Theorem 31prm 48166

Description: 31 is a prime number. In contrast to 37prm 17147, the proof of this theorem is not based on the "blanket" prmlem2 17146, but on isprm7 16733. 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 17147 (1810 characters compared with 1213 for 37prm 17147). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17147). (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 12596	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12492	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12490	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12696	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12589	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12290	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12491	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12285	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12310	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12418	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11299	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12722	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12838	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1354	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4104	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3918	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3457	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4604	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12489	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12286	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11365	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12728	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17089	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5100	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 329	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 48164	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16356	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12355	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5105	. . . . . . . . . . . . . 14 ⊢ (3 ∥ (3 + 1) ↔ 3 ∥ 4)
30	27, 29	bitri 277	. . . . . . . . . . . . 13 ⊢ (3 ∥ ;31 ↔ 3 ∥ 4)
31	26, 30	mtbir 325	. . . . . . . . . . . 12 ⊢ ¬ 3 ∥ ;31
32		breq1 5100	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 329	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 868	. . . . . . . . . 10 ⊢ ((𝑛 = 2 ∨ 𝑛 = 3) → ¬ 𝑛 ∥ ;31)
35	18, 34	sylbi 219	. . . . . . . . 9 ⊢ (𝑛 ∈ {2, 3} → ¬ 𝑛 ∥ ;31)
36	35	adantr 484	. . . . . . . 8 ⊢ ((𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
37	16, 36	sylbi 219	. . . . . . 7 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
38		elin 3918	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4604	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2849	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16719	. . . . . . . . . . . . 13 ⊢ ¬ 4 ∈ ℙ
42	41	pm2.21i 119	. . . . . . . . . . . 12 ⊢ (4 ∈ ℙ → ¬ 𝑛 ∥ ;31)
43	40, 42	biimtrdi 255	. . . . . . . . . . 11 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
44		1nn 12214	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12393	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17090	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5100	. . . . . . . . . . . . 13 ⊢ (𝑛 = 5 → (𝑛 ∥ ;31 ↔ 5 ∥ ;31))
48	46, 47	mtbiri 329	. . . . . . . . . . . 12 ⊢ (𝑛 = 5 → ¬ 𝑛 ∥ ;31)
49	48	a1d 25	. . . . . . . . . . 11 ⊢ (𝑛 = 5 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
50	43, 49	jaoi 868	. . . . . . . . . 10 ⊢ ((𝑛 = 4 ∨ 𝑛 = 5) → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
51	39, 50	sylbi 219	. . . . . . . . 9 ⊢ (𝑛 ∈ {4, 5} → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
52	51	imp 410	. . . . . . . 8 ⊢ ((𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
53	38, 52	sylbi 219	. . . . . . 7 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
54	37, 53	jaoi 868	. . . . . 6 ⊢ ((𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
55	15, 54	sylbi 219	. . . . 5 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
56		indir 4236	. . . . 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 12494	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12298	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12415	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11299	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12384	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12721	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12300	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12398	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12714	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12485	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11176	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12327	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11299	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12722	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 474	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 48163	. . . . . . . . . 10 ⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) → ((;25 ≤ ;31 ∧ ;31 < ;36) ↔ (⌊‘(√‘;31)) = 5))
74	73	bicomd 225	. . . . . . . . 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 721	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7401	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12297	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12588	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12597	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1352	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12291	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11299	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12391	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11299	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 474	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13512	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 721	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 13548	. . . . . . 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 12274	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7401	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13577	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12352	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4693	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2788	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7400	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12276	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7401	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12598	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13577	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12356	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4693	. . . . . . . . 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 4117	. . . . . 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 4166	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2879	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3077	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16733	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 721	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ∪ cun 3900   ∩ cin 3901  {cpr 4581   class class class wbr 5097  ‘cfv 6515  (class class class)co 7390  ℝcr 11065  0cc0 11066  1c1 11067   + caddc 11069   < clt 11209   ≤ cle 11210  2c2 12265  3c3 12266  4c4 12267  5c5 12268  6c6 12269  9c9 12272  ℤcz 12561  ;cdc 12681  ℤ_≥cuz 12832  ...cfz 13505  ⌊cfl 13793  √csqrt 15250   ∥ cdvds 16276  ℙcprime 16695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143  ax-pre-sup 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-2o 8431  df-er 8671  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-sup 9381  df-inf 9382  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-div 11838  df-nn 12204  df-2 12273  df-3 12274  df-4 12275  df-5 12276  df-6 12277  df-7 12278  df-8 12279  df-9 12280  df-n0 12475  df-z 12562  df-dec 12682  df-uz 12833  df-rp 12987  df-fz 13506  df-fl 13795  df-seq 14008  df-exp 14068  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-dvds 16277  df-prm 16696

This theorem is referenced by:  m5prm  48167