Theorem 31prm 48508

Description: 31 is a prime number. In contrast to 37prm 17219, the proof of this theorem is not based on the "blanket" prmlem2 17218, but on isprm7 16805. 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 17219 (1810 characters compared with 1213 for 37prm 17219). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17219). (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 12654	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12550	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12548	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12755	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12647	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12348	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12549	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12343	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12368	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12476	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11361	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12781	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12897	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1360	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4103	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3918	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3457	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4612	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12547	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12344	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11427	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12787	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17161	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 48506	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16428	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12413	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5115	. . . . . . . . . . . . . 14 ⊢ (3 ∥ (3 + 1) ↔ 3 ∥ 4)
30	27, 29	bitri 278	. . . . . . . . . . . . 13 ⊢ (3 ∥ ;31 ↔ 3 ∥ 4)
31	26, 30	mtbir 326	. . . . . . . . . . . 12 ⊢ ¬ 3 ∥ ;31
32		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 871	. . . . . . . . . 10 ⊢ ((𝑛 = 2 ∨ 𝑛 = 3) → ¬ 𝑛 ∥ ;31)
35	18, 34	sylbi 220	. . . . . . . . 9 ⊢ (𝑛 ∈ {2, 3} → ¬ 𝑛 ∥ ;31)
36	35	adantr 486	. . . . . . . 8 ⊢ ((𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
37	16, 36	sylbi 220	. . . . . . 7 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
38		elin 3918	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4612	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2850	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16791	. . . . . . . . . . . . 13 ⊢ ¬ 4 ∈ ℙ
42	41	pm2.21i 120	. . . . . . . . . . . 12 ⊢ (4 ∈ ℙ → ¬ 𝑛 ∥ ;31)
43	40, 42	biimtrdi 256	. . . . . . . . . . 11 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
44		1nn 12272	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12451	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17162	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5110	. . . . . . . . . . . . 13 ⊢ (𝑛 = 5 → (𝑛 ∥ ;31 ↔ 5 ∥ ;31))
48	46, 47	mtbiri 330	. . . . . . . . . . . 12 ⊢ (𝑛 = 5 → ¬ 𝑛 ∥ ;31)
49	48	a1d 26	. . . . . . . . . . 11 ⊢ (𝑛 = 5 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
50	43, 49	jaoi 871	. . . . . . . . . 10 ⊢ ((𝑛 = 4 ∨ 𝑛 = 5) → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
51	39, 50	sylbi 220	. . . . . . . . 9 ⊢ (𝑛 ∈ {4, 5} → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
52	51	imp 412	. . . . . . . 8 ⊢ ((𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
53	38, 52	sylbi 220	. . . . . . 7 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
54	37, 53	jaoi 871	. . . . . 6 ⊢ ((𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
55	15, 54	sylbi 220	. . . . 5 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
56		indir 4235	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2880	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12552	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12356	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12473	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11361	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12442	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12780	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12358	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12456	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12773	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12543	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11238	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12385	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11361	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12781	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 476	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 48505	. . . . . . . . . 10 ⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) → ((;25 ≤ ;31 ∧ ;31 < ;36) ↔ (⌊‘(√‘;31)) = 5))
74	73	bicomd 226	. . . . . . . . 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 724	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7428	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12355	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12646	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12655	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1358	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12349	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11361	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12449	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11361	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 476	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13572	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 724	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 13609	. . . . . . 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 12332	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7428	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13638	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12410	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4701	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2789	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7427	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12334	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7428	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12656	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13638	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12414	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4701	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2785	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2789	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4116	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2789	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4165	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2880	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3080	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16805	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 724	1 ⊢ ;31 ∈ ℙ

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ∪ cun 3900   ∩ cin 3901  {cpr 4589   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  ℝcr 11127  0cc0 11128  1c1 11129   + caddc 11131   < clt 11271   ≤ cle 11272  2c2 12323  3c3 12324  4c4 12325  5c5 12326  6c6 12327  9c9 12330  ℤcz 12619  ;cdc 12740  ℤ_≥cuz 12891  ...cfz 13565  ⌊cfl 13855  √csqrt 15324   ∥ cdvds 16348  ℙcprime 16767

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-sup 9416  df-inf 9417  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-z 12620  df-dec 12741  df-uz 12892  df-rp 13047  df-fz 13566  df-fl 13857  df-seq 14070  df-exp 14130  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-dvds 16349  df-prm 16768

This theorem is used by:  m5prm  48509