Theorem 31prm 48897

Description: 31 is a prime number. In contrast to 37prm 17331, the proof of this theorem is not based on the "blanket" prmlem2 17330, but on isprm7 16916. 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 17331 (1810 characters compared with 1213 for 37prm 17331). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17331). (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 12758	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12654	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12652	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12859	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12751	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12452	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12653	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12447	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12472	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12580	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11463	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12885	. . 3 ⊢ 2 ≤ ;31
13		eluz2 13001	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1360	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4155	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 4096	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3458	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4609	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12651	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12448	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11529	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12891	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17273	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5109	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 48895	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16534	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12517	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5114	. . . . . . . . . . . . . 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 5109	. . . . . . . . . . . 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 4096	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4609	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2852	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16902	. . . . . . . . . . . . 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 12376	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12555	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17274	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5109	. . . . . . . . . . . . 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 4234	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2882	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12656	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12460	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12577	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11463	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12546	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12884	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12462	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12560	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12877	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12647	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11340	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12489	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11463	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12885	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 476	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 48894	. . . . . . . . . 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 7457	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12459	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12750	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12759	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1358	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12453	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11463	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12553	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11463	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 476	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13676	. . . . . . . 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 13714	. . . . . . 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 12436	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7457	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13743	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12514	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4698	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2790	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7456	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12438	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7457	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12760	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13743	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12518	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4698	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2786	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2790	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4168	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2790	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4116	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2882	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3082	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16916	. 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 2147  ∀wral 3080   ∩ cin 4092   ∪ cun 4152  {cpr 4590   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  ℝcr 11229  0cc0 11230  1c1 11231   + caddc 11233   < clt 11373   ≤ cle 11374  2c2 12427  3c3 12428  4c4 12429  5c5 12430  6c6 12431  9c9 12434  ℤcz 12723  ;cdc 12844  ℤ_≥cuz 12995  ...cfz 13669  ⌊cfl 13960  √csqrt 15432   ∥ cdvds 16454  ℙcprime 16878

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307  ax-pre-sup 11308

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-sup 9464  df-inf 9465  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-div 12004  df-nn 12366  df-2 12435  df-3 12436  df-4 12437  df-5 12438  df-6 12439  df-7 12440  df-8 12441  df-9 12442  df-n0 12637  df-z 12724  df-dec 12845  df-uz 12996  df-rp 13151  df-fz 13670  df-fl 13962  df-seq 14175  df-exp 14235  df-cj 15298  df-re 15299  df-im 15300  df-sqrt 15434  df-abs 15435  df-dvds 16455  df-prm 16879

This theorem is used by:  m5prm  48898