Theorem 31prm 47611

Description: 31 is a prime number. In contrast to 37prm 17140, the proof of this theorem is not based on the "blanket" prmlem2 17139, but on isprm7 16727. 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 17140 (1810 characters compared with 1213 for 37prm 17140). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17140). (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 12624	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12519	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12517	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12723	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12617	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12319	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12518	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12314	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12339	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12445	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11358	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12744	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12858	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1342	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4128	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3942	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3463	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4626	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12516	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12315	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11424	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12750	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17083	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5122	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 47609	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16351	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12385	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5127	. . . . . . . . . . . . . 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 5122	. . . . . . . . . . . 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 3942	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4626	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2822	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16714	. . . . . . . . . . . . 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 12251	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12420	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17084	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5122	. . . . . . . . . . . . 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 4261	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2852	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12521	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12327	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12442	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11358	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12412	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12743	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12329	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12425	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12736	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12512	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11237	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12353	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11358	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12744	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 470	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 47608	. . . . . . . . . 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 7416	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12326	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12616	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12625	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1340	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12320	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11358	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12418	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11358	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 470	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13531	. . . . . . . 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 13567	. . . . . . 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 12304	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7416	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13596	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12382	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4713	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2762	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7415	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12306	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7416	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12626	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13596	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12386	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4713	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2758	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2762	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4141	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2762	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4191	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2852	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3053	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16727	. 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 2108  ∀wral 3051   ∪ cun 3924   ∩ cin 3925  {cpr 4603   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  ℝcr 11128  0cc0 11129  1c1 11130   + caddc 11132   < clt 11269   ≤ cle 11270  2c2 12295  3c3 12296  4c4 12297  5c5 12298  6c6 12299  9c9 12302  ℤcz 12588  ;cdc 12708  ℤ_≥cuz 12852  ...cfz 13524  ⌊cfl 13807  √csqrt 15252   ∥ cdvds 16272  ℙcprime 16690

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-rp 13009  df-fz 13525  df-fl 13809  df-seq 14020  df-exp 14080  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-dvds 16273  df-prm 16691

This theorem is referenced by:  m5prm  47612