1259lem5 - Metamath Proof Explorer

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Theorem 1259lem5 16470

Description: Lemma for 1259prm 16471. 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 11713	. . . 4 ⊢ 2 ∈ ℕ
2		3nn0 11918	. . . . 5 ⊢ 3 ∈ ℕ₀
3		4nn0 11919	. . . . 5 ⊢ 4 ∈ ℕ₀
4	2, 3	deccl 12116	. . . 4 ⊢ ;34 ∈ ℕ₀
5		nnexpcl 13445	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;34 ∈ ℕ₀) → (2↑;34) ∈ ℕ)
6	1, 4, 5	mp2an 690	. . 3 ⊢ (2↑;34) ∈ ℕ
7		nnm1nn0 11941	. . 3 ⊢ ((2↑;34) ∈ ℕ → ((2↑;34) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;34) − 1) ∈ ℕ₀
9		8nn0 11923	. . . 4 ⊢ 8 ∈ ℕ₀
10		6nn0 11921	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 12116	. . 3 ⊢ ;86 ∈ ℕ₀
12		9nn0 11924	. . 3 ⊢ 9 ∈ ℕ₀
13	11, 12	deccl 12116	. 2 ⊢ ;;869 ∈ ℕ₀
14		1259prm.1	. . 3 ⊢ 𝑁 = ;;;1259
15		1nn0 11916	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		2nn0 11917	. . . . . 6 ⊢ 2 ∈ ℕ₀
17	15, 16	deccl 12116	. . . . 5 ⊢ ;12 ∈ ℕ₀
18		5nn0 11920	. . . . 5 ⊢ 5 ∈ ℕ₀
19	17, 18	deccl 12116	. . . 4 ⊢ ;;125 ∈ ℕ₀
20		9nn 11738	. . . 4 ⊢ 9 ∈ ℕ
21	19, 20	decnncl 12121	. . 3 ⊢ ;;;1259 ∈ ℕ
22	14, 21	eqeltri 2911	. 2 ⊢ 𝑁 ∈ ℕ
23	14	1259lem2 16467	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
24		6p1e7 11788	. . . . 5 ⊢ (6 + 1) = 7
25		eqid 2823	. . . . 5 ⊢ ;86 = ;86
26	9, 10, 24, 25	decsuc 12132	. . . 4 ⊢ (;86 + 1) = ;87
27		eqid 2823	. . . 4 ⊢ ;;869 = ;;869
28	11, 26, 27	decsucc 12142	. . 3 ⊢ (;;869 + 1) = ;;870
29	22, 6, 15, 13, 23, 28	modsubi 16410	. 2 ⊢ (((2↑;34) − 1) mod 𝑁) = (;;869 mod 𝑁)
30	2, 12	deccl 12116	. . . 4 ⊢ ;39 ∈ ℕ₀
31		0nn0 11915	. . . 4 ⊢ 0 ∈ ℕ₀
32	30, 31	deccl 12116	. . 3 ⊢ ;;390 ∈ ℕ₀
33	9, 12	deccl 12116	. . . 4 ⊢ ;89 ∈ ℕ₀
34	16, 15	deccl 12116	. . . . . 6 ⊢ ;21 ∈ ℕ₀
35	15, 2	deccl 12116	. . . . . . 7 ⊢ ;13 ∈ ℕ₀
36	34	nn0zi 12010	. . . . . . . . 9 ⊢ ;21 ∈ ℤ
37	35	nn0zi 12010	. . . . . . . . 9 ⊢ ;13 ∈ ℤ
38		gcdcom 15864	. . . . . . . . 9 ⊢ ((;21 ∈ ℤ ∧ ;13 ∈ ℤ) → (;21 gcd ;13) = (;13 gcd ;21))
39	36, 37, 38	mp2an 690	. . . . . . . 8 ⊢ (;21 gcd ;13) = (;13 gcd ;21)
40		3nn 11719	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
41	15, 40	decnncl 12121	. . . . . . . . . 10 ⊢ ;13 ∈ ℕ
42		8nn 11735	. . . . . . . . . 10 ⊢ 8 ∈ ℕ
43		eqid 2823	. . . . . . . . . . 11 ⊢ ;13 = ;13
44	9	dec0h 12123	. . . . . . . . . . 11 ⊢ 8 = ;08
45		ax-1cn 10597	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46	45	mulid1i 10647	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
47	45	addid2i 10830	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
48	46, 47	oveq12i 7170	. . . . . . . . . . . 12 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
49		1p1e2 11765	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
50	48, 49	eqtri 2846	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 1)) = 2
51		3cn 11721	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
52	51	mulid1i 10647	. . . . . . . . . . . . 13 ⊢ (3 · 1) = 3
53	52	oveq1i 7168	. . . . . . . . . . . 12 ⊢ ((3 · 1) + 8) = (3 + 8)
54		8cn 11737	. . . . . . . . . . . . 13 ⊢ 8 ∈ ℂ
55		8p3e11 12182	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
56	54, 51, 55	addcomli 10834	. . . . . . . . . . . 12 ⊢ (3 + 8) = ;11
57	53, 56	eqtri 2846	. . . . . . . . . . 11 ⊢ ((3 · 1) + 8) = ;11
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 12153	. . . . . . . . . 10 ⊢ ((;13 · 1) + 8) = ;21
59		1nn 11651	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
60		8lt10 12233	. . . . . . . . . . 11 ⊢ 8 < ;10
61	59, 2, 9, 60	declti 12139	. . . . . . . . . 10 ⊢ 8 < ;13
62	41, 15, 42, 58, 61	ndvdsi 15765	. . . . . . . . 9 ⊢ ¬ ;13 ∥ ;21
63		13prm 16451	. . . . . . . . . 10 ⊢ ;13 ∈ ℙ
64		coprm 16057	. . . . . . . . . 10 ⊢ ((;13 ∈ ℙ ∧ ;21 ∈ ℤ) → (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1))
65	63, 36, 64	mp2an 690	. . . . . . . . 9 ⊢ (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1)
66	62, 65	mpbi 232	. . . . . . . 8 ⊢ (;13 gcd ;21) = 1
67	39, 66	eqtri 2846	. . . . . . 7 ⊢ (;21 gcd ;13) = 1
68		eqid 2823	. . . . . . . 8 ⊢ ;21 = ;21
69		2cn 11715	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
70	69	mulid2i 10648	. . . . . . . . . 10 ⊢ (1 · 2) = 2
71	45	addid1i 10829	. . . . . . . . . 10 ⊢ (1 + 0) = 1
72	70, 71	oveq12i 7170	. . . . . . . . 9 ⊢ ((1 · 2) + (1 + 0)) = (2 + 1)
73		2p1e3 11782	. . . . . . . . 9 ⊢ (2 + 1) = 3
74	72, 73	eqtri 2846	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 0)) = 3
75	46	oveq1i 7168	. . . . . . . . 9 ⊢ ((1 · 1) + 3) = (1 + 3)
76		3p1e4 11785	. . . . . . . . . 10 ⊢ (3 + 1) = 4
77	51, 45, 76	addcomli 10834	. . . . . . . . 9 ⊢ (1 + 3) = 4
78	3	dec0h 12123	. . . . . . . . 9 ⊢ 4 = ;04
79	75, 77, 78	3eqtri 2850	. . . . . . . 8 ⊢ ((1 · 1) + 3) = ;04
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 12154	. . . . . . 7 ⊢ ((1 · ;21) + ;13) = ;34
81	15, 35, 34, 67, 80	gcdi 16411	. . . . . 6 ⊢ (;34 gcd ;21) = 1
82		eqid 2823	. . . . . . 7 ⊢ ;34 = ;34
83		3t2e6 11806	. . . . . . . . . 10 ⊢ (3 · 2) = 6
84	51, 69, 83	mulcomli 10652	. . . . . . . . 9 ⊢ (2 · 3) = 6
85	69	addid1i 10829	. . . . . . . . 9 ⊢ (2 + 0) = 2
86	84, 85	oveq12i 7170	. . . . . . . 8 ⊢ ((2 · 3) + (2 + 0)) = (6 + 2)
87		6p2e8 11799	. . . . . . . 8 ⊢ (6 + 2) = 8
88	86, 87	eqtri 2846	. . . . . . 7 ⊢ ((2 · 3) + (2 + 0)) = 8
89		4cn 11725	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
90		4t2e8 11808	. . . . . . . . . 10 ⊢ (4 · 2) = 8
91	89, 69, 90	mulcomli 10652	. . . . . . . . 9 ⊢ (2 · 4) = 8
92	91	oveq1i 7168	. . . . . . . 8 ⊢ ((2 · 4) + 1) = (8 + 1)
93		8p1e9 11790	. . . . . . . 8 ⊢ (8 + 1) = 9
94	12	dec0h 12123	. . . . . . . 8 ⊢ 9 = ;09
95	92, 93, 94	3eqtri 2850	. . . . . . 7 ⊢ ((2 · 4) + 1) = ;09
96	2, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95	decma2c 12154	. . . . . 6 ⊢ ((2 · ;34) + ;21) = ;89
97	16, 34, 4, 81, 96	gcdi 16411	. . . . 5 ⊢ (;89 gcd ;34) = 1
98		eqid 2823	. . . . . 6 ⊢ ;89 = ;89
99		4p3e7 11794	. . . . . . . . 9 ⊢ (4 + 3) = 7
100	89, 51, 99	addcomli 10834	. . . . . . . 8 ⊢ (3 + 4) = 7
101	100	oveq2i 7169	. . . . . . 7 ⊢ ((4 · 8) + (3 + 4)) = ((4 · 8) + 7)
102		7nn0 11922	. . . . . . . 8 ⊢ 7 ∈ ℕ₀
103		8t4e32 12218	. . . . . . . . 9 ⊢ (8 · 4) = ;32
104	54, 89, 103	mulcomli 10652	. . . . . . . 8 ⊢ (4 · 8) = ;32
105		7cn 11734	. . . . . . . . 9 ⊢ 7 ∈ ℂ
106		7p2e9 11801	. . . . . . . . 9 ⊢ (7 + 2) = 9
107	105, 69, 106	addcomli 10834	. . . . . . . 8 ⊢ (2 + 7) = 9
108	2, 16, 102, 104, 107	decaddi 12161	. . . . . . 7 ⊢ ((4 · 8) + 7) = ;39
109	101, 108	eqtri 2846	. . . . . 6 ⊢ ((4 · 8) + (3 + 4)) = ;39
110		9cn 11740	. . . . . . . 8 ⊢ 9 ∈ ℂ
111		9t4e36 12225	. . . . . . . 8 ⊢ (9 · 4) = ;36
112	110, 89, 111	mulcomli 10652	. . . . . . 7 ⊢ (4 · 9) = ;36
113		6p4e10 12173	. . . . . . 7 ⊢ (6 + 4) = ;10
114	2, 10, 3, 112, 76, 113	decaddci2 12163	. . . . . 6 ⊢ ((4 · 9) + 4) = ;40
115	9, 12, 2, 3, 98, 82, 3, 31, 3, 109, 114	decma2c 12154	. . . . 5 ⊢ ((4 · ;89) + ;34) = ;;390
116	3, 4, 33, 97, 115	gcdi 16411	. . . 4 ⊢ (;;390 gcd ;89) = 1
117		eqid 2823	. . . . 5 ⊢ ;;390 = ;;390
118		eqid 2823	. . . . . 6 ⊢ ;39 = ;39
119	54	addid1i 10829	. . . . . . 7 ⊢ (8 + 0) = 8
120	119, 44	eqtri 2846	. . . . . 6 ⊢ (8 + 0) = ;08
121	69	addid2i 10830	. . . . . . . 8 ⊢ (0 + 2) = 2
122	84, 121	oveq12i 7170	. . . . . . 7 ⊢ ((2 · 3) + (0 + 2)) = (6 + 2)
123	122, 87	eqtri 2846	. . . . . 6 ⊢ ((2 · 3) + (0 + 2)) = 8
124		9t2e18 12223	. . . . . . . 8 ⊢ (9 · 2) = ;18
125	110, 69, 124	mulcomli 10652	. . . . . . 7 ⊢ (2 · 9) = ;18
126		8p8e16 12187	. . . . . . 7 ⊢ (8 + 8) = ;16
127	15, 9, 9, 125, 49, 10, 126	decaddci 12162	. . . . . 6 ⊢ ((2 · 9) + 8) = ;26
128	2, 12, 31, 9, 118, 120, 16, 10, 16, 123, 127	decma2c 12154	. . . . 5 ⊢ ((2 · ;39) + (8 + 0)) = ;86
129		2t0e0 11809	. . . . . . 7 ⊢ (2 · 0) = 0
130	129	oveq1i 7168	. . . . . 6 ⊢ ((2 · 0) + 9) = (0 + 9)
131	110	addid2i 10830	. . . . . 6 ⊢ (0 + 9) = 9
132	130, 131, 94	3eqtri 2850	. . . . 5 ⊢ ((2 · 0) + 9) = ;09
133	30, 31, 9, 12, 117, 98, 16, 12, 31, 128, 132	decma2c 12154	. . . 4 ⊢ ((2 · ;;390) + ;89) = ;;869
134	16, 33, 32, 116, 133	gcdi 16411	. . 3 ⊢ (;;869 gcd ;;390) = 1
135	30	nn0cni 11912	. . . . . . 7 ⊢ ;39 ∈ ℂ
136	135	addid1i 10829	. . . . . 6 ⊢ (;39 + 0) = ;39
137	54	mulid2i 10648	. . . . . . . 8 ⊢ (1 · 8) = 8
138	137, 76	oveq12i 7170	. . . . . . 7 ⊢ ((1 · 8) + (3 + 1)) = (8 + 4)
139		8p4e12 12183	. . . . . . 7 ⊢ (8 + 4) = ;12
140	138, 139	eqtri 2846	. . . . . 6 ⊢ ((1 · 8) + (3 + 1)) = ;12
141		6cn 11731	. . . . . . . . 9 ⊢ 6 ∈ ℂ
142	141	mulid2i 10648	. . . . . . . 8 ⊢ (1 · 6) = 6
143	142	oveq1i 7168	. . . . . . 7 ⊢ ((1 · 6) + 9) = (6 + 9)
144		9p6e15 12192	. . . . . . . 8 ⊢ (9 + 6) = ;15
145	110, 141, 144	addcomli 10834	. . . . . . 7 ⊢ (6 + 9) = ;15
146	143, 145	eqtri 2846	. . . . . 6 ⊢ ((1 · 6) + 9) = ;15
147	9, 10, 2, 12, 25, 136, 15, 18, 15, 140, 146	decma2c 12154	. . . . 5 ⊢ ((1 · ;86) + (;39 + 0)) = ;;125
148	110	mulid2i 10648	. . . . . . 7 ⊢ (1 · 9) = 9
149	148	oveq1i 7168	. . . . . 6 ⊢ ((1 · 9) + 0) = (9 + 0)
150	110	addid1i 10829	. . . . . 6 ⊢ (9 + 0) = 9
151	149, 150, 94	3eqtri 2850	. . . . 5 ⊢ ((1 · 9) + 0) = ;09
152	11, 12, 30, 31, 27, 117, 15, 12, 31, 147, 151	decma2c 12154	. . . 4 ⊢ ((1 · ;;869) + ;;390) = ;;;1259
153	152, 14	eqtr4i 2849	. . 3 ⊢ ((1 · ;;869) + ;;390) = 𝑁
154	15, 32, 13, 134, 153	gcdi 16411	. 2 ⊢ (𝑁 gcd ;;869) = 1
155	8, 13, 22, 29, 154	gcdmodi 16412	1 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 − cmin 10872 ℕcn 11640 2c2 11695 3c3 11696 4c4 11697 5c5 11698 6c6 11699 7c7 11700 8c8 11701 9c9 11702 ℕ₀cn0 11900 ℤcz 11984 cdc 12101 ↑cexp 13432 ∥ cdvds 15609 gcd cgcd 15845 ℙcprime 16017

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-gcd 15846 df-prm 16018

This theorem is referenced by: 1259prm 16471

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