1259lem5 - Intuitionistic Logic Explorer

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Theorem 1259lem5 13269

Description: Lemma for 1259prm 13270. 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 9471	. . . 4 ⊢ 2 ∈ ℕ
2		3nn0 9586	. . . . 5 ⊢ 3 ∈ ℕ₀
3		4nn0 9587	. . . . 5 ⊢ 4 ∈ ℕ₀
4	2, 3	deccl 9796	. . . 4 ⊢ ;34 ∈ ℕ₀
5		nnexpcl 11004	. . . 4 ⊢ ((2 ∈ ℕ ∧ ;34 ∈ ℕ₀) → (2↑;34) ∈ ℕ)
6	1, 4, 5	mp2an 430	. . 3 ⊢ (2↑;34) ∈ ℕ
7		nnm1nn0 9609	. . 3 ⊢ ((2↑;34) ∈ ℕ → ((2↑;34) − 1) ∈ ℕ₀)
8	6, 7	ax-mp 5	. 2 ⊢ ((2↑;34) − 1) ∈ ℕ₀
9		8nn0 9591	. . . 4 ⊢ 8 ∈ ℕ₀
10		6nn0 9589	. . . 4 ⊢ 6 ∈ ℕ₀
11	9, 10	deccl 9796	. . 3 ⊢ ;86 ∈ ℕ₀
12		9nn0 9592	. . 3 ⊢ 9 ∈ ℕ₀
13	11, 12	deccl 9796	. 2 ⊢ ;;869 ∈ ℕ₀
14		1259prm.1	. . 3 ⊢ 𝑁 = ;;;1259
15		1nn0 9584	. . . . . 6 ⊢ 1 ∈ ℕ₀
16		2nn0 9585	. . . . . 6 ⊢ 2 ∈ ℕ₀
17	15, 16	deccl 9796	. . . . 5 ⊢ ;12 ∈ ℕ₀
18		5nn0 9588	. . . . 5 ⊢ 5 ∈ ℕ₀
19	17, 18	deccl 9796	. . . 4 ⊢ ;;125 ∈ ℕ₀
20		9nn 9478	. . . 4 ⊢ 9 ∈ ℕ
21	19, 20	decnncl 9805	. . 3 ⊢ ;;;1259 ∈ ℕ
22	14, 21	eqeltri 2311	. 2 ⊢ 𝑁 ∈ ℕ
23	14	1259lem2 13266	. . 3 ⊢ ((2↑;34) mod 𝑁) = (;;870 mod 𝑁)
24		6p1e7 9446	. . . . 5 ⊢ (6 + 1) = 7
25		eqid 2238	. . . . 5 ⊢ ;86 = ;86
26	9, 10, 24, 25	decsuc 9817	. . . 4 ⊢ (;86 + 1) = ;87
27		eqid 2238	. . . 4 ⊢ ;;869 = ;;869
28	11, 26, 27	decsucc 9827	. . 3 ⊢ (;;869 + 1) = ;;870
29	22, 6, 15, 13, 23, 28	modsubi 13222	. 2 ⊢ (((2↑;34) − 1) mod 𝑁) = (;;869 mod 𝑁)
30	2, 12	deccl 9796	. . . 4 ⊢ ;39 ∈ ℕ₀
31		0nn0 9583	. . . 4 ⊢ 0 ∈ ℕ₀
32	30, 31	deccl 9796	. . 3 ⊢ ;;390 ∈ ℕ₀
33	9, 12	deccl 9796	. . . 4 ⊢ ;89 ∈ ℕ₀
34	16, 15	deccl 9796	. . . . . 6 ⊢ ;21 ∈ ℕ₀
35	15, 2	deccl 9796	. . . . . . 7 ⊢ ;13 ∈ ℕ₀
36	34	nn0zi 9671	. . . . . . . . 9 ⊢ ;21 ∈ ℤ
37	35	nn0zi 9671	. . . . . . . . 9 ⊢ ;13 ∈ ℤ
38		gcdcom 12769	. . . . . . . . 9 ⊢ ((;21 ∈ ℤ ∧ ;13 ∈ ℤ) → (;21 gcd ;13) = (;13 gcd ;21))
39	36, 37, 38	mp2an 430	. . . . . . . 8 ⊢ (;21 gcd ;13) = (;13 gcd ;21)
40		3nn 9472	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
41	15, 40	decnncl 9805	. . . . . . . . . 10 ⊢ ;13 ∈ ℕ
42		8nn 9477	. . . . . . . . . 10 ⊢ 8 ∈ ℕ
43		eqid 2238	. . . . . . . . . . 11 ⊢ ;13 = ;13
44	9	dec0h 9808	. . . . . . . . . . 11 ⊢ 8 = ;08
45		ax-1cn 8273	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46	45	mulridi 8329	. . . . . . . . . . . . 13 ⊢ (1 · 1) = 1
47	45	addlidi 8471	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
48	46, 47	oveq12i 6097	. . . . . . . . . . . 12 ⊢ ((1 · 1) + (0 + 1)) = (1 + 1)
49		1p1e2 9424	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
50	48, 49	eqtri 2259	. . . . . . . . . . 11 ⊢ ((1 · 1) + (0 + 1)) = 2
51		3cn 9382	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
52	51	mulridi 8329	. . . . . . . . . . . . 13 ⊢ (3 · 1) = 3
53	52	oveq1i 6095	. . . . . . . . . . . 12 ⊢ ((3 · 1) + 8) = (3 + 8)
54		8cn 9393	. . . . . . . . . . . . 13 ⊢ 8 ∈ ℂ
55		8p3e11 9867	. . . . . . . . . . . . 13 ⊢ (8 + 3) = ;11
56	54, 51, 55	addcomli 8473	. . . . . . . . . . . 12 ⊢ (3 + 8) = ;11
57	53, 56	eqtri 2259	. . . . . . . . . . 11 ⊢ ((3 · 1) + 8) = ;11
58	15, 2, 31, 9, 43, 44, 15, 15, 15, 50, 57	decmac 9838	. . . . . . . . . 10 ⊢ ((;13 · 1) + 8) = ;21
59		1nn 9318	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
60		8lt10 9918	. . . . . . . . . . 11 ⊢ 8 < ;10
61	59, 2, 9, 60	declti 9824	. . . . . . . . . 10 ⊢ 8 < ;13
62	41, 15, 42, 58, 61	ndvdsi 12719	. . . . . . . . 9 ⊢ ¬ ;13 ∥ ;21
63		13prm 13253	. . . . . . . . . 10 ⊢ ;13 ∈ ℙ
64		coprm 12942	. . . . . . . . . 10 ⊢ ((;13 ∈ ℙ ∧ ;21 ∈ ℤ) → (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1))
65	63, 36, 64	mp2an 430	. . . . . . . . 9 ⊢ (¬ ;13 ∥ ;21 ↔ (;13 gcd ;21) = 1)
66	62, 65	mpbi 145	. . . . . . . 8 ⊢ (;13 gcd ;21) = 1
67	39, 66	eqtri 2259	. . . . . . 7 ⊢ (;21 gcd ;13) = 1
68		eqid 2238	. . . . . . . 8 ⊢ ;21 = ;21
69		2cn 9378	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
70	69	mullidi 8330	. . . . . . . . . 10 ⊢ (1 · 2) = 2
71	45	addridi 8470	. . . . . . . . . 10 ⊢ (1 + 0) = 1
72	70, 71	oveq12i 6097	. . . . . . . . 9 ⊢ ((1 · 2) + (1 + 0)) = (2 + 1)
73		2p1e3 9441	. . . . . . . . 9 ⊢ (2 + 1) = 3
74	72, 73	eqtri 2259	. . . . . . . 8 ⊢ ((1 · 2) + (1 + 0)) = 3
75	46	oveq1i 6095	. . . . . . . . 9 ⊢ ((1 · 1) + 3) = (1 + 3)
76		3p1e4 9443	. . . . . . . . . 10 ⊢ (3 + 1) = 4
77	51, 45, 76	addcomli 8473	. . . . . . . . 9 ⊢ (1 + 3) = 4
78	3	dec0h 9808	. . . . . . . . 9 ⊢ 4 = ;04
79	75, 77, 78	3eqtri 2263	. . . . . . . 8 ⊢ ((1 · 1) + 3) = ;04
80	16, 15, 15, 2, 68, 43, 15, 3, 31, 74, 79	decma2c 9839	. . . . . . 7 ⊢ ((1 · ;21) + ;13) = ;34
81	15, 35, 34, 67, 80	gcdi 13223	. . . . . 6 ⊢ (;34 gcd ;21) = 1
82		eqid 2238	. . . . . . 7 ⊢ ;34 = ;34
83		2t3e6 9465	. . . . . . . . 9 ⊢ (2 · 3) = 6
84	69	addridi 8470	. . . . . . . . 9 ⊢ (2 + 0) = 2
85	83, 84	oveq12i 6097	. . . . . . . 8 ⊢ ((2 · 3) + (2 + 0)) = (6 + 2)
86		6p2e8 9457	. . . . . . . 8 ⊢ (6 + 2) = 8
87	85, 86	eqtri 2259	. . . . . . 7 ⊢ ((2 · 3) + (2 + 0)) = 8
88		2t4e8 9468	. . . . . . . . 9 ⊢ (2 · 4) = 8
89	88	oveq1i 6095	. . . . . . . 8 ⊢ ((2 · 4) + 1) = (8 + 1)
90		8p1e9 9448	. . . . . . . 8 ⊢ (8 + 1) = 9
91	12	dec0h 9808	. . . . . . . 8 ⊢ 9 = ;09
92	89, 90, 91	3eqtri 2263	. . . . . . 7 ⊢ ((2 · 4) + 1) = ;09
93	2, 3, 16, 15, 82, 68, 16, 12, 31, 87, 92	decma2c 9839	. . . . . 6 ⊢ ((2 · ;34) + ;21) = ;89
94	16, 34, 4, 81, 93	gcdi 13223	. . . . 5 ⊢ (;89 gcd ;34) = 1
95		eqid 2238	. . . . . 6 ⊢ ;89 = ;89
96		4cn 9385	. . . . . . . . 9 ⊢ 4 ∈ ℂ
97		4p3e7 9452	. . . . . . . . 9 ⊢ (4 + 3) = 7
98	96, 51, 97	addcomli 8473	. . . . . . . 8 ⊢ (3 + 4) = 7
99	98	oveq2i 6096	. . . . . . 7 ⊢ ((4 · 8) + (3 + 4)) = ((4 · 8) + 7)
100		7nn0 9590	. . . . . . . 8 ⊢ 7 ∈ ℕ₀
101		8t4e32 9903	. . . . . . . . 9 ⊢ (8 · 4) = ;32
102	54, 96, 101	mulcomli 8334	. . . . . . . 8 ⊢ (4 · 8) = ;32
103		7cn 9391	. . . . . . . . 9 ⊢ 7 ∈ ℂ
104		7p2e9 9459	. . . . . . . . 9 ⊢ (7 + 2) = 9
105	103, 69, 104	addcomli 8473	. . . . . . . 8 ⊢ (2 + 7) = 9
106	2, 16, 100, 102, 105	decaddi 9846	. . . . . . 7 ⊢ ((4 · 8) + 7) = ;39
107	99, 106	eqtri 2259	. . . . . 6 ⊢ ((4 · 8) + (3 + 4)) = ;39
108		9cn 9395	. . . . . . . 8 ⊢ 9 ∈ ℂ
109		9t4e36 9910	. . . . . . . 8 ⊢ (9 · 4) = ;36
110	108, 96, 109	mulcomli 8334	. . . . . . 7 ⊢ (4 · 9) = ;36
111		6p4e10 9858	. . . . . . 7 ⊢ (6 + 4) = ;10
112	2, 10, 3, 110, 76, 111	decaddci2 9848	. . . . . 6 ⊢ ((4 · 9) + 4) = ;40
113	9, 12, 2, 3, 95, 82, 3, 31, 3, 107, 112	decma2c 9839	. . . . 5 ⊢ ((4 · ;89) + ;34) = ;;390
114	3, 4, 33, 94, 113	gcdi 13223	. . . 4 ⊢ (;;390 gcd ;89) = 1
115		eqid 2238	. . . . 5 ⊢ ;;390 = ;;390
116		eqid 2238	. . . . . 6 ⊢ ;39 = ;39
117	54	addridi 8470	. . . . . . 7 ⊢ (8 + 0) = 8
118	117, 44	eqtri 2259	. . . . . 6 ⊢ (8 + 0) = ;08
119	69	addlidi 8471	. . . . . . . 8 ⊢ (0 + 2) = 2
120	83, 119	oveq12i 6097	. . . . . . 7 ⊢ ((2 · 3) + (0 + 2)) = (6 + 2)
121	120, 86	eqtri 2259	. . . . . 6 ⊢ ((2 · 3) + (0 + 2)) = 8
122		9t2e18 9908	. . . . . . . 8 ⊢ (9 · 2) = ;18
123	108, 69, 122	mulcomli 8334	. . . . . . 7 ⊢ (2 · 9) = ;18
124		8p8e16 9872	. . . . . . 7 ⊢ (8 + 8) = ;16
125	15, 9, 9, 123, 49, 10, 124	decaddci 9847	. . . . . 6 ⊢ ((2 · 9) + 8) = ;26
126	2, 12, 31, 9, 116, 118, 16, 10, 16, 121, 125	decma2c 9839	. . . . 5 ⊢ ((2 · ;39) + (8 + 0)) = ;86
127		2t0e0 9469	. . . . . . 7 ⊢ (2 · 0) = 0
128	127	oveq1i 6095	. . . . . 6 ⊢ ((2 · 0) + 9) = (0 + 9)
129	108	addlidi 8471	. . . . . 6 ⊢ (0 + 9) = 9
130	128, 129, 91	3eqtri 2263	. . . . 5 ⊢ ((2 · 0) + 9) = ;09
131	30, 31, 9, 12, 115, 95, 16, 12, 31, 126, 130	decma2c 9839	. . . 4 ⊢ ((2 · ;;390) + ;89) = ;;869
132	16, 33, 32, 114, 131	gcdi 13223	. . 3 ⊢ (;;869 gcd ;;390) = 1
133	30	nn0cni 9580	. . . . . . 7 ⊢ ;39 ∈ ℂ
134	133	addridi 8470	. . . . . 6 ⊢ (;39 + 0) = ;39
135	54	mullidi 8330	. . . . . . . 8 ⊢ (1 · 8) = 8
136	135, 76	oveq12i 6097	. . . . . . 7 ⊢ ((1 · 8) + (3 + 1)) = (8 + 4)
137		8p4e12 9868	. . . . . . 7 ⊢ (8 + 4) = ;12
138	136, 137	eqtri 2259	. . . . . 6 ⊢ ((1 · 8) + (3 + 1)) = ;12
139		6cn 9389	. . . . . . . . 9 ⊢ 6 ∈ ℂ
140	139	mullidi 8330	. . . . . . . 8 ⊢ (1 · 6) = 6
141	140	oveq1i 6095	. . . . . . 7 ⊢ ((1 · 6) + 9) = (6 + 9)
142		9p6e15 9877	. . . . . . . 8 ⊢ (9 + 6) = ;15
143	108, 139, 142	addcomli 8473	. . . . . . 7 ⊢ (6 + 9) = ;15
144	141, 143	eqtri 2259	. . . . . 6 ⊢ ((1 · 6) + 9) = ;15
145	9, 10, 2, 12, 25, 134, 15, 18, 15, 138, 144	decma2c 9839	. . . . 5 ⊢ ((1 · ;86) + (;39 + 0)) = ;;125
146	108	mullidi 8330	. . . . . . 7 ⊢ (1 · 9) = 9
147	146	oveq1i 6095	. . . . . 6 ⊢ ((1 · 9) + 0) = (9 + 0)
148	108	addridi 8470	. . . . . 6 ⊢ (9 + 0) = 9
149	147, 148, 91	3eqtri 2263	. . . . 5 ⊢ ((1 · 9) + 0) = ;09
150	11, 12, 30, 31, 27, 115, 15, 12, 31, 145, 149	decma2c 9839	. . . 4 ⊢ ((1 · ;;869) + ;;390) = ;;;1259
151	150, 14	eqtr4i 2262	. . 3 ⊢ ((1 · ;;869) + ;;390) = 𝑁
152	15, 32, 13, 132, 151	gcdi 13223	. 2 ⊢ (𝑁 gcd ;;869) = 1
153	8, 13, 22, 29, 152	gcdmodi 13224	1 ⊢ (((2↑;34) − 1) gcd 𝑁) = 1

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 105 = wceq 1402 ∈ wcel 2209 class class class wbr 4130 (class class class)co 6085 0cc0 8180 1c1 8181 + caddc 8183 · cmul 8185 − cmin 8499 ℕcn 9307 2c2 9358 3c3 9359 4c4 9360 5c5 9361 6c6 9362 7c7 9363 8c8 9364 9c9 9365 ℕ₀cn0 9568 ℤcz 9649 cdc 9782 ↑cexp 10990 ∥ cdvds 12573 gcd cgcd 12749 ℙcprime 12904

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-2o 6688 df-er 6807 df-en 7023 df-sup 7325 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-5 9369 df-6 9370 df-7 9371 df-8 9372 df-9 9373 df-n0 9569 df-z 9650 df-dec 9783 df-uz 9932 df-q 10030 df-rp 10066 df-fz 10423 df-fzo 10561 df-fl 10716 df-mod 10775 df-seqfrec 10900 df-exp 10991 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-dvds 12574 df-gcd 12750 df-prm 12905

This theorem is used by: 1259prm 13270

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