1259lem5 - Metamath Proof Explorer

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Theorem 1259lem5 17173

Description: Lemma for 1259prm 17174. Calculate the GCD of 2↑34 − 1≡869 with 𝑁 = 1259. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)

**Hypothesis**
Ref	Expression
1259prm.1	⊢ 𝑁 = ;;;1259

**Assertion**
Ref	Expression
1259lem5	⊢ (((2↑;34) − 1) gcd 𝑁) = 1

**Proof of Theorem 1259lem5**
Step	Hyp	Ref	Expression
1		2nn 12340	. . . 4 ⊢ 2 ∈ ℕ
2		3nn0 12546	. . . . 5 ⊢ 3 ∈ ℕ₀
3		4nn0 12547	. . . . 5 ⊢ 4 ∈ ℕ₀
4	2, 3	deccl 12750	. . . 4 ⊢ ;34 ∈ ℕ₀
5		nnexpcl 14116	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;34 ∈ ℕ₀) → (2↑;34) ∈ ℕ)
6	1, 4, 5	mp2an 692	. . 3 ⊢ (2↑;34) ∈ ℕ
7		nnm1nn0 12569	. . 3 ⊢ ((2↑;34) ∈ ℕ → ((2↑;34) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;34) − 1) ∈ ℕ₀
9		8nn0 12551	. . . 4 ⊢ 8 ∈ ℕ₀
10		6nn0 12549	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12750	. . 3 ⊢ ;86 ∈ ℕ₀
12		9nn0 12552	. . 3 ⊢ 9 ∈ ℕ₀
13	11, 12	deccl 12750	. 2 ⊢ ;;869 ∈ ℕ₀
14		1259prm.1	. . 3 ⊢ 𝑁 = ;;;1259
15		1nn0 12544	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		2nn0 12545	. . . . . 6 ⊢ 2 ∈ ℕ₀
17	15, 16	deccl 12750	. . . . 5 ⊢ ;12 ∈ ℕ₀
18		5nn0 12548	. . . . 5 ⊢ 5 ∈ ℕ₀
19	17, 18	deccl 12750	. . . 4 ⊢ ;;125 ∈ ℕ₀
20		9nn 12365	. . . 4 ⊢ 9 ∈ ℕ
21	19, 20	decnncl 12755	. . 3 ⊢ ;;;1259 ∈ ℕ
22	14, 21	eqeltri 2836	. 2 ⊢ 𝑁 ∈ ℕ
23	14	1259lem2 17170	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
24		6p1e7 12415	. . . . 5 ⊢ (6 + 1) = 7
25		eqid 2736	. . . . 5 ⊢ ;86 = ;86
26	9, 10, 24, 25	decsuc 12766	. . . 4 ⊢ (;86 + 1) = ;87
27		eqid 2736	. . . 4 ⊢ ;;869 = ;;869
28	11, 26, 27	decsucc 12776	. . 3 ⊢ (;;869 + 1) = ;;870
29	22, 6, 15, 13, 23, 28	modsubi 17111	. 2 ⊢ (((2↑;34) − 1) mod 𝑁) = (;;869 mod 𝑁)
30	2, 12	deccl 12750	. . . 4 ⊢ ;39 ∈ ℕ₀
31		0nn0 12543	. . . 4 ⊢ 0 ∈ ℕ₀
32	30, 31	deccl 12750	. . 3 ⊢ ;;390 ∈ ℕ₀
33	9, 12	deccl 12750	. . . 4 ⊢ ;89 ∈ ℕ₀
34	16, 15	deccl 12750	. . . . . 6 ⊢ ;21 ∈ ℕ₀
35	15, 2	deccl 12750	. . . . . . 7 ⊢ ;13 ∈ ℕ₀
36	34	nn0zi 12644	. . . . . . . . 9 ⊢ ;21 ∈ ℤ
37	35	nn0zi 12644	. . . . . . . . 9 ⊢ ;13 ∈ ℤ
38		gcdcom 16551	. . . . . . . . 9 ⊢ ((;21 ∈ ℤ ∧ ;13 ∈ ℤ) → (;21 gcd ;13) = (;13 gcd ;21))
39	36, 37, 38	mp2an 692	. . . . . . . 8 ⊢ (;21 gcd ;13) = (;13 gcd ;21)
40		3nn 12346	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
41	15, 40	decnncl 12755	. . . . . . . . . 10 ⊢ ;13 ∈ ℕ
42		8nn 12362	. . . . . . . . . 10 ⊢ 8 ∈ ℕ
43		eqid 2736	. . . . . . . . . . 11 ⊢ ;13 = ;13
44	9	dec0h 12757	. . . . . . . . . . 11 ⊢ 8 = ;08
45		ax-1cn 11214	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46	45	mulridi 11266	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
47	45	addlidi 11450	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
48	46, 47	oveq12i 7444	. . . . . . . . . . . 12 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
49		1p1e2 12392	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
50	48, 49	eqtri 2764	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 1)) = 2
51		3cn 12348	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
52	51	mulridi 11266	. . . . . . . . . . . . 13 ⊢ (3 · 1) = 3
53	52	oveq1i 7442	. . . . . . . . . . . 12 ⊢ ((3 · 1) + 8) = (3 + 8)
54		8cn 12364	. . . . . . . . . . . . 13 ⊢ 8 ∈ ℂ
55		8p3e11 12816	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
56	54, 51, 55	addcomli 11454	. . . . . . . . . . . 12 ⊢ (3 + 8) = ;11
57	53, 56	eqtri 2764	. . . . . . . . . . 11 ⊢ ((3 · 1) + 8) = ;11
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 12787	. . . . . . . . . 10 ⊢ ((;13 · 1) + 8) = ;21
59		1nn 12278	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
60		8lt10 12867	. . . . . . . . . . 11 ⊢ 8 < ;10
61	59, 2, 9, 60	declti 12773	. . . . . . . . . 10 ⊢ 8 < ;13
62	41, 15, 42, 58, 61	ndvdsi 16450	. . . . . . . . 9 ⊢ ¬ ;13 ∥ ;21
63		13prm 17154	. . . . . . . . . 10 ⊢ ;13 ∈ ℙ
64		coprm 16749	. . . . . . . . . 10 ⊢ ((;13 ∈ ℙ ∧ ;21 ∈ ℤ) → (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1))
65	63, 36, 64	mp2an 692	. . . . . . . . 9 ⊢ (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1)
66	62, 65	mpbi 230	. . . . . . . 8 ⊢ (;13 gcd ;21) = 1
67	39, 66	eqtri 2764	. . . . . . 7 ⊢ (;21 gcd ;13) = 1
68		eqid 2736	. . . . . . . 8 ⊢ ;21 = ;21
69		2cn 12342	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
70	69	mullidi 11267	. . . . . . . . . 10 ⊢ (1 · 2) = 2
71	45	addridi 11449	. . . . . . . . . 10 ⊢ (1 + 0) = 1
72	70, 71	oveq12i 7444	. . . . . . . . 9 ⊢ ((1 · 2) + (1 + 0)) = (2 + 1)
73		2p1e3 12409	. . . . . . . . 9 ⊢ (2 + 1) = 3
74	72, 73	eqtri 2764	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 0)) = 3
75	46	oveq1i 7442	. . . . . . . . 9 ⊢ ((1 · 1) + 3) = (1 + 3)
76		3p1e4 12412	. . . . . . . . . 10 ⊢ (3 + 1) = 4
77	51, 45, 76	addcomli 11454	. . . . . . . . 9 ⊢ (1 + 3) = 4
78	3	dec0h 12757	. . . . . . . . 9 ⊢ 4 = ;04
79	75, 77, 78	3eqtri 2768	. . . . . . . 8 ⊢ ((1 · 1) + 3) = ;04
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 12788	. . . . . . 7 ⊢ ((1 · ;21) + ;13) = ;34
81	15, 35, 34, 67, 80	gcdi 17112	. . . . . 6 ⊢ (;34 gcd ;21) = 1
82		eqid 2736	. . . . . . 7 ⊢ ;34 = ;34
83		3t2e6 12433	. . . . . . . . . 10 ⊢ (3 · 2) = 6
84	51, 69, 83	mulcomli 11271	. . . . . . . . 9 ⊢ (2 · 3) = 6
85	69	addridi 11449	. . . . . . . . 9 ⊢ (2 + 0) = 2
86	84, 85	oveq12i 7444	. . . . . . . 8 ⊢ ((2 · 3) + (2 + 0)) = (6 + 2)
87		6p2e8 12426	. . . . . . . 8 ⊢ (6 + 2) = 8
88	86, 87	eqtri 2764	. . . . . . 7 ⊢ ((2 · 3) + (2 + 0)) = 8
89		4cn 12352	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
90		4t2e8 12435	. . . . . . . . . 10 ⊢ (4 · 2) = 8
91	89, 69, 90	mulcomli 11271	. . . . . . . . 9 ⊢ (2 · 4) = 8
92	91	oveq1i 7442	. . . . . . . 8 ⊢ ((2 · 4) + 1) = (8 + 1)
93		8p1e9 12417	. . . . . . . 8 ⊢ (8 + 1) = 9
94	12	dec0h 12757	. . . . . . . 8 ⊢ 9 = ;09
95	92, 93, 94	3eqtri 2768	. . . . . . 7 ⊢ ((2 · 4) + 1) = ;09
96	2, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95	decma2c 12788	. . . . . 6 ⊢ ((2 · ;34) + ;21) = ;89
97	16, 34, 4, 81, 96	gcdi 17112	. . . . 5 ⊢ (;89 gcd ;34) = 1
98		eqid 2736	. . . . . 6 ⊢ ;89 = ;89
99		4p3e7 12421	. . . . . . . . 9 ⊢ (4 + 3) = 7
100	89, 51, 99	addcomli 11454	. . . . . . . 8 ⊢ (3 + 4) = 7
101	100	oveq2i 7443	. . . . . . 7 ⊢ ((4 · 8) + (3 + 4)) = ((4 · 8) + 7)
102		7nn0 12550	. . . . . . . 8 ⊢ 7 ∈ ℕ₀
103		8t4e32 12852	. . . . . . . . 9 ⊢ (8 · 4) = ;32
104	54, 89, 103	mulcomli 11271	. . . . . . . 8 ⊢ (4 · 8) = ;32
105		7cn 12361	. . . . . . . . 9 ⊢ 7 ∈ ℂ
106		7p2e9 12428	. . . . . . . . 9 ⊢ (7 + 2) = 9
107	105, 69, 106	addcomli 11454	. . . . . . . 8 ⊢ (2 + 7) = 9
108	2, 16, 102, 104, 107	decaddi 12795	. . . . . . 7 ⊢ ((4 · 8) + 7) = ;39
109	101, 108	eqtri 2764	. . . . . 6 ⊢ ((4 · 8) + (3 + 4)) = ;39
110		9cn 12367	. . . . . . . 8 ⊢ 9 ∈ ℂ
111		9t4e36 12859	. . . . . . . 8 ⊢ (9 · 4) = ;36
112	110, 89, 111	mulcomli 11271	. . . . . . 7 ⊢ (4 · 9) = ;36
113		6p4e10 12807	. . . . . . 7 ⊢ (6 + 4) = ;10
114	2, 10, 3, 112, 76, 113	decaddci2 12797	. . . . . 6 ⊢ ((4 · 9) + 4) = ;40
115	9, 12, 2, 3, 98, 82, 3, 31, 3, 109, 114	decma2c 12788	. . . . 5 ⊢ ((4 · ;89) + ;34) = ;;390
116	3, 4, 33, 97, 115	gcdi 17112	. . . 4 ⊢ (;;390 gcd ;89) = 1
117		eqid 2736	. . . . 5 ⊢ ;;390 = ;;390
118		eqid 2736	. . . . . 6 ⊢ ;39 = ;39
119	54	addridi 11449	. . . . . . 7 ⊢ (8 + 0) = 8
120	119, 44	eqtri 2764	. . . . . 6 ⊢ (8 + 0) = ;08
121	69	addlidi 11450	. . . . . . . 8 ⊢ (0 + 2) = 2
122	84, 121	oveq12i 7444	. . . . . . 7 ⊢ ((2 · 3) + (0 + 2)) = (6 + 2)
123	122, 87	eqtri 2764	. . . . . 6 ⊢ ((2 · 3) + (0 + 2)) = 8
124		9t2e18 12857	. . . . . . . 8 ⊢ (9 · 2) = ;18
125	110, 69, 124	mulcomli 11271	. . . . . . 7 ⊢ (2 · 9) = ;18
126		8p8e16 12821	. . . . . . 7 ⊢ (8 + 8) = ;16
127	15, 9, 9, 125, 49, 10, 126	decaddci 12796	. . . . . 6 ⊢ ((2 · 9) + 8) = ;26
128	2, 12, 31, 9, 118, 120, 16, 10, 16, 123, 127	decma2c 12788	. . . . 5 ⊢ ((2 · ;39) + (8 + 0)) = ;86
129		2t0e0 12436	. . . . . . 7 ⊢ (2 · 0) = 0
130	129	oveq1i 7442	. . . . . 6 ⊢ ((2 · 0) + 9) = (0 + 9)
131	110	addlidi 11450	. . . . . 6 ⊢ (0 + 9) = 9
132	130, 131, 94	3eqtri 2768	. . . . 5 ⊢ ((2 · 0) + 9) = ;09
133	30, 31, 9, 12, 117, 98, 16, 12, 31, 128, 132	decma2c 12788	. . . 4 ⊢ ((2 · ;;390) + ;89) = ;;869
134	16, 33, 32, 116, 133	gcdi 17112	. . 3 ⊢ (;;869 gcd ;;390) = 1
135	30	nn0cni 12540	. . . . . . 7 ⊢ ;39 ∈ ℂ
136	135	addridi 11449	. . . . . 6 ⊢ (;39 + 0) = ;39
137	54	mullidi 11267	. . . . . . . 8 ⊢ (1 · 8) = 8
138	137, 76	oveq12i 7444	. . . . . . 7 ⊢ ((1 · 8) + (3 + 1)) = (8 + 4)
139		8p4e12 12817	. . . . . . 7 ⊢ (8 + 4) = ;12
140	138, 139	eqtri 2764	. . . . . 6 ⊢ ((1 · 8) + (3 + 1)) = ;12
141		6cn 12358	. . . . . . . . 9 ⊢ 6 ∈ ℂ
142	141	mullidi 11267	. . . . . . . 8 ⊢ (1 · 6) = 6
143	142	oveq1i 7442	. . . . . . 7 ⊢ ((1 · 6) + 9) = (6 + 9)
144		9p6e15 12826	. . . . . . . 8 ⊢ (9 + 6) = ;15
145	110, 141, 144	addcomli 11454	. . . . . . 7 ⊢ (6 + 9) = ;15
146	143, 145	eqtri 2764	. . . . . 6 ⊢ ((1 · 6) + 9) = ;15
147	9, 10, 2, 12, 25, 136, 15, 18, 15, 140, 146	decma2c 12788	. . . . 5 ⊢ ((1 · ;86) + (;39 + 0)) = ;;125
148	110	mullidi 11267	. . . . . . 7 ⊢ (1 · 9) = 9
149	148	oveq1i 7442	. . . . . 6 ⊢ ((1 · 9) + 0) = (9 + 0)
150	110	addridi 11449	. . . . . 6 ⊢ (9 + 0) = 9
151	149, 150, 94	3eqtri 2768	. . . . 5 ⊢ ((1 · 9) + 0) = ;09
152	11, 12, 30, 31, 27, 117, 15, 12, 31, 147, 151	decma2c 12788	. . . 4 ⊢ ((1 · ;;869) + ;;390) = ;;;1259
153	152, 14	eqtr4i 2767	. . 3 ⊢ ((1 · ;;869) + ;;390) = 𝑁
154	15, 32, 13, 134, 153	gcdi 17112	. 2 ⊢ (𝑁 gcd ;;869) = 1
155	8, 13, 22, 29, 154	gcdmodi 17113	1 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 − cmin 11493 ℕcn 12267 2c2 12322 3c3 12323 4c4 12324 5c5 12325 6c6 12326 7c7 12327 8c8 12328 9c9 12329 ℕ₀cn0 12528 ℤcz 12615 cdc 12735 ↑cexp 14103 ∥ cdvds 16291 gcd cgcd 16532 ℙcprime 16709

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-fz 13549 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-dvds 16292 df-gcd 16533 df-prm 16710

This theorem is referenced by: 1259prm 17174

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