Theorem 31prm 48626

Description: 31 is a prime number. In contrast to 37prm 17279, the proof of this theorem is not based on the "blanket" prmlem2 17278, but on isprm7 16864. 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 17279 (1810 characters compared with 1213 for 37prm 17279). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17279). (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 12709	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12605	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12603	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12810	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12702	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12403	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12604	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12398	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12423	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12531	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11414	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12836	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12952	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1360	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4100	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3915	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3455	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4609	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12602	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12399	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11480	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12842	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17221	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5106	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 48624	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16482	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12468	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5111	. . . . . . . . . . . . . 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 5106	. . . . . . . . . . . 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 3915	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4609	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16850	. . . . . . . . . . . . 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 12327	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12506	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17222	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5106	. . . . . . . . . . . . 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 4232	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2879	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12607	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12411	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12528	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11414	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12497	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12835	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12413	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12511	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12828	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12598	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11291	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12440	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11414	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12836	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 476	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 48623	. . . . . . . . . 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 7423	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12410	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12701	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12710	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1358	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12404	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11414	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12504	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11414	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 476	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13627	. . . . . . . 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 13664	. . . . . . 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 12387	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7423	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13693	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12465	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4698	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2788	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7422	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12389	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7423	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12711	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13693	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12469	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4698	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2784	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2788	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4113	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2788	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4162	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2879	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3079	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16864	. 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 1570   ∈ wcel 2145  ∀wral 3077   ∪ cun 3897   ∩ cin 3898  {cpr 4586   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  ℝcr 11180  0cc0 11181  1c1 11182   + caddc 11184   < clt 11324   ≤ cle 11325  2c2 12378  3c3 12379  4c4 12380  5c5 12381  6c6 12382  9c9 12385  ℤcz 12674  ;cdc 12795  ℤ_≥cuz 12946  ...cfz 13620  ⌊cfl 13910  √csqrt 15380   ∥ cdvds 16402  ℙcprime 16826

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-dec 12796  df-uz 12947  df-rp 13102  df-fz 13621  df-fl 13912  df-seq 14125  df-exp 14185  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-dvds 16403  df-prm 16827

This theorem is used by:  m5prm  48627