Theorem 31prm 48332

Description: 31 is a prime number. In contrast to 37prm 17182, the proof of this theorem is not based on the "blanket" prmlem2 17181, but on isprm7 16768. 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 17182 (1810 characters compared with 1213 for 37prm 17182). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17182). (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 12627	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12523	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12521	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12727	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12620	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12321	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12522	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12316	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12341	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12449	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11334	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12753	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12869	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1360	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4108	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3922	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3459	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4615	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12520	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12317	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11400	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12759	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17124	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5113	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 48330	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16391	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12386	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5118	. . . . . . . . . . . . . 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 5113	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 870	. . . . . . . . . 10 ⊢ ((𝑛 = 2 ∨ 𝑛 = 3) → ¬ 𝑛 ∥ ;31)
35	18, 34	sylbi 220	. . . . . . . . 9 ⊢ (𝑛 ∈ {2, 3} → ¬ 𝑛 ∥ ;31)
36	35	adantr 485	. . . . . . . 8 ⊢ ((𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
37	16, 36	sylbi 220	. . . . . . 7 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
38		elin 3922	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4615	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2851	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16754	. . . . . . . . . . . . 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 12245	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12424	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17125	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5113	. . . . . . . . . . . . 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 870	. . . . . . . . . 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 411	. . . . . . . 8 ⊢ ((𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
53	38, 52	sylbi 220	. . . . . . 7 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
54	37, 53	jaoi 870	. . . . . 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 4240	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2881	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12525	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12329	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12446	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11334	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12415	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12752	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12331	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12429	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12745	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12516	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11211	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12358	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11334	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12753	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 475	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 48329	. . . . . . . . . 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 723	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7423	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12328	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12619	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12628	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1358	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12322	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11334	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12422	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11334	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 475	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13543	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 723	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 13580	. . . . . . 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 12305	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7423	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13609	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12383	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4704	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2790	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7422	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12307	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7423	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12629	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13609	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12387	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4704	. . . . . . . . 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 4121	. . . . . 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 4170	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2881	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3081	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16768	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 723	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∪ cun 3904   ∩ cin 3905  {cpr 4592   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412  ℝcr 11100  0cc0 11101  1c1 11102   + caddc 11104   < clt 11244   ≤ cle 11245  2c2 12296  3c3 12297  4c4 12298  5c5 12299  6c6 12300  9c9 12303  ℤcz 12592  ;cdc 12712  ℤ_≥cuz 12863  ...cfz 13536  ⌊cfl 13825  √csqrt 15286   ∥ cdvds 16311  ℙcprime 16730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-pre-sup 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-inf 9404  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-7 12309  df-8 12310  df-9 12311  df-n0 12506  df-z 12593  df-dec 12713  df-uz 12864  df-rp 13018  df-fz 13537  df-fl 13827  df-seq 14040  df-exp 14100  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-dvds 16312  df-prm 16731

This theorem is referenced by:  m5prm  48333