Theorem 31prm 43780

Description: 31 is a prime number. In contrast to 37prm 16454, the proof of this theorem is not based on the "blanket" prmlem2 16453, but on isprm7 16052. Although the checks for non-divisibility by the primes 7 to 23 are not needed, the proof is much longer (regarding size) than the proof of 37prm 16454 (1810 characters compared with 1213 for 37prm 16454). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 16454). (Contributed by AV, 17-Aug-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
31prm	⊢ ;31 ∈ ℙ

Proof of Theorem 31prm

Step	Hyp	Ref	Expression
1		2z 12015	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 11916	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 11914	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12114	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12008	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 11717	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 11915	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 11712	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 11737	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 11843	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 10763	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12135	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12250	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1337	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4125	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 4169	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3497	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4590	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 11913	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 11713	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 10829	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12141	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 16399	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5069	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 329	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 43778	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 15681	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 11783	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5074	. . . . . . . . . . . . . 14 ⊢ (3 ∥ (3 + 1) ↔ 3 ∥ 4)
30	27, 29	bitri 277	. . . . . . . . . . . . 13 ⊢ (3 ∥ ;31 ↔ 3 ∥ 4)
31	26, 30	mtbir 325	. . . . . . . . . . . 12 ⊢ ¬ 3 ∥ ;31
32		breq1 5069	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 329	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 853	. . . . . . . . . 10 ⊢ ((𝑛 = 2 ∨ 𝑛 = 3) → ¬ 𝑛 ∥ ;31)
35	18, 34	sylbi 219	. . . . . . . . 9 ⊢ (𝑛 ∈ {2, 3} → ¬ 𝑛 ∥ ;31)
36	35	adantr 483	. . . . . . . 8 ⊢ ((𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
37	16, 36	sylbi 219	. . . . . . 7 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
38		elin 4169	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4590	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2900	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16039	. . . . . . . . . . . . 13 ⊢ ¬ 4 ∈ ℙ
42	41	pm2.21i 119	. . . . . . . . . . . 12 ⊢ (4 ∈ ℙ → ¬ 𝑛 ∥ ;31)
43	40, 42	syl6bi 255	. . . . . . . . . . 11 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
44		1nn 11649	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 11818	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 16400	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5069	. . . . . . . . . . . . 13 ⊢ (𝑛 = 5 → (𝑛 ∥ ;31 ↔ 5 ∥ ;31))
48	46, 47	mtbiri 329	. . . . . . . . . . . 12 ⊢ (𝑛 = 5 → ¬ 𝑛 ∥ ;31)
49	48	a1d 25	. . . . . . . . . . 11 ⊢ (𝑛 = 5 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
50	43, 49	jaoi 853	. . . . . . . . . 10 ⊢ ((𝑛 = 4 ∨ 𝑛 = 5) → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
51	39, 50	sylbi 219	. . . . . . . . 9 ⊢ (𝑛 ∈ {4, 5} → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
52	51	imp 409	. . . . . . . 8 ⊢ ((𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
53	38, 52	sylbi 219	. . . . . . 7 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
54	37, 53	jaoi 853	. . . . . 6 ⊢ ((𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
55	15, 54	sylbi 219	. . . . 5 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
56		indir 4252	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2931	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 11918	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 11725	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 11840	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 10763	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 11810	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12134	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 11727	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 11823	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12127	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 11909	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 10643	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 11751	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 10763	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12135	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 473	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 43777	. . . . . . . . . 10 ⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) → ((;25 ≤ ;31 ∧ ;31 < ;36) ↔ (⌊‘(√‘;31)) = 5))
74	73	bicomd 225	. . . . . . . . 9 ⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) → ((⌊‘(√‘;31)) = 5 ↔ (;25 ≤ ;31 ∧ ;31 < ;36)))
75	72, 74	ax-mp 5	. . . . . . . 8 ⊢ ((⌊‘(√‘;31)) = 5 ↔ (;25 ≤ ;31 ∧ ;31 < ;36))
76	63, 66, 75	mpbir2an 709	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7167	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 11724	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12007	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12016	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1335	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 11718	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 10763	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 11816	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 10763	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 473	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 12900	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 709	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 12934	. . . . . . 7 ⊢ (3 ∈ (2...5) → (2...5) = ((2...3) ∪ ((3 + 1)...5)))
90	88, 89	ax-mp 5	. . . . . 6 ⊢ (2...5) = ((2...3) ∪ ((3 + 1)...5))
91		df-3 11702	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7167	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 12963	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 11780	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4673	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2848	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7166	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 11704	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7167	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12017	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 12963	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 11784	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4673	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2844	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2848	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4137	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2848	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4185	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2931	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3148	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16052	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 709	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ∪ cun 3934   ∩ cin 3935  {cpr 4569   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  ℝcr 10536  0cc0 10537  1c1 10538   + caddc 10540   < clt 10675   ≤ cle 10676  2c2 11693  3c3 11694  4c4 11695  5c5 11696  6c6 11697  9c9 11700  ℤcz 11982  ;cdc 12099  ℤ_≥cuz 12244  ...cfz 12893  ⌊cfl 13161  √csqrt 14592   ∥ cdvds 15607  ℙcprime 16015

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 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-rp 12391  df-fz 12894  df-fl 13163  df-seq 13371  df-exp 13431  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-dvds 15608  df-prm 16016

This theorem is referenced by:  m5prm  43781