Theorem 31prm 47954

Description: 31 is a prime number. In contrast to 37prm 17060, the proof of this theorem is not based on the "blanket" prmlem2 17059, but on isprm7 16647. 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 17060 (1810 characters compared with 1213 for 37prm 17060). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17060). (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 12535	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12431	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12429	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12634	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12528	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12236	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12430	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12231	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12256	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12357	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11268	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12655	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12769	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1343	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4107	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3919	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3446	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4607	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12428	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12232	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11334	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12661	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 17003	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5103	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 47952	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16271	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12297	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5108	. . . . . . . . . . . . . 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 5103	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 858	. . . . . . . . . 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 3919	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4607	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16634	. . . . . . . . . . . . 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 12168	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12332	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 17004	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5103	. . . . . . . . . . . . 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 858	. . . . . . . . . 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 858	. . . . . 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 4240	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2855	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12433	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12244	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12354	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11268	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12324	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12654	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12246	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12337	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12647	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12424	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11146	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12270	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11268	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12655	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 470	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 47951	. . . . . . . . . 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 712	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7379	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12243	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12527	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12536	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1341	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12237	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11268	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12330	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11268	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 470	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13442	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 712	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 13478	. . . . . . 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 12221	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7379	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13507	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12294	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4696	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2764	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7378	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12223	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7379	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12537	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13507	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12298	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4696	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2760	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2764	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4120	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2764	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4170	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2855	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3054	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16647	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 712	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∪ cun 3901   ∩ cin 3902  {cpr 4584   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   < clt 11178   ≤ cle 11179  2c2 12212  3c3 12213  4c4 12214  5c5 12215  6c6 12216  9c9 12219  ℤcz 12500  ;cdc 12619  ℤ_≥cuz 12763  ...cfz 13435  ⌊cfl 13722  √csqrt 15168   ∥ cdvds 16191  ℙcprime 16610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-rp 12918  df-fz 13436  df-fl 13724  df-seq 13937  df-exp 13997  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-dvds 16192  df-prm 16611

This theorem is referenced by:  m5prm  47955