257prm - Mathbox for Alexander van der Vekens

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Theorem 257prm 48464

Description: 257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)

**Assertion**
Ref	Expression
257prm	⊢ ;;257 ∈ ℙ

**Proof of Theorem 257prm**
Step	Hyp	Ref	Expression
1		2nn0 12545	. . . 4 ⊢ 2 ∈ ℕ₀
2		5nn0 12548	. . . 4 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12751	. . 3 ⊢ ;25 ∈ ℕ₀
4		7nn 12357	. . 3 ⊢ 7 ∈ ℕ
5	3, 4	decnncl 12760	. 2 ⊢ ;;257 ∈ ℕ
6		8nn0 12551	. . 3 ⊢ 8 ∈ ℕ₀
7		4nn0 12547	. . 3 ⊢ 4 ∈ ℕ₀
8		7nn0 12550	. . 3 ⊢ 7 ∈ ℕ₀
9		1nn0 12544	. . 3 ⊢ 1 ∈ ℕ₀
10		2lt8 12464	. . 3 ⊢ 2 < 8
11		5lt10 12877	. . 3 ⊢ 5 < ;10
12		7lt10 12875	. . 3 ⊢ 7 < ;10
13	1, 6, 2, 7, 8, 9, 10, 11, 12	3decltc 12774	. 2 ⊢ ;;257 < ;;841
14		5nn 12351	. . . 4 ⊢ 5 ∈ ℕ
15	1, 14	decnncl 12760	. . 3 ⊢ ;25 ∈ ℕ
16		1lt10 12881	. . 3 ⊢ 1 < ;10
17	15, 8, 9, 16	declti 12779	. 2 ⊢ 1 < ;;257
18		3nn0 12546	. . 3 ⊢ 3 ∈ ℕ₀
19		3t2e6 12430	. . 3 ⊢ (3 · 2) = 6
20		df-7 12332	. . 3 ⊢ 7 = (6 + 1)
21	3, 18, 19, 20	dec2dvds 17155	. 2 ⊢ ¬ 2 ∥ ;;257
22		3nn 12344	. . . 4 ⊢ 3 ∈ ℕ
23		2nn 12338	. . . 4 ⊢ 2 ∈ ℕ
24		3cn 12346	. . . . . . 7 ⊢ 3 ∈ ℂ
25	24	mulridi 11237	. . . . . 6 ⊢ (3 · 1) = 3
26	25	oveq1i 7423	. . . . 5 ⊢ ((3 · 1) + 2) = (3 + 2)
27		3p2e5 12415	. . . . 5 ⊢ (3 + 2) = 5
28	26, 27	eqtri 2783	. . . 4 ⊢ ((3 · 1) + 2) = 5
29		2lt3 12438	. . . 4 ⊢ 2 < 3
30	22, 9, 23, 28, 29	ndvdsi 16502	. . 3 ⊢ ¬ 3 ∥ 5
31	1, 2, 8	3dvds2dec 16423	. . . 4 ⊢ (3 ∥ ;;257 ↔ 3 ∥ ((2 + 5) + 7))
32		5cn 12353	. . . . . . . 8 ⊢ 5 ∈ ℂ
33		2cn 12340	. . . . . . . 8 ⊢ 2 ∈ ℂ
34		5p2e7 12420	. . . . . . . 8 ⊢ (5 + 2) = 7
35	32, 33, 34	addcomli 11426	. . . . . . 7 ⊢ (2 + 5) = 7
36	35	oveq1i 7423	. . . . . 6 ⊢ ((2 + 5) + 7) = (7 + 7)
37		7p7e14 12820	. . . . . 6 ⊢ (7 + 7) = ;14
38	36, 37	eqtri 2783	. . . . 5 ⊢ ((2 + 5) + 7) = ;14
39	38	breq2i 5111	. . . 4 ⊢ (3 ∥ ((2 + 5) + 7) ↔ 3 ∥ ;14)
40	9, 7	3dvdsdec 16422	. . . . 5 ⊢ (3 ∥ ;14 ↔ 3 ∥ (1 + 4))
41		4cn 12350	. . . . . . 7 ⊢ 4 ∈ ℂ
42		ax-1cn 11182	. . . . . . 7 ⊢ 1 ∈ ℂ
43		4p1e5 12410	. . . . . . 7 ⊢ (4 + 1) = 5
44	41, 42, 43	addcomli 11426	. . . . . 6 ⊢ (1 + 4) = 5
45	44	breq2i 5111	. . . . 5 ⊢ (3 ∥ (1 + 4) ↔ 3 ∥ 5)
46	40, 45	bitri 278	. . . 4 ⊢ (3 ∥ ;14 ↔ 3 ∥ 5)
47	31, 39, 46	3bitri 300	. . 3 ⊢ (3 ∥ ;;257 ↔ 3 ∥ 5)
48	30, 47	mtbir 326	. 2 ⊢ ¬ 3 ∥ ;;257
49		2lt5 12446	. . 3 ⊢ 2 < 5
50	3, 23, 49, 34	dec5dvds2 17157	. 2 ⊢ ¬ 5 ∥ ;;257
51		6nn0 12549	. . . 4 ⊢ 6 ∈ ℕ₀
52	18, 51	deccl 12751	. . 3 ⊢ ;36 ∈ ℕ₀
53		eqid 2760	. . . . 5 ⊢ ;36 = ;36
54		7t3e21 12851	. . . . . 6 ⊢ (7 · 3) = ;21
55	1, 9, 7, 54, 44	decaddi 12801	. . . . 5 ⊢ ((7 · 3) + 4) = ;25
56		7t6e42 12854	. . . . 5 ⊢ (7 · 6) = ;42
57	8, 18, 51, 53, 1, 7, 55, 56	decmul2c 12807	. . . 4 ⊢ (7 · ;36) = ;;252
58	3, 1, 2, 57, 35	decaddi 12801	. . 3 ⊢ ((7 · ;36) + 5) = ;;257
59		5lt7 12454	. . 3 ⊢ 5 < 7
60	4, 52, 14, 58, 59	ndvdsi 16502	. 2 ⊢ ¬ 7 ∥ ;;257
61		1nn 12268	. . . 4 ⊢ 1 ∈ ℕ
62	9, 61	decnncl 12760	. . 3 ⊢ ;11 ∈ ℕ
63	1, 18	deccl 12751	. . 3 ⊢ ;23 ∈ ℕ₀
64		4nn 12348	. . 3 ⊢ 4 ∈ ℕ
65	9, 9	deccl 12751	. . . . 5 ⊢ ;11 ∈ ℕ₀
66		eqid 2760	. . . . 5 ⊢ ;23 = ;23
67	65	nn0cni 12540	. . . . . . . 8 ⊢ ;11 ∈ ℂ
68	67, 33	mulcomi 11241	. . . . . . 7 ⊢ (;11 · 2) = (2 · ;11)
69	68	oveq1i 7423	. . . . . 6 ⊢ ((;11 · 2) + 3) = ((2 · ;11) + 3)
70	1	11multnc 12809	. . . . . . 7 ⊢ (2 · ;11) = ;22
71	24, 33, 27	addcomli 11426	. . . . . . 7 ⊢ (2 + 3) = 5
72	1, 1, 18, 70, 71	decaddi 12801	. . . . . 6 ⊢ ((2 · ;11) + 3) = ;25
73	69, 72	eqtri 2783	. . . . 5 ⊢ ((;11 · 2) + 3) = ;25
74	18	11multnc 12809	. . . . . 6 ⊢ (3 · ;11) = ;33
75	24, 67, 74	mulcomli 11242	. . . . 5 ⊢ (;11 · 3) = ;33
76	65, 1, 18, 66, 18, 18, 73, 75	decmul2c 12807	. . . 4 ⊢ (;11 · ;23) = ;;253
77		4p3e7 12418	. . . . 5 ⊢ (4 + 3) = 7
78	41, 24, 77	addcomli 11426	. . . 4 ⊢ (3 + 4) = 7
79	3, 18, 7, 76, 78	decaddi 12801	. . 3 ⊢ ((;11 · ;23) + 4) = ;;257
80		4lt10 12878	. . . 4 ⊢ 4 < ;10
81	61, 9, 7, 80	declti 12779	. . 3 ⊢ 4 < ;11
82	62, 63, 64, 79, 81	ndvdsi 16502	. 2 ⊢ ¬ ;11 ∥ ;;257
83	9, 22	decnncl 12760	. . 3 ⊢ ;13 ∈ ℕ
84		9nn0 12552	. . . 4 ⊢ 9 ∈ ℕ₀
85	9, 84	deccl 12751	. . 3 ⊢ ;19 ∈ ℕ₀
86		10nn 12756	. . 3 ⊢ ;10 ∈ ℕ
87	9, 18	deccl 12751	. . . . . . 7 ⊢ ;13 ∈ ℕ₀
88	87	nn0cni 12540	. . . . . 6 ⊢ ;13 ∈ ℂ
89	85	nn0cni 12540	. . . . . 6 ⊢ ;19 ∈ ℂ
90	88, 89	mulcomi 11241	. . . . 5 ⊢ (;13 · ;19) = (;19 · ;13)
91	90	oveq1i 7423	. . . 4 ⊢ ((;13 · ;19) + ;10) = ((;19 · ;13) + ;10)
92		0nn0 12543	. . . . 5 ⊢ 0 ∈ ℕ₀
93		eqid 2760	. . . . 5 ⊢ ;19 = ;19
94		eqid 2760	. . . . 5 ⊢ ;10 = ;10
95	88	mullidi 11238	. . . . . 6 ⊢ (1 · ;13) = ;13
96		1p1e2 12388	. . . . . . . 8 ⊢ (1 + 1) = 2
97		eqid 2760	. . . . . . . 8 ⊢ ;11 = ;11
98	9, 9, 96, 97	decsuc 12772	. . . . . . 7 ⊢ (;11 + 1) = ;12
99	67, 42, 98	addcomli 11426	. . . . . 6 ⊢ (1 + ;11) = ;12
100	9, 18, 9, 1, 95, 99, 96, 27	decadd 12795	. . . . 5 ⊢ ((1 · ;13) + (1 + ;11)) = ;25
101		eqid 2760	. . . . . . . 8 ⊢ ;13 = ;13
102		9cn 12365	. . . . . . . . . . 11 ⊢ 9 ∈ ℂ
103	102	mulridi 11237	. . . . . . . . . 10 ⊢ (9 · 1) = 9
104	103	oveq1i 7423	. . . . . . . . 9 ⊢ ((9 · 1) + 2) = (9 + 2)
105		9p2e11 12828	. . . . . . . . 9 ⊢ (9 + 2) = ;11
106	104, 105	eqtri 2783	. . . . . . . 8 ⊢ ((9 · 1) + 2) = ;11
107		9t3e27 12864	. . . . . . . 8 ⊢ (9 · 3) = ;27
108	84, 9, 18, 101, 8, 1, 106, 107	decmul2c 12807	. . . . . . 7 ⊢ (9 · ;13) = ;;117
109	108	oveq1i 7423	. . . . . 6 ⊢ ((9 · ;13) + 0) = (;;117 + 0)
110	65, 8	deccl 12751	. . . . . . . 8 ⊢ ;;117 ∈ ℕ₀
111	110	nn0cni 12540	. . . . . . 7 ⊢ ;;117 ∈ ℂ
112	111	addridi 11421	. . . . . 6 ⊢ (;;117 + 0) = ;;117
113	109, 112	eqtri 2783	. . . . 5 ⊢ ((9 · ;13) + 0) = ;;117
114	9, 84, 9, 92, 93, 94, 87, 8, 65, 100, 113	decmac 12793	. . . 4 ⊢ ((;19 · ;13) + ;10) = ;;257
115	91, 114	eqtri 2783	. . 3 ⊢ ((;13 · ;19) + ;10) = ;;257
116		3pos 12373	. . . 4 ⊢ 0 < 3
117	9, 92, 22, 116	declt 12769	. . 3 ⊢ ;10 < ;13
118	83, 85, 86, 115, 117	ndvdsi 16502	. 2 ⊢ ¬ ;13 ∥ ;;257
119	9, 4	decnncl 12760	. . 3 ⊢ ;17 ∈ ℕ
120	9, 2	deccl 12751	. . 3 ⊢ ;15 ∈ ℕ₀
121	9, 8	deccl 12751	. . . . 5 ⊢ ;17 ∈ ℕ₀
122		eqid 2760	. . . . 5 ⊢ ;15 = ;15
123	121	nn0cni 12540	. . . . . . 7 ⊢ ;17 ∈ ℂ
124	123	mulridi 11237	. . . . . 6 ⊢ (;17 · 1) = ;17
125		8cn 12362	. . . . . . 7 ⊢ 8 ∈ ℂ
126		7cn 12359	. . . . . . 7 ⊢ 7 ∈ ℂ
127		8p7e15 12826	. . . . . . 7 ⊢ (8 + 7) = ;15
128	125, 126, 127	addcomli 11426	. . . . . 6 ⊢ (7 + 8) = ;15
129	9, 8, 6, 124, 96, 2, 128	decaddci 12802	. . . . 5 ⊢ ((;17 · 1) + 8) = ;25
130		eqid 2760	. . . . . 6 ⊢ ;17 = ;17
131	32	mullidi 11238	. . . . . . . 8 ⊢ (1 · 5) = 5
132	131	oveq1i 7423	. . . . . . 7 ⊢ ((1 · 5) + 3) = (5 + 3)
133		5p3e8 12421	. . . . . . 7 ⊢ (5 + 3) = 8
134	132, 133	eqtri 2783	. . . . . 6 ⊢ ((1 · 5) + 3) = 8
135		7t5e35 12853	. . . . . 6 ⊢ (7 · 5) = ;35
136	2, 9, 8, 130, 2, 18, 134, 135	decmul1c 12806	. . . . 5 ⊢ (;17 · 5) = ;85
137	121, 9, 2, 122, 2, 6, 129, 136	decmul2c 12807	. . . 4 ⊢ (;17 · ;15) = ;;255
138	3, 2, 1, 137, 34	decaddi 12801	. . 3 ⊢ ((;17 · ;15) + 2) = ;;257
139		2lt10 12880	. . . 4 ⊢ 2 < ;10
140	61, 8, 1, 139	declti 12779	. . 3 ⊢ 2 < ;17
141	119, 120, 23, 138, 140	ndvdsi 16502	. 2 ⊢ ¬ ;17 ∥ ;;257
142		9nn 12363	. . . 4 ⊢ 9 ∈ ℕ
143	9, 142	decnncl 12760	. . 3 ⊢ ;19 ∈ ℕ
144		9pos 12381	. . . 4 ⊢ 0 < 9
145	9, 92, 142, 144	declt 12769	. . 3 ⊢ ;10 < ;19
146	143, 87, 86, 114, 145	ndvdsi 16502	. 2 ⊢ ¬ ;19 ∥ ;;257
147	1, 22	decnncl 12760	. . 3 ⊢ ;23 ∈ ℕ
148	65, 1, 18, 66, 18, 18, 72, 74	decmul1c 12806	. . . 4 ⊢ (;23 · ;11) = ;;253
149	3, 18, 7, 148, 78	decaddi 12801	. . 3 ⊢ ((;23 · ;11) + 4) = ;;257
150	23, 18, 7, 80	declti 12779	. . 3 ⊢ 4 < ;23
151	147, 65, 64, 149, 150	ndvdsi 16502	. 2 ⊢ ¬ ;23 ∥ ;;257
152	5, 13, 17, 21, 48, 50, 60, 82, 118, 141, 146, 151	prmlem2 17212	1 ⊢ ;;257 ∈ ℙ

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∈ wcel 2145 class class class wbr 5103 (class class class)co 7413 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 2c2 12319 3c3 12320 4c4 12321 5c5 12322 6c6 12323 7c7 12324 8c8 12325 9c9 12326 cdc 12736 ∥ cdvds 16342 ℙcprime 16761

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-rp 13043 df-fz 13562 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-dvds 16343 df-prm 16762

This theorem is used by: fmtno3prm 48465

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