1259lem5 - Metamath Proof Explorer

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Theorem 1259lem5 15680

Description: Lemma for 1259prm 15681. 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 11062	. . . 4 ⊢ 2 ∈ ℕ
2		3nn0 11187	. . . . 5 ⊢ 3 ∈ ℕ₀
3		4nn0 11188	. . . . 5 ⊢ 4 ∈ ℕ₀
4	2, 3	deccl 11388	. . . 4 ⊢ ;34 ∈ ℕ₀
5		nnexpcl 12735	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;34 ∈ ℕ₀) → (2↑;34) ∈ ℕ)
6	1, 4, 5	mp2an 704	. . 3 ⊢ (2↑;34) ∈ ℕ
7		nnm1nn0 11211	. . 3 ⊢ ((2↑;34) ∈ ℕ → ((2↑;34) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;34) − 1) ∈ ℕ₀
9		8nn0 11192	. . . 4 ⊢ 8 ∈ ℕ₀
10		6nn0 11190	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 11388	. . 3 ⊢ ;86 ∈ ℕ₀
12		9nn0 11193	. . 3 ⊢ 9 ∈ ℕ₀
13	11, 12	deccl 11388	. 2 ⊢ ;;869 ∈ ℕ₀
14		1259prm.1	. . 3 ⊢ 𝑁 = ;;;1259
15		1nn0 11185	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		2nn0 11186	. . . . . 6 ⊢ 2 ∈ ℕ₀
17	15, 16	deccl 11388	. . . . 5 ⊢ ;12 ∈ ℕ₀
18		5nn0 11189	. . . . 5 ⊢ 5 ∈ ℕ₀
19	17, 18	deccl 11388	. . . 4 ⊢ ;;125 ∈ ℕ₀
20		9nn 11069	. . . 4 ⊢ 9 ∈ ℕ
21	19, 20	decnncl 11394	. . 3 ⊢ ;;;1259 ∈ ℕ
22	14, 21	eqeltri 2684	. 2 ⊢ 𝑁 ∈ ℕ
23	14	1259lem2 15677	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
24		6p1e7 11033	. . . . 5 ⊢ (6 + 1) = 7
25		eqid 2610	. . . . 5 ⊢ ;86 = ;86
26	9, 10, 24, 25	decsuc 11411	. . . 4 ⊢ (;86 + 1) = ;87
27		eqid 2610	. . . 4 ⊢ ;;869 = ;;869
28	11, 26, 27	decsucc 11426	. . 3 ⊢ (;;869 + 1) = ;;870
29	22, 6, 15, 13, 23, 28	modsubi 15614	. 2 ⊢ (((2↑;34) − 1) mod 𝑁) = (;;869 mod 𝑁)
30	2, 12	deccl 11388	. . . 4 ⊢ ;39 ∈ ℕ₀
31		0nn0 11184	. . . 4 ⊢ 0 ∈ ℕ₀
32	30, 31	deccl 11388	. . 3 ⊢ ;;390 ∈ ℕ₀
33	9, 12	deccl 11388	. . . 4 ⊢ ;89 ∈ ℕ₀
34	16, 15	deccl 11388	. . . . . 6 ⊢ ;21 ∈ ℕ₀
35	15, 2	deccl 11388	. . . . . . 7 ⊢ ;13 ∈ ℕ₀
36	34	nn0zi 11279	. . . . . . . . 9 ⊢ ;21 ∈ ℤ
37	35	nn0zi 11279	. . . . . . . . 9 ⊢ ;13 ∈ ℤ
38		gcdcom 15073	. . . . . . . . 9 ⊢ ((;21 ∈ ℤ ∧ ;13 ∈ ℤ) → (;21 gcd ;13) = (;13 gcd ;21))
39	36, 37, 38	mp2an 704	. . . . . . . 8 ⊢ (;21 gcd ;13) = (;13 gcd ;21)
40		3nn 11063	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
41	15, 40	decnncl 11394	. . . . . . . . . 10 ⊢ ;13 ∈ ℕ
42		8nn 11068	. . . . . . . . . 10 ⊢ 8 ∈ ℕ
43		eqid 2610	. . . . . . . . . . 11 ⊢ ;13 = ;13
44	9	dec0h 11398	. . . . . . . . . . 11 ⊢ 8 = ;08
45		ax-1cn 9873	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46	45	mulid1i 9921	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
47	45	addid2i 10103	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
48	46, 47	oveq12i 6561	. . . . . . . . . . . 12 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
49		1p1e2 11011	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
50	48, 49	eqtri 2632	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 1)) = 2
51		3cn 10972	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
52	51	mulid1i 9921	. . . . . . . . . . . . 13 ⊢ (3 · 1) = 3
53	52	oveq1i 6559	. . . . . . . . . . . 12 ⊢ ((3 · 1) + 8) = (3 + 8)
54		8cn 10983	. . . . . . . . . . . . 13 ⊢ 8 ∈ ℂ
55		8p3e11 11488	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
56	54, 51, 55	addcomli 10107	. . . . . . . . . . . 12 ⊢ (3 + 8) = ;11
57	53, 56	eqtri 2632	. . . . . . . . . . 11 ⊢ ((3 · 1) + 8) = ;11
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 11442	. . . . . . . . . 10 ⊢ ((;13 · 1) + 8) = ;21
59		1nn 10908	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
60		8lt10 11550	. . . . . . . . . . 11 ⊢ 8 < ;10
61	59, 2, 9, 60	declti 11422	. . . . . . . . . 10 ⊢ 8 < ;13
62	41, 15, 42, 58, 61	ndvdsi 14974	. . . . . . . . 9 ⊢ ¬ ;13 ∥ ;21
63		13prm 15661	. . . . . . . . . 10 ⊢ ;13 ∈ ℙ
64		coprm 15261	. . . . . . . . . 10 ⊢ ((;13 ∈ ℙ ∧ ;21 ∈ ℤ) → (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1))
65	63, 36, 64	mp2an 704	. . . . . . . . 9 ⊢ (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1)
66	62, 65	mpbi 219	. . . . . . . 8 ⊢ (;13 gcd ;21) = 1
67	39, 66	eqtri 2632	. . . . . . 7 ⊢ (;21 gcd ;13) = 1
68		eqid 2610	. . . . . . . 8 ⊢ ;21 = ;21
69		2cn 10968	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
70	69	mulid2i 9922	. . . . . . . . . 10 ⊢ (1 · 2) = 2
71	45	addid1i 10102	. . . . . . . . . 10 ⊢ (1 + 0) = 1
72	70, 71	oveq12i 6561	. . . . . . . . 9 ⊢ ((1 · 2) + (1 + 0)) = (2 + 1)
73		2p1e3 11028	. . . . . . . . 9 ⊢ (2 + 1) = 3
74	72, 73	eqtri 2632	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 0)) = 3
75	46	oveq1i 6559	. . . . . . . . 9 ⊢ ((1 · 1) + 3) = (1 + 3)
76		3p1e4 11030	. . . . . . . . . 10 ⊢ (3 + 1) = 4
77	51, 45, 76	addcomli 10107	. . . . . . . . 9 ⊢ (1 + 3) = 4
78	3	dec0h 11398	. . . . . . . . 9 ⊢ 4 = ;04
79	75, 77, 78	3eqtri 2636	. . . . . . . 8 ⊢ ((1 · 1) + 3) = ;04
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 11444	. . . . . . 7 ⊢ ((1 · ;21) + ;13) = ;34
81	15, 35, 34, 67, 80	gcdi 15615	. . . . . 6 ⊢ (;34 gcd ;21) = 1
82		eqid 2610	. . . . . . 7 ⊢ ;34 = ;34
83		3t2e6 11056	. . . . . . . . . 10 ⊢ (3 · 2) = 6
84	51, 69, 83	mulcomli 9926	. . . . . . . . 9 ⊢ (2 · 3) = 6
85	69	addid1i 10102	. . . . . . . . 9 ⊢ (2 + 0) = 2
86	84, 85	oveq12i 6561	. . . . . . . 8 ⊢ ((2 · 3) + (2 + 0)) = (6 + 2)
87		6p2e8 11046	. . . . . . . 8 ⊢ (6 + 2) = 8
88	86, 87	eqtri 2632	. . . . . . 7 ⊢ ((2 · 3) + (2 + 0)) = 8
89		4cn 10975	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
90		4t2e8 11058	. . . . . . . . . 10 ⊢ (4 · 2) = 8
91	89, 69, 90	mulcomli 9926	. . . . . . . . 9 ⊢ (2 · 4) = 8
92	91	oveq1i 6559	. . . . . . . 8 ⊢ ((2 · 4) + 1) = (8 + 1)
93		8p1e9 11035	. . . . . . . 8 ⊢ (8 + 1) = 9
94	12	dec0h 11398	. . . . . . . 8 ⊢ 9 = ;09
95	92, 93, 94	3eqtri 2636	. . . . . . 7 ⊢ ((2 · 4) + 1) = ;09
96	2, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95	decma2c 11444	. . . . . 6 ⊢ ((2 · ;34) + ;21) = ;89
97	16, 34, 4, 81, 96	gcdi 15615	. . . . 5 ⊢ (;89 gcd ;34) = 1
98		eqid 2610	. . . . . 6 ⊢ ;89 = ;89
99		4p3e7 11040	. . . . . . . . 9 ⊢ (4 + 3) = 7
100	89, 51, 99	addcomli 10107	. . . . . . . 8 ⊢ (3 + 4) = 7
101	100	oveq2i 6560	. . . . . . 7 ⊢ ((4 · 8) + (3 + 4)) = ((4 · 8) + 7)
102		7nn0 11191	. . . . . . . 8 ⊢ 7 ∈ ℕ₀
103		8t4e32 11532	. . . . . . . . 9 ⊢ (8 · 4) = ;32
104	54, 89, 103	mulcomli 9926	. . . . . . . 8 ⊢ (4 · 8) = ;32
105		7cn 10981	. . . . . . . . 9 ⊢ 7 ∈ ℂ
106		7p2e9 11049	. . . . . . . . 9 ⊢ (7 + 2) = 9
107	105, 69, 106	addcomli 10107	. . . . . . . 8 ⊢ (2 + 7) = 9
108	2, 16, 102, 104, 107	decaddi 11455	. . . . . . 7 ⊢ ((4 · 8) + 7) = ;39
109	101, 108	eqtri 2632	. . . . . 6 ⊢ ((4 · 8) + (3 + 4)) = ;39
110		9cn 10985	. . . . . . . 8 ⊢ 9 ∈ ℂ
111		9t4e36 11541	. . . . . . . 8 ⊢ (9 · 4) = ;36
112	110, 89, 111	mulcomli 9926	. . . . . . 7 ⊢ (4 · 9) = ;36
113		6p4e10 11474	. . . . . . 7 ⊢ (6 + 4) = ;10
114	2, 10, 3, 112, 76, 113	decaddci2 11457	. . . . . 6 ⊢ ((4 · 9) + 4) = ;40
115	9, 12, 2, 3, 98, 82, 3, 31, 3, 109, 114	decma2c 11444	. . . . 5 ⊢ ((4 · ;89) + ;34) = ;;390
116	3, 4, 33, 97, 115	gcdi 15615	. . . 4 ⊢ (;;390 gcd ;89) = 1
117		eqid 2610	. . . . 5 ⊢ ;;390 = ;;390
118		eqid 2610	. . . . . 6 ⊢ ;39 = ;39
119	54	addid1i 10102	. . . . . . 7 ⊢ (8 + 0) = 8
120	119, 44	eqtri 2632	. . . . . 6 ⊢ (8 + 0) = ;08
121	69	addid2i 10103	. . . . . . . 8 ⊢ (0 + 2) = 2
122	84, 121	oveq12i 6561	. . . . . . 7 ⊢ ((2 · 3) + (0 + 2)) = (6 + 2)
123	122, 87	eqtri 2632	. . . . . 6 ⊢ ((2 · 3) + (0 + 2)) = 8
124		9t2e18 11539	. . . . . . . 8 ⊢ (9 · 2) = ;18
125	110, 69, 124	mulcomli 9926	. . . . . . 7 ⊢ (2 · 9) = ;18
126		8p8e16 11494	. . . . . . 7 ⊢ (8 + 8) = ;16
127	15, 9, 9, 125, 49, 10, 126	decaddci 11456	. . . . . 6 ⊢ ((2 · 9) + 8) = ;26
128	2, 12, 31, 9, 118, 120, 16, 10, 16, 123, 127	decma2c 11444	. . . . 5 ⊢ ((2 · ;39) + (8 + 0)) = ;86
129		2t0e0 11060	. . . . . . 7 ⊢ (2 · 0) = 0
130	129	oveq1i 6559	. . . . . 6 ⊢ ((2 · 0) + 9) = (0 + 9)
131	110	addid2i 10103	. . . . . 6 ⊢ (0 + 9) = 9
132	130, 131, 94	3eqtri 2636	. . . . 5 ⊢ ((2 · 0) + 9) = ;09
133	30, 31, 9, 12, 117, 98, 16, 12, 31, 128, 132	decma2c 11444	. . . 4 ⊢ ((2 · ;;390) + ;89) = ;;869
134	16, 33, 32, 116, 133	gcdi 15615	. . 3 ⊢ (;;869 gcd ;;390) = 1
135	30	nn0cni 11181	. . . . . . 7 ⊢ ;39 ∈ ℂ
136	135	addid1i 10102	. . . . . 6 ⊢ (;39 + 0) = ;39
137	54	mulid2i 9922	. . . . . . . 8 ⊢ (1 · 8) = 8
138	137, 76	oveq12i 6561	. . . . . . 7 ⊢ ((1 · 8) + (3 + 1)) = (8 + 4)
139		8p4e12 11490	. . . . . . 7 ⊢ (8 + 4) = ;12
140	138, 139	eqtri 2632	. . . . . 6 ⊢ ((1 · 8) + (3 + 1)) = ;12
141		6cn 10979	. . . . . . . . 9 ⊢ 6 ∈ ℂ
142	141	mulid2i 9922	. . . . . . . 8 ⊢ (1 · 6) = 6
143	142	oveq1i 6559	. . . . . . 7 ⊢ ((1 · 6) + 9) = (6 + 9)
144		9p6e15 11500	. . . . . . . 8 ⊢ (9 + 6) = ;15
145	110, 141, 144	addcomli 10107	. . . . . . 7 ⊢ (6 + 9) = ;15
146	143, 145	eqtri 2632	. . . . . 6 ⊢ ((1 · 6) + 9) = ;15
147	9, 10, 2, 12, 25, 136, 15, 18, 15, 140, 146	decma2c 11444	. . . . 5 ⊢ ((1 · ;86) + (;39 + 0)) = ;;125
148	110	mulid2i 9922	. . . . . . 7 ⊢ (1 · 9) = 9
149	148	oveq1i 6559	. . . . . 6 ⊢ ((1 · 9) + 0) = (9 + 0)
150	110	addid1i 10102	. . . . . 6 ⊢ (9 + 0) = 9
151	149, 150, 94	3eqtri 2636	. . . . 5 ⊢ ((1 · 9) + 0) = ;09
152	11, 12, 30, 31, 27, 117, 15, 12, 31, 147, 151	decma2c 11444	. . . 4 ⊢ ((1 · ;;869) + ;;390) = ;;;1259
153	152, 14	eqtr4i 2635	. . 3 ⊢ ((1 · ;;869) + ;;390) = 𝑁
154	15, 32, 13, 134, 153	gcdi 15615	. 2 ⊢ (𝑁 gcd ;;869) = 1
155	8, 13, 22, 29, 154	gcdmodi 15616	1 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 ℕcn 10897 2c2 10947 3c3 10948 4c4 10949 5c5 10950 6c6 10951 7c7 10952 8c8 10953 9c9 10954 ℕ₀cn0 11169 ℤcz 11254 cdc 11369 ↑cexp 12722 ∥ cdvds 14821 gcd cgcd 15054 ℙcprime 15223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224

This theorem is referenced by: 1259prm 15681

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