Theorem 31prm 47471

Description: 31 is a prime number. In contrast to 37prm 17168, the proof of this theorem is not based on the "blanket" prmlem2 17167, but on isprm7 16755. 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 17168 (1810 characters compared with 1213 for 37prm 17168). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17168). (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 12675	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12571	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12569	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12773	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12668	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12372	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12570	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12367	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12392	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12498	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11413	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12794	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12909	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1341	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4176	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3992	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3492	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4672	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12568	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12368	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11479	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12800	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17110	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5169	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 47469	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16380	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12438	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5174	. . . . . . . . . . . . . 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 5169	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 856	. . . . . . . . . 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 3992	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4672	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2832	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16742	. . . . . . . . . . . . 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 12304	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12473	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17111	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5169	. . . . . . . . . . . . 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 856	. . . . . . . . . 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 856	. . . . . 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 4305	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2862	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12573	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12380	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12495	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11413	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12465	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12793	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12382	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12478	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12786	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12564	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11292	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12406	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11413	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12794	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 470	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 47468	. . . . . . . . . 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 710	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7459	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12379	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12667	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12676	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1339	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12373	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11413	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12471	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11413	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 470	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13574	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 710	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 13610	. . . . . . 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 12357	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7459	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13639	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12435	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4762	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2772	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7458	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12359	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7459	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12677	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13639	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12439	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4762	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2768	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2772	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4189	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2772	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4237	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2862	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3069	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16755	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 710	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∪ cun 3974   ∩ cin 3975  {cpr 4650   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325  2c2 12348  3c3 12349  4c4 12350  5c5 12351  6c6 12352  9c9 12355  ℤcz 12639  ;cdc 12758  ℤ_≥cuz 12903  ...cfz 13567  ⌊cfl 13841  √csqrt 15282   ∥ cdvds 16302  ℙcprime 16718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-rp 13058  df-fz 13568  df-fl 13843  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-dvds 16303  df-prm 16719

This theorem is referenced by:  m5prm  47472