	eqtri 2760	. . . . . . . . . . . 12 ⊢ ((3 · ;32) + 6) = ;;102
60	17, 1, 17	decmulnc 12702	. . . . . . . . . . . . 13 ⊢ (2 · ;32) = ;(2 · 3)(2 · 2)
61	43, 29, 55	mulcomli 11145	. . . . . . . . . . . . . 14 ⊢ (2 · 3) = 6
62		2t2e4 12331	. . . . . . . . . . . . . 14 ⊢ (2 · 2) = 4
63	61, 62	deceq12i 12644	. . . . . . . . . . . . 13 ⊢ ;(2 · 3)(2 · 2) = ;64
64	60, 63	eqtri 2760	. . . . . . . . . . . 12 ⊢ (2 · ;32) = ;64
65	18, 1, 17, 51, 2, 20, 59, 64	decmul1c 12700	. . . . . . . . . . 11 ⊢ (;32 · ;32) = ;;;1024
66	35, 50, 65	3eqtr4i 2770	. . . . . . . . . 10 ⊢ ((3 · ;;341) + 1) = (;32 · ;32)
67	5, 6, 14, 16, 18, 7, 27, 31, 66	mod2xi 17031	. . . . . . . . 9 ⊢ ((2↑;10) mod ;;341) = (1 mod ;;341)
68	17, 7, 10	decmulnc 12702	. . . . . . . . . 10 ⊢ (2 · ;10) = ;(2 · 1)(2 · 0)
69	29	mulridi 11140	. . . . . . . . . . 11 ⊢ (2 · 1) = 2
70		2t0e0 12336	. . . . . . . . . . 11 ⊢ (2 · 0) = 0
71	69, 70	deceq12i 12644	. . . . . . . . . 10 ⊢ ;(2 · 1)(2 · 0) = ;20
72	68, 71	eqtri 2760	. . . . . . . . 9 ⊢ (2 · ;10) = ;20
73		0p1e1 12289	. . . . . . . . . 10 ⊢ (0 + 1) = 1
74	5	nncni 12175	. . . . . . . . . . . 12 ⊢ ;;341 ∈ ℂ
75	74	mul02i 11326	. . . . . . . . . . 11 ⊢ (0 · ;;341) = 0
76	75	oveq1i 7370	. . . . . . . . . 10 ⊢ ((0 · ;;341) + 1) = (0 + 1)
77		1t1e1 12329	. . . . . . . . . 10 ⊢ (1 · 1) = 1
78	73, 76, 77	3eqtr4i 2770	. . . . . . . . 9 ⊢ ((0 · ;;341) + 1) = (1 · 1)
79	5, 6, 25, 12, 7, 7, 67, 72, 78	mod2xi 17031	. . . . . . . 8 ⊢ ((2↑;20) mod ;;341) = (1 mod ;;341)
80		eqid 2737	. . . . . . . . 9 ⊢ ;20 = ;20
81	17, 10, 73, 80	decsuc 12666	. . . . . . . 8 ⊢ (;20 + 1) = ;21
82	29	addlidi 11325	. . . . . . . . 9 ⊢ (0 + 2) = 2
83	75	oveq1i 7370	. . . . . . . . 9 ⊢ ((0 · ;;341) + 2) = (0 + 2)
84	29	mullidi 11141	. . . . . . . . 9 ⊢ (1 · 2) = 2
85	82, 83, 84	3eqtr4i 2770	. . . . . . . 8 ⊢ ((0 · ;;341) + 2) = (1 · 2)
86	5, 6, 24, 12, 7, 17, 79, 81, 85	modxp1i 17032	. . . . . . 7 ⊢ ((2↑;21) mod ;;341) = (2 mod ;;341)
87	17, 17, 7	decmulnc 12702	. . . . . . . 8 ⊢ (2 · ;21) = ;(2 · 2)(2 · 1)
88	62, 69	deceq12i 12644	. . . . . . . 8 ⊢ ;(2 · 2)(2 · 1) = ;42
89	87, 88	eqtri 2760	. . . . . . 7 ⊢ (2 · ;21) = ;42
90	42	addlidi 11325	. . . . . . . 8 ⊢ (0 + 4) = 4
91	75	oveq1i 7370	. . . . . . . 8 ⊢ ((0 · ;;341) + 4) = (0 + 4)
92	90, 91, 62	3eqtr4i 2770	. . . . . . 7 ⊢ ((0 · ;;341) + 4) = (2 · 2)
93	5, 6, 23, 12, 17, 2, 86, 89, 92	mod2xi 17031	. . . . . 6 ⊢ ((2↑;42) mod ;;341) = (4 mod ;;341)
94	17, 2, 17	decmulnc 12702	. . . . . . 7 ⊢ (2 · ;42) = ;(2 · 4)(2 · 2)
95		4t2e8 12335	. . . . . . . . 9 ⊢ (4 · 2) = 8
96	42, 29, 95	mulcomli 11145	. . . . . . . 8 ⊢ (2 · 4) = 8
97	96, 62	deceq12i 12644	. . . . . . 7 ⊢ ;(2 · 4)(2 · 2) = ;84
98	94, 97	eqtri 2760	. . . . . 6 ⊢ (2 · ;42) = ;84
99	21	nn0cni 12440	. . . . . . . 8 ⊢ ;16 ∈ ℂ
100	99	addlidi 11325	. . . . . . 7 ⊢ (0 + ;16) = ;16
101	75	oveq1i 7370	. . . . . . 7 ⊢ ((0 · ;;341) + ;16) = (0 + ;16)
102		4t4e16 12734	. . . . . . 7 ⊢ (4 · 4) = ;16
103	100, 101, 102	3eqtr4i 2770	. . . . . 6 ⊢ ((0 · ;;341) + ;16) = (4 · 4)
104	5, 6, 22, 12, 2, 21, 93, 98, 103	mod2xi 17031	. . . . 5 ⊢ ((2↑;84) mod ;;341) = (;16 mod ;;341)
105		4p1e5 12313	. . . . . 6 ⊢ (4 + 1) = 5
106		eqid 2737	. . . . . 6 ⊢ ;84 = ;84
107	13, 2, 105, 106	decsuc 12666	. . . . 5 ⊢ (;84 + 1) = ;85
108	18	nn0cni 12440	. . . . . . 7 ⊢ ;32 ∈ ℂ
109	108	addlidi 11325	. . . . . 6 ⊢ (0 + ;32) = ;32
110	75	oveq1i 7370	. . . . . 6 ⊢ ((0 · ;;341) + ;32) = (0 + ;32)
111		eqid 2737	. . . . . . 7 ⊢ ;16 = ;16
112	84	oveq1i 7370	. . . . . . . 8 ⊢ ((1 · 2) + 1) = (2 + 1)
113		2p1e3 12309	. . . . . . . 8 ⊢ (2 + 1) = 3
114	112, 113	eqtri 2760	. . . . . . 7 ⊢ ((1 · 2) + 1) = 3
115		6t2e12 12739	. . . . . . 7 ⊢ (6 · 2) = ;12
116	17, 7, 20, 111, 17, 7, 114, 115	decmul1c 12700	. . . . . 6 ⊢ (;16 · 2) = ;32
117	109, 110, 116	3eqtr4i 2770	. . . . 5 ⊢ ((0 · ;;341) + ;32) = (;16 · 2)
118	5, 6, 19, 12, 21, 18, 104, 107, 117	modxp1i 17032	. . . 4 ⊢ ((2↑;85) mod ;;341) = (;32 mod ;;341)
119		eqid 2737	. . . . 5 ⊢ ;85 = ;85
120		6p1e7 12315	. . . . . 6 ⊢ (6 + 1) = 7
121		8cn 12269	. . . . . . 7 ⊢ 8 ∈ ℂ
122		8t2e16 12750	. . . . . . 7 ⊢ (8 · 2) = ;16
123	121, 29, 122	mulcomli 11145	. . . . . 6 ⊢ (2 · 8) = ;16
124	7, 20, 120, 123	decsuc 12666	. . . . 5 ⊢ ((2 · 8) + 1) = ;17
125	17, 13, 14, 119, 10, 7, 124, 31	decmul2c 12701	. . . 4 ⊢ (2 · ;85) = ;;170
126	5, 6, 15, 16, 18, 7, 118, 125, 66	mod2xi 17031	. . 3 ⊢ ((2↑;;170) mod ;;341) = (1 mod ;;341)
127	17, 9, 10	decmulnc 12702	. . . 4 ⊢ (2 · ;;170) = ;(2 · ;17)(2 · 0)
128		eqid 2737	. . . . . 6 ⊢ ;17 = ;17
129	69	oveq1i 7370	. . . . . . 7 ⊢ ((2 · 1) + 1) = (2 + 1)
130	129, 113	eqtri 2760	. . . . . 6 ⊢ ((2 · 1) + 1) = 3
131		7cn 12266	. . . . . . 7 ⊢ 7 ∈ ℂ
132		7t2e14 12744	. . . . . . 7 ⊢ (7 · 2) = ;14
133	131, 29, 132	mulcomli 11145	. . . . . 6 ⊢ (2 · 7) = ;14
134	17, 7, 8, 128, 2, 7, 130, 133	decmul2c 12701	. . . . 5 ⊢ (2 · ;17) = ;34
135	134, 70	deceq12i 12644	. . . 4 ⊢ ;(2 · ;17)(2 · 0) = ;;340
136	127, 135	eqtri 2760	. . 3 ⊢ (2 · ;;170) = ;;340
137	5, 6, 11, 12, 7, 7, 126, 136, 78	mod2xi 17031	. 2 ⊢ ((2↑;;340) mod ;;341) = (1 mod ;;341)
138		1re 11135	. . 3 ⊢ 1 ∈ ℝ
139		nnrp 12945	. . . 4 ⊢ (;;341 ∈ ℕ → ;;341 ∈ ℝ⁺)
140	5, 139	ax-mp 5	. . 3 ⊢ ;;341 ∈ ℝ⁺
141		0le1 11664	. . 3 ⊢ 0 ≤ 1
142		4nn 12255	. . . . 5 ⊢ 4 ∈ ℕ
143	1, 142	decnncl 12655	. . . 4 ⊢ ;34 ∈ ℕ
144		9re 12271	. . . . 5 ⊢ 9 ∈ ℝ
145		1lt9 12373	. . . . 5 ⊢ 1 < 9
146	138, 144, 145	ltleii 11260	. . . 4 ⊢ 1 ≤ 9
147	143, 7, 7, 146	decltdi 12674	. . 3 ⊢ 1 < ;;341
148		modid 13846	. . 3 ⊢ (((1 ∈ ℝ ∧ ;;341 ∈ ℝ⁺) ∧ (0 ≤ 1 ∧ 1 < ;;341)) → (1 mod ;;341) = 1)
149	138, 140, 141, 147, 148	mp4an 694	. 2 ⊢ (1 mod ;;341) = 1
150	137, 149	eqtri 2760	1 ⊢ ((2↑;;340) mod ;;341) = 1

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 class class class wbr 5086 (class class class)co 7360 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 ≤ cle 11171 ℕcn 12165 2c2 12227 3c3 12228 4c4 12229 5c5 12230 6c6 12231 7c7 12232 8c8 12233 9c9 12234 cdc 12635 ℝ⁺crp 12933 mod cmo 13819 ↑cexp 14014

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9348 df-inf 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015

This theorem is referenced by: 341fppr2 48222

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