Theorem 31prm 45060

Description: 31 is a prime number. In contrast to 37prm 16831, the proof of this theorem is not based on the "blanket" prmlem2 16830, but on isprm7 16422. 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 16831 (1810 characters compared with 1213 for 37prm 16831). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 16831). (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 12361	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12260	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12258	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12461	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12354	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12061	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12259	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12056	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12081	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12187	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11107	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12482	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12597	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1340	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4084	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3904	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3437	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4585	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12257	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12057	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11173	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12488	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 16773	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5078	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 45058	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16050	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12127	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5083	. . . . . . . . . . . . . 14 ⊢ (3 ∥ (3 + 1) ↔ 3 ∥ 4)
30	27, 29	bitri 274	. . . . . . . . . . . . 13 ⊢ (3 ∥ ;31 ↔ 3 ∥ 4)
31	26, 30	mtbir 323	. . . . . . . . . . . 12 ⊢ ¬ 3 ∥ ;31
32		breq1 5078	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 854	. . . . . . . . . 10 ⊢ ((𝑛 = 2 ∨ 𝑛 = 3) → ¬ 𝑛 ∥ ;31)
35	18, 34	sylbi 216	. . . . . . . . 9 ⊢ (𝑛 ∈ {2, 3} → ¬ 𝑛 ∥ ;31)
36	35	adantr 481	. . . . . . . 8 ⊢ ((𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
37	16, 36	sylbi 216	. . . . . . 7 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
38		elin 3904	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4585	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2827	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16409	. . . . . . . . . . . . 13 ⊢ ¬ 4 ∈ ℙ
42	41	pm2.21i 119	. . . . . . . . . . . 12 ⊢ (4 ∈ ℙ → ¬ 𝑛 ∥ ;31)
43	40, 42	syl6bi 252	. . . . . . . . . . 11 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
44		1nn 11993	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12162	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 16774	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5078	. . . . . . . . . . . . 13 ⊢ (𝑛 = 5 → (𝑛 ∥ ;31 ↔ 5 ∥ ;31))
48	46, 47	mtbiri 327	. . . . . . . . . . . 12 ⊢ (𝑛 = 5 → ¬ 𝑛 ∥ ;31)
49	48	a1d 25	. . . . . . . . . . 11 ⊢ (𝑛 = 5 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
50	43, 49	jaoi 854	. . . . . . . . . 10 ⊢ ((𝑛 = 4 ∨ 𝑛 = 5) → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
51	39, 50	sylbi 216	. . . . . . . . 9 ⊢ (𝑛 ∈ {4, 5} → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
52	51	imp 407	. . . . . . . 8 ⊢ ((𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
53	38, 52	sylbi 216	. . . . . . 7 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
54	37, 53	jaoi 854	. . . . . 6 ⊢ ((𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
55	15, 54	sylbi 216	. . . . 5 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
56		indir 4210	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2858	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12262	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12069	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12184	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11107	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12154	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12481	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12071	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12167	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12474	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12253	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 10986	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12095	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11107	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12482	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 471	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 45057	. . . . . . . . . 10 ⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) → ((;25 ≤ ;31 ∧ ;31 < ;36) ↔ (⌊‘(√‘;31)) = 5))
74	73	bicomd 222	. . . . . . . . 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 708	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7295	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12068	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12353	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12362	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1338	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12062	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11107	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12160	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11107	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 471	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13255	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 708	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 13291	. . . . . . 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 12046	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7295	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13320	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12124	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4674	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2771	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7294	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12048	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7295	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12363	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13320	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12128	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4674	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2767	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2771	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4096	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2771	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4143	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2858	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3075	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16422	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 708	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3065   ∪ cun 3886   ∩ cin 3887  {cpr 4564   class class class wbr 5075  ‘cfv 6437  (class class class)co 7284  ℝcr 10879  0cc0 10880  1c1 10881   + caddc 10883   < clt 11018   ≤ cle 11019  2c2 12037  3c3 12038  4c4 12039  5c5 12040  6c6 12041  9c9 12044  ℤcz 12328  ;cdc 12446  ℤ_≥cuz 12591  ...cfz 13248  ⌊cfl 13519  √csqrt 14953   ∥ cdvds 15972  ℙcprime 16385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957  ax-pre-sup 10958

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-2o 8307  df-er 8507  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-sup 9210  df-inf 9211  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-div 11642  df-nn 11983  df-2 12045  df-3 12046  df-4 12047  df-5 12048  df-6 12049  df-7 12050  df-8 12051  df-9 12052  df-n0 12243  df-z 12329  df-dec 12447  df-uz 12592  df-rp 12740  df-fz 13249  df-fl 13521  df-seq 13731  df-exp 13792  df-cj 14819  df-re 14820  df-im 14821  df-sqrt 14955  df-abs 14956  df-dvds 15973  df-prm 16386

This theorem is referenced by:  m5prm  45061