Theorem 31prm 43767

Description: 31 is a prime number. In contrast to 37prm 16456, the proof of this theorem is not based on the "blanket" prmlem2 16455, but on isprm7 16054. 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 16456 (1810 characters compared with 1213 for 37prm 16456). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 16456). (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 12017	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 11918	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 11916	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12116	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12010	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 11719	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 11917	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 11714	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 11739	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 11845	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 10765	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12137	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12252	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1337	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4127	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 4171	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3499	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4592	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 11915	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 11715	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 10831	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12143	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 16401	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5071	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 329	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 43765	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 15683	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 11785	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5076	. . . . . . . . . . . . . 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 5071	. . . . . . . . . . . 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 4171	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4592	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2902	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16041	. . . . . . . . . . . . 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 11651	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 11820	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 16402	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5071	. . . . . . . . . . . . 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 4254	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2933	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 11920	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 11727	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 11842	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 10765	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 11812	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12136	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 11729	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 11825	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12129	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 11911	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 10645	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 11753	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 10765	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12137	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 473	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 43764	. . . . . . . . . 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 7169	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 11726	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12009	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12018	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1335	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 11720	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 10765	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 11818	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 10765	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 473	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 12902	. . . . . . . 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 12936	. . . . . . 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 11704	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7169	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 12965	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 11782	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4675	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2850	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7168	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 11706	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7169	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12019	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 12965	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 11786	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4675	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2846	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2850	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4139	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2850	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4187	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2933	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3150	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16054	. 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 3140   ∪ cun 3936   ∩ cin 3937  {cpr 4571   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  ℝcr 10538  0cc0 10539  1c1 10540   + caddc 10542   < clt 10677   ≤ cle 10678  2c2 11695  3c3 11696  4c4 11697  5c5 11698  6c6 11699  9c9 11702  ℤcz 11984  ;cdc 12101  ℤ_≥cuz 12246  ...cfz 12895  ⌊cfl 13163  √csqrt 14594   ∥ cdvds 15609  ℙcprime 16017

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 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-inf 8909  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-rp 12393  df-fz 12896  df-fl 13165  df-seq 13373  df-exp 13433  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-dvds 15610  df-prm 16018

This theorem is referenced by:  m5prm  43768