1259lem5 - Metamath Proof Explorer

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Theorem 1259lem5 16764

Description: Lemma for 1259prm 16765. 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 11976	. . . 4 ⊢ 2 ∈ ℕ
2		3nn0 12181	. . . . 5 ⊢ 3 ∈ ℕ₀
3		4nn0 12182	. . . . 5 ⊢ 4 ∈ ℕ₀
4	2, 3	deccl 12381	. . . 4 ⊢ ;34 ∈ ℕ₀
5		nnexpcl 13723	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;34 ∈ ℕ₀) → (2↑;34) ∈ ℕ)
6	1, 4, 5	mp2an 688	. . 3 ⊢ (2↑;34) ∈ ℕ
7		nnm1nn0 12204	. . 3 ⊢ ((2↑;34) ∈ ℕ → ((2↑;34) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;34) − 1) ∈ ℕ₀
9		8nn0 12186	. . . 4 ⊢ 8 ∈ ℕ₀
10		6nn0 12184	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12381	. . 3 ⊢ ;86 ∈ ℕ₀
12		9nn0 12187	. . 3 ⊢ 9 ∈ ℕ₀
13	11, 12	deccl 12381	. 2 ⊢ ;;869 ∈ ℕ₀
14		1259prm.1	. . 3 ⊢ 𝑁 = ;;;1259
15		1nn0 12179	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		2nn0 12180	. . . . . 6 ⊢ 2 ∈ ℕ₀
17	15, 16	deccl 12381	. . . . 5 ⊢ ;12 ∈ ℕ₀
18		5nn0 12183	. . . . 5 ⊢ 5 ∈ ℕ₀
19	17, 18	deccl 12381	. . . 4 ⊢ ;;125 ∈ ℕ₀
20		9nn 12001	. . . 4 ⊢ 9 ∈ ℕ
21	19, 20	decnncl 12386	. . 3 ⊢ ;;;1259 ∈ ℕ
22	14, 21	eqeltri 2835	. 2 ⊢ 𝑁 ∈ ℕ
23	14	1259lem2 16761	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
24		6p1e7 12051	. . . . 5 ⊢ (6 + 1) = 7
25		eqid 2738	. . . . 5 ⊢ ;86 = ;86
26	9, 10, 24, 25	decsuc 12397	. . . 4 ⊢ (;86 + 1) = ;87
27		eqid 2738	. . . 4 ⊢ ;;869 = ;;869
28	11, 26, 27	decsucc 12407	. . 3 ⊢ (;;869 + 1) = ;;870
29	22, 6, 15, 13, 23, 28	modsubi 16701	. 2 ⊢ (((2↑;34) − 1) mod 𝑁) = (;;869 mod 𝑁)
30	2, 12	deccl 12381	. . . 4 ⊢ ;39 ∈ ℕ₀
31		0nn0 12178	. . . 4 ⊢ 0 ∈ ℕ₀
32	30, 31	deccl 12381	. . 3 ⊢ ;;390 ∈ ℕ₀
33	9, 12	deccl 12381	. . . 4 ⊢ ;89 ∈ ℕ₀
34	16, 15	deccl 12381	. . . . . 6 ⊢ ;21 ∈ ℕ₀
35	15, 2	deccl 12381	. . . . . . 7 ⊢ ;13 ∈ ℕ₀
36	34	nn0zi 12275	. . . . . . . . 9 ⊢ ;21 ∈ ℤ
37	35	nn0zi 12275	. . . . . . . . 9 ⊢ ;13 ∈ ℤ
38		gcdcom 16148	. . . . . . . . 9 ⊢ ((;21 ∈ ℤ ∧ ;13 ∈ ℤ) → (;21 gcd ;13) = (;13 gcd ;21))
39	36, 37, 38	mp2an 688	. . . . . . . 8 ⊢ (;21 gcd ;13) = (;13 gcd ;21)
40		3nn 11982	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
41	15, 40	decnncl 12386	. . . . . . . . . 10 ⊢ ;13 ∈ ℕ
42		8nn 11998	. . . . . . . . . 10 ⊢ 8 ∈ ℕ
43		eqid 2738	. . . . . . . . . . 11 ⊢ ;13 = ;13
44	9	dec0h 12388	. . . . . . . . . . 11 ⊢ 8 = ;08
45		ax-1cn 10860	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46	45	mulid1i 10910	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
47	45	addid2i 11093	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
48	46, 47	oveq12i 7267	. . . . . . . . . . . 12 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
49		1p1e2 12028	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
50	48, 49	eqtri 2766	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 1)) = 2
51		3cn 11984	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
52	51	mulid1i 10910	. . . . . . . . . . . . 13 ⊢ (3 · 1) = 3
53	52	oveq1i 7265	. . . . . . . . . . . 12 ⊢ ((3 · 1) + 8) = (3 + 8)
54		8cn 12000	. . . . . . . . . . . . 13 ⊢ 8 ∈ ℂ
55		8p3e11 12447	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
56	54, 51, 55	addcomli 11097	. . . . . . . . . . . 12 ⊢ (3 + 8) = ;11
57	53, 56	eqtri 2766	. . . . . . . . . . 11 ⊢ ((3 · 1) + 8) = ;11
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 12418	. . . . . . . . . 10 ⊢ ((;13 · 1) + 8) = ;21
59		1nn 11914	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
60		8lt10 12498	. . . . . . . . . . 11 ⊢ 8 < ;10
61	59, 2, 9, 60	declti 12404	. . . . . . . . . 10 ⊢ 8 < ;13
62	41, 15, 42, 58, 61	ndvdsi 16049	. . . . . . . . 9 ⊢ ¬ ;13 ∥ ;21
63		13prm 16745	. . . . . . . . . 10 ⊢ ;13 ∈ ℙ
64		coprm 16344	. . . . . . . . . 10 ⊢ ((;13 ∈ ℙ ∧ ;21 ∈ ℤ) → (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1))
65	63, 36, 64	mp2an 688	. . . . . . . . 9 ⊢ (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1)
66	62, 65	mpbi 229	. . . . . . . 8 ⊢ (;13 gcd ;21) = 1
67	39, 66	eqtri 2766	. . . . . . 7 ⊢ (;21 gcd ;13) = 1
68		eqid 2738	. . . . . . . 8 ⊢ ;21 = ;21
69		2cn 11978	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
70	69	mulid2i 10911	. . . . . . . . . 10 ⊢ (1 · 2) = 2
71	45	addid1i 11092	. . . . . . . . . 10 ⊢ (1 + 0) = 1
72	70, 71	oveq12i 7267	. . . . . . . . 9 ⊢ ((1 · 2) + (1 + 0)) = (2 + 1)
73		2p1e3 12045	. . . . . . . . 9 ⊢ (2 + 1) = 3
74	72, 73	eqtri 2766	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 0)) = 3
75	46	oveq1i 7265	. . . . . . . . 9 ⊢ ((1 · 1) + 3) = (1 + 3)
76		3p1e4 12048	. . . . . . . . . 10 ⊢ (3 + 1) = 4
77	51, 45, 76	addcomli 11097	. . . . . . . . 9 ⊢ (1 + 3) = 4
78	3	dec0h 12388	. . . . . . . . 9 ⊢ 4 = ;04
79	75, 77, 78	3eqtri 2770	. . . . . . . 8 ⊢ ((1 · 1) + 3) = ;04
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 12419	. . . . . . 7 ⊢ ((1 · ;21) + ;13) = ;34
81	15, 35, 34, 67, 80	gcdi 16702	. . . . . 6 ⊢ (;34 gcd ;21) = 1
82		eqid 2738	. . . . . . 7 ⊢ ;34 = ;34
83		3t2e6 12069	. . . . . . . . . 10 ⊢ (3 · 2) = 6
84	51, 69, 83	mulcomli 10915	. . . . . . . . 9 ⊢ (2 · 3) = 6
85	69	addid1i 11092	. . . . . . . . 9 ⊢ (2 + 0) = 2
86	84, 85	oveq12i 7267	. . . . . . . 8 ⊢ ((2 · 3) + (2 + 0)) = (6 + 2)
87		6p2e8 12062	. . . . . . . 8 ⊢ (6 + 2) = 8
88	86, 87	eqtri 2766	. . . . . . 7 ⊢ ((2 · 3) + (2 + 0)) = 8
89		4cn 11988	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
90		4t2e8 12071	. . . . . . . . . 10 ⊢ (4 · 2) = 8
91	89, 69, 90	mulcomli 10915	. . . . . . . . 9 ⊢ (2 · 4) = 8
92	91	oveq1i 7265	. . . . . . . 8 ⊢ ((2 · 4) + 1) = (8 + 1)
93		8p1e9 12053	. . . . . . . 8 ⊢ (8 + 1) = 9
94	12	dec0h 12388	. . . . . . . 8 ⊢ 9 = ;09
95	92, 93, 94	3eqtri 2770	. . . . . . 7 ⊢ ((2 · 4) + 1) = ;09
96	2, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95	decma2c 12419	. . . . . 6 ⊢ ((2 · ;34) + ;21) = ;89
97	16, 34, 4, 81, 96	gcdi 16702	. . . . 5 ⊢ (;89 gcd ;34) = 1
98		eqid 2738	. . . . . 6 ⊢ ;89 = ;89
99		4p3e7 12057	. . . . . . . . 9 ⊢ (4 + 3) = 7
100	89, 51, 99	addcomli 11097	. . . . . . . 8 ⊢ (3 + 4) = 7
101	100	oveq2i 7266	. . . . . . 7 ⊢ ((4 · 8) + (3 + 4)) = ((4 · 8) + 7)
102		7nn0 12185	. . . . . . . 8 ⊢ 7 ∈ ℕ₀
103		8t4e32 12483	. . . . . . . . 9 ⊢ (8 · 4) = ;32
104	54, 89, 103	mulcomli 10915	. . . . . . . 8 ⊢ (4 · 8) = ;32
105		7cn 11997	. . . . . . . . 9 ⊢ 7 ∈ ℂ
106		7p2e9 12064	. . . . . . . . 9 ⊢ (7 + 2) = 9
107	105, 69, 106	addcomli 11097	. . . . . . . 8 ⊢ (2 + 7) = 9
108	2, 16, 102, 104, 107	decaddi 12426	. . . . . . 7 ⊢ ((4 · 8) + 7) = ;39
109	101, 108	eqtri 2766	. . . . . 6 ⊢ ((4 · 8) + (3 + 4)) = ;39
110		9cn 12003	. . . . . . . 8 ⊢ 9 ∈ ℂ
111		9t4e36 12490	. . . . . . . 8 ⊢ (9 · 4) = ;36
112	110, 89, 111	mulcomli 10915	. . . . . . 7 ⊢ (4 · 9) = ;36
113		6p4e10 12438	. . . . . . 7 ⊢ (6 + 4) = ;10
114	2, 10, 3, 112, 76, 113	decaddci2 12428	. . . . . 6 ⊢ ((4 · 9) + 4) = ;40
115	9, 12, 2, 3, 98, 82, 3, 31, 3, 109, 114	decma2c 12419	. . . . 5 ⊢ ((4 · ;89) + ;34) = ;;390
116	3, 4, 33, 97, 115	gcdi 16702	. . . 4 ⊢ (;;390 gcd ;89) = 1
117		eqid 2738	. . . . 5 ⊢ ;;390 = ;;390
118		eqid 2738	. . . . . 6 ⊢ ;39 = ;39
119	54	addid1i 11092	. . . . . . 7 ⊢ (8 + 0) = 8
120	119, 44	eqtri 2766	. . . . . 6 ⊢ (8 + 0) = ;08
121	69	addid2i 11093	. . . . . . . 8 ⊢ (0 + 2) = 2
122	84, 121	oveq12i 7267	. . . . . . 7 ⊢ ((2 · 3) + (0 + 2)) = (6 + 2)
123	122, 87	eqtri 2766	. . . . . 6 ⊢ ((2 · 3) + (0 + 2)) = 8
124		9t2e18 12488	. . . . . . . 8 ⊢ (9 · 2) = ;18
125	110, 69, 124	mulcomli 10915	. . . . . . 7 ⊢ (2 · 9) = ;18
126		8p8e16 12452	. . . . . . 7 ⊢ (8 + 8) = ;16
127	15, 9, 9, 125, 49, 10, 126	decaddci 12427	. . . . . 6 ⊢ ((2 · 9) + 8) = ;26
128	2, 12, 31, 9, 118, 120, 16, 10, 16, 123, 127	decma2c 12419	. . . . 5 ⊢ ((2 · ;39) + (8 + 0)) = ;86
129		2t0e0 12072	. . . . . . 7 ⊢ (2 · 0) = 0
130	129	oveq1i 7265	. . . . . 6 ⊢ ((2 · 0) + 9) = (0 + 9)
131	110	addid2i 11093	. . . . . 6 ⊢ (0 + 9) = 9
132	130, 131, 94	3eqtri 2770	. . . . 5 ⊢ ((2 · 0) + 9) = ;09
133	30, 31, 9, 12, 117, 98, 16, 12, 31, 128, 132	decma2c 12419	. . . 4 ⊢ ((2 · ;;390) + ;89) = ;;869
134	16, 33, 32, 116, 133	gcdi 16702	. . 3 ⊢ (;;869 gcd ;;390) = 1
135	30	nn0cni 12175	. . . . . . 7 ⊢ ;39 ∈ ℂ
136	135	addid1i 11092	. . . . . 6 ⊢ (;39 + 0) = ;39
137	54	mulid2i 10911	. . . . . . . 8 ⊢ (1 · 8) = 8
138	137, 76	oveq12i 7267	. . . . . . 7 ⊢ ((1 · 8) + (3 + 1)) = (8 + 4)
139		8p4e12 12448	. . . . . . 7 ⊢ (8 + 4) = ;12
140	138, 139	eqtri 2766	. . . . . 6 ⊢ ((1 · 8) + (3 + 1)) = ;12
141		6cn 11994	. . . . . . . . 9 ⊢ 6 ∈ ℂ
142	141	mulid2i 10911	. . . . . . . 8 ⊢ (1 · 6) = 6
143	142	oveq1i 7265	. . . . . . 7 ⊢ ((1 · 6) + 9) = (6 + 9)
144		9p6e15 12457	. . . . . . . 8 ⊢ (9 + 6) = ;15
145	110, 141, 144	addcomli 11097	. . . . . . 7 ⊢ (6 + 9) = ;15
146	143, 145	eqtri 2766	. . . . . 6 ⊢ ((1 · 6) + 9) = ;15
147	9, 10, 2, 12, 25, 136, 15, 18, 15, 140, 146	decma2c 12419	. . . . 5 ⊢ ((1 · ;86) + (;39 + 0)) = ;;125
148	110	mulid2i 10911	. . . . . . 7 ⊢ (1 · 9) = 9
149	148	oveq1i 7265	. . . . . 6 ⊢ ((1 · 9) + 0) = (9 + 0)
150	110	addid1i 11092	. . . . . 6 ⊢ (9 + 0) = 9
151	149, 150, 94	3eqtri 2770	. . . . 5 ⊢ ((1 · 9) + 0) = ;09
152	11, 12, 30, 31, 27, 117, 15, 12, 31, 147, 151	decma2c 12419	. . . 4 ⊢ ((1 · ;;869) + ;;390) = ;;;1259
153	152, 14	eqtr4i 2769	. . 3 ⊢ ((1 · ;;869) + ;;390) = 𝑁
154	15, 32, 13, 134, 153	gcdi 16702	. 2 ⊢ (𝑁 gcd ;;869) = 1
155	8, 13, 22, 29, 154	gcdmodi 16703	1 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 (class class class)co 7255 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 − cmin 11135 ℕcn 11903 2c2 11958 3c3 11959 4c4 11960 5c5 11961 6c6 11962 7c7 11963 8c8 11964 9c9 11965 ℕ₀cn0 12163 ℤcz 12249 cdc 12366 ↑cexp 13710 ∥ cdvds 15891 gcd cgcd 16129 ℙcprime 16304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-prm 16305

This theorem is referenced by: 1259prm 16765

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