1259lem5 - Metamath Proof Explorer

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Theorem 1259lem5 15842

Description: Lemma for 1259prm 15843. 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 11185	. . . 4 ⊢ 2 ∈ ℕ
2		3nn0 11310	. . . . 5 ⊢ 3 ∈ ℕ₀
3		4nn0 11311	. . . . 5 ⊢ 4 ∈ ℕ₀
4	2, 3	deccl 11512	. . . 4 ⊢ ;34 ∈ ℕ₀
5		nnexpcl 12873	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;34 ∈ ℕ₀) → (2↑;34) ∈ ℕ)
6	1, 4, 5	mp2an 708	. . 3 ⊢ (2↑;34) ∈ ℕ
7		nnm1nn0 11334	. . 3 ⊢ ((2↑;34) ∈ ℕ → ((2↑;34) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;34) − 1) ∈ ℕ₀
9		8nn0 11315	. . . 4 ⊢ 8 ∈ ℕ₀
10		6nn0 11313	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 11512	. . 3 ⊢ ;86 ∈ ℕ₀
12		9nn0 11316	. . 3 ⊢ 9 ∈ ℕ₀
13	11, 12	deccl 11512	. 2 ⊢ ;;869 ∈ ℕ₀
14		1259prm.1	. . 3 ⊢ 𝑁 = ;;;1259
15		1nn0 11308	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		2nn0 11309	. . . . . 6 ⊢ 2 ∈ ℕ₀
17	15, 16	deccl 11512	. . . . 5 ⊢ ;12 ∈ ℕ₀
18		5nn0 11312	. . . . 5 ⊢ 5 ∈ ℕ₀
19	17, 18	deccl 11512	. . . 4 ⊢ ;;125 ∈ ℕ₀
20		9nn 11192	. . . 4 ⊢ 9 ∈ ℕ
21	19, 20	decnncl 11518	. . 3 ⊢ ;;;1259 ∈ ℕ
22	14, 21	eqeltri 2697	. 2 ⊢ 𝑁 ∈ ℕ
23	14	1259lem2 15839	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
24		6p1e7 11156	. . . . 5 ⊢ (6 + 1) = 7
25		eqid 2622	. . . . 5 ⊢ ;86 = ;86
26	9, 10, 24, 25	decsuc 11535	. . . 4 ⊢ (;86 + 1) = ;87
27		eqid 2622	. . . 4 ⊢ ;;869 = ;;869
28	11, 26, 27	decsucc 11550	. . 3 ⊢ (;;869 + 1) = ;;870
29	22, 6, 15, 13, 23, 28	modsubi 15776	. 2 ⊢ (((2↑;34) − 1) mod 𝑁) = (;;869 mod 𝑁)
30	2, 12	deccl 11512	. . . 4 ⊢ ;39 ∈ ℕ₀
31		0nn0 11307	. . . 4 ⊢ 0 ∈ ℕ₀
32	30, 31	deccl 11512	. . 3 ⊢ ;;390 ∈ ℕ₀
33	9, 12	deccl 11512	. . . 4 ⊢ ;89 ∈ ℕ₀
34	16, 15	deccl 11512	. . . . . 6 ⊢ ;21 ∈ ℕ₀
35	15, 2	deccl 11512	. . . . . . 7 ⊢ ;13 ∈ ℕ₀
36	34	nn0zi 11402	. . . . . . . . 9 ⊢ ;21 ∈ ℤ
37	35	nn0zi 11402	. . . . . . . . 9 ⊢ ;13 ∈ ℤ
38		gcdcom 15235	. . . . . . . . 9 ⊢ ((;21 ∈ ℤ ∧ ;13 ∈ ℤ) → (;21 gcd ;13) = (;13 gcd ;21))
39	36, 37, 38	mp2an 708	. . . . . . . 8 ⊢ (;21 gcd ;13) = (;13 gcd ;21)
40		3nn 11186	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
41	15, 40	decnncl 11518	. . . . . . . . . 10 ⊢ ;13 ∈ ℕ
42		8nn 11191	. . . . . . . . . 10 ⊢ 8 ∈ ℕ
43		eqid 2622	. . . . . . . . . . 11 ⊢ ;13 = ;13
44	9	dec0h 11522	. . . . . . . . . . 11 ⊢ 8 = ;08
45		ax-1cn 9994	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46	45	mulid1i 10042	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
47	45	addid2i 10224	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
48	46, 47	oveq12i 6662	. . . . . . . . . . . 12 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
49		1p1e2 11134	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
50	48, 49	eqtri 2644	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 1)) = 2
51		3cn 11095	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
52	51	mulid1i 10042	. . . . . . . . . . . . 13 ⊢ (3 · 1) = 3
53	52	oveq1i 6660	. . . . . . . . . . . 12 ⊢ ((3 · 1) + 8) = (3 + 8)
54		8cn 11106	. . . . . . . . . . . . 13 ⊢ 8 ∈ ℂ
55		8p3e11 11612	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
56	54, 51, 55	addcomli 10228	. . . . . . . . . . . 12 ⊢ (3 + 8) = ;11
57	53, 56	eqtri 2644	. . . . . . . . . . 11 ⊢ ((3 · 1) + 8) = ;11
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 11566	. . . . . . . . . 10 ⊢ ((;13 · 1) + 8) = ;21
59		1nn 11031	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
60		8lt10 11674	. . . . . . . . . . 11 ⊢ 8 < ;10
61	59, 2, 9, 60	declti 11546	. . . . . . . . . 10 ⊢ 8 < ;13
62	41, 15, 42, 58, 61	ndvdsi 15136	. . . . . . . . 9 ⊢ ¬ ;13 ∥ ;21
63		13prm 15823	. . . . . . . . . 10 ⊢ ;13 ∈ ℙ
64		coprm 15423	. . . . . . . . . 10 ⊢ ((;13 ∈ ℙ ∧ ;21 ∈ ℤ) → (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1))
65	63, 36, 64	mp2an 708	. . . . . . . . 9 ⊢ (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1)
66	62, 65	mpbi 220	. . . . . . . 8 ⊢ (;13 gcd ;21) = 1
67	39, 66	eqtri 2644	. . . . . . 7 ⊢ (;21 gcd ;13) = 1
68		eqid 2622	. . . . . . . 8 ⊢ ;21 = ;21
69		2cn 11091	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
70	69	mulid2i 10043	. . . . . . . . . 10 ⊢ (1 · 2) = 2
71	45	addid1i 10223	. . . . . . . . . 10 ⊢ (1 + 0) = 1
72	70, 71	oveq12i 6662	. . . . . . . . 9 ⊢ ((1 · 2) + (1 + 0)) = (2 + 1)
73		2p1e3 11151	. . . . . . . . 9 ⊢ (2 + 1) = 3
74	72, 73	eqtri 2644	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 0)) = 3
75	46	oveq1i 6660	. . . . . . . . 9 ⊢ ((1 · 1) + 3) = (1 + 3)
76		3p1e4 11153	. . . . . . . . . 10 ⊢ (3 + 1) = 4
77	51, 45, 76	addcomli 10228	. . . . . . . . 9 ⊢ (1 + 3) = 4
78	3	dec0h 11522	. . . . . . . . 9 ⊢ 4 = ;04
79	75, 77, 78	3eqtri 2648	. . . . . . . 8 ⊢ ((1 · 1) + 3) = ;04
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 11568	. . . . . . 7 ⊢ ((1 · ;21) + ;13) = ;34
81	15, 35, 34, 67, 80	gcdi 15777	. . . . . 6 ⊢ (;34 gcd ;21) = 1
82		eqid 2622	. . . . . . 7 ⊢ ;34 = ;34
83		3t2e6 11179	. . . . . . . . . 10 ⊢ (3 · 2) = 6
84	51, 69, 83	mulcomli 10047	. . . . . . . . 9 ⊢ (2 · 3) = 6
85	69	addid1i 10223	. . . . . . . . 9 ⊢ (2 + 0) = 2
86	84, 85	oveq12i 6662	. . . . . . . 8 ⊢ ((2 · 3) + (2 + 0)) = (6 + 2)
87		6p2e8 11169	. . . . . . . 8 ⊢ (6 + 2) = 8
88	86, 87	eqtri 2644	. . . . . . 7 ⊢ ((2 · 3) + (2 + 0)) = 8
89		4cn 11098	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
90		4t2e8 11181	. . . . . . . . . 10 ⊢ (4 · 2) = 8
91	89, 69, 90	mulcomli 10047	. . . . . . . . 9 ⊢ (2 · 4) = 8
92	91	oveq1i 6660	. . . . . . . 8 ⊢ ((2 · 4) + 1) = (8 + 1)
93		8p1e9 11158	. . . . . . . 8 ⊢ (8 + 1) = 9
94	12	dec0h 11522	. . . . . . . 8 ⊢ 9 = ;09
95	92, 93, 94	3eqtri 2648	. . . . . . 7 ⊢ ((2 · 4) + 1) = ;09
96	2, 3, 16, 15, 82, 68, 16, 12, 31, 88, 95	decma2c 11568	. . . . . 6 ⊢ ((2 · ;34) + ;21) = ;89
97	16, 34, 4, 81, 96	gcdi 15777	. . . . 5 ⊢ (;89 gcd ;34) = 1
98		eqid 2622	. . . . . 6 ⊢ ;89 = ;89
99		4p3e7 11163	. . . . . . . . 9 ⊢ (4 + 3) = 7
100	89, 51, 99	addcomli 10228	. . . . . . . 8 ⊢ (3 + 4) = 7
101	100	oveq2i 6661	. . . . . . 7 ⊢ ((4 · 8) + (3 + 4)) = ((4 · 8) + 7)
102		7nn0 11314	. . . . . . . 8 ⊢ 7 ∈ ℕ₀
103		8t4e32 11656	. . . . . . . . 9 ⊢ (8 · 4) = ;32
104	54, 89, 103	mulcomli 10047	. . . . . . . 8 ⊢ (4 · 8) = ;32
105		7cn 11104	. . . . . . . . 9 ⊢ 7 ∈ ℂ
106		7p2e9 11172	. . . . . . . . 9 ⊢ (7 + 2) = 9
107	105, 69, 106	addcomli 10228	. . . . . . . 8 ⊢ (2 + 7) = 9
108	2, 16, 102, 104, 107	decaddi 11579	. . . . . . 7 ⊢ ((4 · 8) + 7) = ;39
109	101, 108	eqtri 2644	. . . . . 6 ⊢ ((4 · 8) + (3 + 4)) = ;39
110		9cn 11108	. . . . . . . 8 ⊢ 9 ∈ ℂ
111		9t4e36 11665	. . . . . . . 8 ⊢ (9 · 4) = ;36
112	110, 89, 111	mulcomli 10047	. . . . . . 7 ⊢ (4 · 9) = ;36
113		6p4e10 11598	. . . . . . 7 ⊢ (6 + 4) = ;10
114	2, 10, 3, 112, 76, 113	decaddci2 11581	. . . . . 6 ⊢ ((4 · 9) + 4) = ;40
115	9, 12, 2, 3, 98, 82, 3, 31, 3, 109, 114	decma2c 11568	. . . . 5 ⊢ ((4 · ;89) + ;34) = ;;390
116	3, 4, 33, 97, 115	gcdi 15777	. . . 4 ⊢ (;;390 gcd ;89) = 1
117		eqid 2622	. . . . 5 ⊢ ;;390 = ;;390
118		eqid 2622	. . . . . 6 ⊢ ;39 = ;39
119	54	addid1i 10223	. . . . . . 7 ⊢ (8 + 0) = 8
120	119, 44	eqtri 2644	. . . . . 6 ⊢ (8 + 0) = ;08
121	69	addid2i 10224	. . . . . . . 8 ⊢ (0 + 2) = 2
122	84, 121	oveq12i 6662	. . . . . . 7 ⊢ ((2 · 3) + (0 + 2)) = (6 + 2)
123	122, 87	eqtri 2644	. . . . . 6 ⊢ ((2 · 3) + (0 + 2)) = 8
124		9t2e18 11663	. . . . . . . 8 ⊢ (9 · 2) = ;18
125	110, 69, 124	mulcomli 10047	. . . . . . 7 ⊢ (2 · 9) = ;18
126		8p8e16 11618	. . . . . . 7 ⊢ (8 + 8) = ;16
127	15, 9, 9, 125, 49, 10, 126	decaddci 11580	. . . . . 6 ⊢ ((2 · 9) + 8) = ;26
128	2, 12, 31, 9, 118, 120, 16, 10, 16, 123, 127	decma2c 11568	. . . . 5 ⊢ ((2 · ;39) + (8 + 0)) = ;86
129		2t0e0 11183	. . . . . . 7 ⊢ (2 · 0) = 0
130	129	oveq1i 6660	. . . . . 6 ⊢ ((2 · 0) + 9) = (0 + 9)
131	110	addid2i 10224	. . . . . 6 ⊢ (0 + 9) = 9
132	130, 131, 94	3eqtri 2648	. . . . 5 ⊢ ((2 · 0) + 9) = ;09
133	30, 31, 9, 12, 117, 98, 16, 12, 31, 128, 132	decma2c 11568	. . . 4 ⊢ ((2 · ;;390) + ;89) = ;;869
134	16, 33, 32, 116, 133	gcdi 15777	. . 3 ⊢ (;;869 gcd ;;390) = 1
135	30	nn0cni 11304	. . . . . . 7 ⊢ ;39 ∈ ℂ
136	135	addid1i 10223	. . . . . 6 ⊢ (;39 + 0) = ;39
137	54	mulid2i 10043	. . . . . . . 8 ⊢ (1 · 8) = 8
138	137, 76	oveq12i 6662	. . . . . . 7 ⊢ ((1 · 8) + (3 + 1)) = (8 + 4)
139		8p4e12 11614	. . . . . . 7 ⊢ (8 + 4) = ;12
140	138, 139	eqtri 2644	. . . . . 6 ⊢ ((1 · 8) + (3 + 1)) = ;12
141		6cn 11102	. . . . . . . . 9 ⊢ 6 ∈ ℂ
142	141	mulid2i 10043	. . . . . . . 8 ⊢ (1 · 6) = 6
143	142	oveq1i 6660	. . . . . . 7 ⊢ ((1 · 6) + 9) = (6 + 9)
144		9p6e15 11624	. . . . . . . 8 ⊢ (9 + 6) = ;15
145	110, 141, 144	addcomli 10228	. . . . . . 7 ⊢ (6 + 9) = ;15
146	143, 145	eqtri 2644	. . . . . 6 ⊢ ((1 · 6) + 9) = ;15
147	9, 10, 2, 12, 25, 136, 15, 18, 15, 140, 146	decma2c 11568	. . . . 5 ⊢ ((1 · ;86) + (;39 + 0)) = ;;125
148	110	mulid2i 10043	. . . . . . 7 ⊢ (1 · 9) = 9
149	148	oveq1i 6660	. . . . . 6 ⊢ ((1 · 9) + 0) = (9 + 0)
150	110	addid1i 10223	. . . . . 6 ⊢ (9 + 0) = 9
151	149, 150, 94	3eqtri 2648	. . . . 5 ⊢ ((1 · 9) + 0) = ;09
152	11, 12, 30, 31, 27, 117, 15, 12, 31, 147, 151	decma2c 11568	. . . 4 ⊢ ((1 · ;;869) + ;;390) = ;;;1259
153	152, 14	eqtr4i 2647	. . 3 ⊢ ((1 · ;;869) + ;;390) = 𝑁
154	15, 32, 13, 134, 153	gcdi 15777	. 2 ⊢ (𝑁 gcd ;;869) = 1
155	8, 13, 22, 29, 154	gcdmodi 15778	1 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 ℕcn 11020 2c2 11070 3c3 11071 4c4 11072 5c5 11073 6c6 11074 7c7 11075 8c8 11076 9c9 11077 ℕ₀cn0 11292 ℤcz 11377 cdc 11493 ↑cexp 12860 ∥ cdvds 14983 gcd cgcd 15216 ℙcprime 15385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386

This theorem is referenced by: 1259prm 15843

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