Theorem 31prm 43196

Description: 31 is a prime number. In contrast to 37prm 16271, the proof of this theorem is not based on the "blanket" prmlem2 16270, but on isprm7 15869. 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 16271 (1810 characters compared with 1213 for 37prm 16271). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 16271). (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 11852	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 11752	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 11750	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 11951	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 11845	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 11553	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 11751	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 11548	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 11573	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 11679	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 10599	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 11972	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12088	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1332	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4041	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 4085	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3435	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4489	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 11749	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 11549	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 10665	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 11978	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 16216	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 4959	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 328	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 43194	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 15502	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 11619	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 4964	. . . . . . . . . . . . . 14 ⊢ (3 ∥ (3 + 1) ↔ 3 ∥ 4)
30	27, 29	bitri 276	. . . . . . . . . . . . 13 ⊢ (3 ∥ ;31 ↔ 3 ∥ 4)
31	26, 30	mtbir 324	. . . . . . . . . . . 12 ⊢ ¬ 3 ∥ ;31
32		breq1 4959	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 328	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 852	. . . . . . . . . 10 ⊢ ((𝑛 = 2 ∨ 𝑛 = 3) → ¬ 𝑛 ∥ ;31)
35	18, 34	sylbi 218	. . . . . . . . 9 ⊢ (𝑛 ∈ {2, 3} → ¬ 𝑛 ∥ ;31)
36	35	adantr 481	. . . . . . . 8 ⊢ ((𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
37	16, 36	sylbi 218	. . . . . . 7 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
38		elin 4085	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4489	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2868	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 15856	. . . . . . . . . . . . 13 ⊢ ¬ 4 ∈ ℙ
42	41	pm2.21i 119	. . . . . . . . . . . 12 ⊢ (4 ∈ ℙ → ¬ 𝑛 ∥ ;31)
43	40, 42	syl6bi 254	. . . . . . . . . . 11 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
44		1nn 11486	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 11654	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 16217	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 4959	. . . . . . . . . . . . 13 ⊢ (𝑛 = 5 → (𝑛 ∥ ;31 ↔ 5 ∥ ;31))
48	46, 47	mtbiri 328	. . . . . . . . . . . 12 ⊢ (𝑛 = 5 → ¬ 𝑛 ∥ ;31)
49	48	a1d 25	. . . . . . . . . . 11 ⊢ (𝑛 = 5 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
50	43, 49	jaoi 852	. . . . . . . . . 10 ⊢ ((𝑛 = 4 ∨ 𝑛 = 5) → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
51	39, 50	sylbi 218	. . . . . . . . 9 ⊢ (𝑛 ∈ {4, 5} → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
52	51	imp 407	. . . . . . . 8 ⊢ ((𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
53	38, 52	sylbi 218	. . . . . . 7 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
54	37, 53	jaoi 852	. . . . . 6 ⊢ ((𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
55	15, 54	sylbi 218	. . . . 5 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
56		indir 4167	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2899	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 11754	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 11561	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 11676	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 10599	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 11646	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 11971	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 11563	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 11659	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 11964	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 11745	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 10478	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 11587	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 10599	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 11972	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 471	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 43193	. . . . . . . . . 10 ⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) → ((;25 ≤ ;31 ∧ ;31 < ;36) ↔ (⌊‘(√‘;31)) = 5))
74	73	bicomd 224	. . . . . . . . 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 707	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7018	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 11560	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 11844	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 11853	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1330	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 11554	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 10599	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 11652	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 10599	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 471	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 12738	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 707	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 12772	. . . . . . 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 11538	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7018	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 12801	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 11616	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4574	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2821	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7017	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 11540	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7018	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 11854	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 12801	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 11620	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4574	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2817	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2821	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4053	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2821	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4100	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2899	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3113	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 15869	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 707	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1078   = wceq 1520   ∈ wcel 2079  ∀wral 3103   ∪ cun 3852   ∩ cin 3853  {cpr 4468   class class class wbr 4956  ‘cfv 6217  (class class class)co 7007  ℝcr 10371  0cc0 10372  1c1 10373   + caddc 10375   < clt 10510   ≤ cle 10511  2c2 11529  3c3 11530  4c4 11531  5c5 11532  6c6 11533  9c9 11536  ℤcz 11818  ;cdc 11936  ℤ_≥cuz 12082  ...cfz 12731  ⌊cfl 12998  √csqrt 14414   ∥ cdvds 15428  ℙcprime 15832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449  ax-pre-sup 10450

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-2o 7945  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-sup 8742  df-inf 8743  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-div 11135  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-5 11540  df-6 11541  df-7 11542  df-8 11543  df-9 11544  df-n0 11735  df-z 11819  df-dec 11937  df-uz 12083  df-rp 12229  df-fz 12732  df-fl 13000  df-seq 13208  df-exp 13268  df-cj 14280  df-re 14281  df-im 14282  df-sqrt 14416  df-abs 14417  df-dvds 15429  df-prm 15833

This theorem is referenced by:  m5prm  43197