Theorem 31prm 47598

Description: 31 is a prime number. In contrast to 37prm 17091, the proof of this theorem is not based on the "blanket" prmlem2 17090, but on isprm7 16678. 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 17091 (1810 characters compared with 1213 for 37prm 17091). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17091). (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 12565	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12460	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12458	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12664	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12558	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12265	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12459	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12260	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12285	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12386	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11297	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12685	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12799	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1342	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4116	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3930	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3451	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4614	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12457	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12261	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11363	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12691	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17034	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 47596	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16302	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12326	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5115	. . . . . . . . . . . . . 14 ⊢ (3 ∥ (3 + 1) ↔ 3 ∥ 4)
30	27, 29	bitri 275	. . . . . . . . . . . . 13 ⊢ (3 ∥ ;31 ↔ 3 ∥ 4)
31	26, 30	mtbir 323	. . . . . . . . . . . 12 ⊢ ¬ 3 ∥ ;31
32		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 857	. . . . . . . . . 10 ⊢ ((𝑛 = 2 ∨ 𝑛 = 3) → ¬ 𝑛 ∥ ;31)
35	18, 34	sylbi 217	. . . . . . . . 9 ⊢ (𝑛 ∈ {2, 3} → ¬ 𝑛 ∥ ;31)
36	35	adantr 480	. . . . . . . 8 ⊢ ((𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
37	16, 36	sylbi 217	. . . . . . 7 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
38		elin 3930	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4614	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2816	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16665	. . . . . . . . . . . . 13 ⊢ ¬ 4 ∈ ℙ
42	41	pm2.21i 119	. . . . . . . . . . . 12 ⊢ (4 ∈ ℙ → ¬ 𝑛 ∥ ;31)
43	40, 42	biimtrdi 253	. . . . . . . . . . 11 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
44		1nn 12197	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12361	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17035	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5110	. . . . . . . . . . . . 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 857	. . . . . . . . . 10 ⊢ ((𝑛 = 4 ∨ 𝑛 = 5) → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
51	39, 50	sylbi 217	. . . . . . . . 9 ⊢ (𝑛 ∈ {4, 5} → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
52	51	imp 406	. . . . . . . 8 ⊢ ((𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
53	38, 52	sylbi 217	. . . . . . 7 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
54	37, 53	jaoi 857	. . . . . 6 ⊢ ((𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
55	15, 54	sylbi 217	. . . . 5 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
56		indir 4249	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2846	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12462	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12273	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12383	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11297	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12353	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12684	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12275	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12366	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12677	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12453	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11176	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12299	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11297	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12685	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 470	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 47595	. . . . . . . . . 10 ⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) → ((;25 ≤ ;31 ∧ ;31 < ;36) ↔ (⌊‘(√‘;31)) = 5))
74	73	bicomd 223	. . . . . . . . 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 711	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7398	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12272	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12557	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12566	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1340	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12266	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11297	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12359	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11297	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 470	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13475	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 711	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 13511	. . . . . . 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 12250	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7398	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13540	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12323	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4701	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2756	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7397	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12252	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7398	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12567	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13540	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12327	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4701	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2752	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2756	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4129	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2756	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4179	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2846	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3046	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16678	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 711	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∪ cun 3912   ∩ cin 3913  {cpr 4591   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  ℝcr 11067  0cc0 11068  1c1 11069   + caddc 11071   < clt 11208   ≤ cle 11209  2c2 12241  3c3 12242  4c4 12243  5c5 12244  6c6 12245  9c9 12248  ℤcz 12529  ;cdc 12649  ℤ_≥cuz 12793  ...cfz 13468  ⌊cfl 13752  √csqrt 15199   ∥ cdvds 16222  ℙcprime 16641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-rp 12952  df-fz 13469  df-fl 13754  df-seq 13967  df-exp 14027  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-dvds 16223  df-prm 16642

This theorem is referenced by:  m5prm  47599