Theorem 31prm 48776

Description: 31 is a prime number. In contrast to 37prm 17305, the proof of this theorem is not based on the "blanket" prmlem2 17304, but on isprm7 16890. 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 17305 (1810 characters compared with 1213 for 37prm 17305). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17305). (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 12732	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12628	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12626	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12833	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12725	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12426	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12627	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12421	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12446	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12554	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11437	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12859	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12975	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1360	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4152	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 4093	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3454	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4605	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12625	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12422	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11503	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12865	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17247	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5105	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 48774	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16508	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12491	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5110	. . . . . . . . . . . . . 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 5105	. . . . . . . . . . . 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 4093	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4605	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2848	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16876	. . . . . . . . . . . . 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 12350	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12529	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17248	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5105	. . . . . . . . . . . . 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 4231	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2878	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12630	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12434	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12551	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11437	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12520	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12858	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12436	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12534	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12851	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12621	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11314	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12463	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11437	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12859	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 476	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 48773	. . . . . . . . . 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 7432	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12433	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12724	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12733	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1358	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12427	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11437	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12527	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11437	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 476	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13650	. . . . . . . 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 13688	. . . . . . 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 12410	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7432	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13717	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12488	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4694	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2787	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7431	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12412	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7432	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12734	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13717	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12492	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4694	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2783	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2787	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4165	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2787	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4113	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2878	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3078	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16890	. 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 1569   ∈ wcel 2144  ∀wral 3076   ∩ cin 4089   ∪ cun 4149  {cpr 4586   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  ℝcr 11203  0cc0 11204  1c1 11205   + caddc 11207   < clt 11347   ≤ cle 11348  2c2 12401  3c3 12402  4c4 12403  5c5 12404  6c6 12405  9c9 12408  ℤcz 12697  ;cdc 12818  ℤ_≥cuz 12969  ...cfz 13643  ⌊cfl 13934  √csqrt 15406   ∥ cdvds 16428  ℙcprime 16852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281  ax-pre-sup 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-sup 9438  df-inf 9439  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-div 11978  df-nn 12340  df-2 12409  df-3 12410  df-4 12411  df-5 12412  df-6 12413  df-7 12414  df-8 12415  df-9 12416  df-n0 12611  df-z 12698  df-dec 12819  df-uz 12970  df-rp 13125  df-fz 13644  df-fl 13936  df-seq 14149  df-exp 14209  df-cj 15272  df-re 15273  df-im 15274  df-sqrt 15408  df-abs 15409  df-dvds 16429  df-prm 16853

This theorem is used by:  m5prm  48777