Theorem 31prm 42188

Description: 31 is a prime number. In contrast to 37prm 16101, the proof of this theorem is not based on the "blanket" prmlem2 16100, but on isprm7 15699. 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 16101 (1810 characters compared with 1213 for 37prm 16101). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 16101). (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 11656	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 11558	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 11556	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 11755	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 11649	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 11351	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 11557	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 11346	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 11377	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 11483	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 10414	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 11777	. . 3 ⊢ 2 ≤ ;31
13		eluz2 11892	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1441	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 3915	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3958	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3353	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4357	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 11555	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 11347	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 10479	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 11783	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 16046	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 4812	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 318	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 42186	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 15338	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 11423	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 4817	. . . . . . . . . . . . . 14 ⊢ (3 ∥ (3 + 1) ↔ 3 ∥ 4)
30	27, 29	bitri 266	. . . . . . . . . . . . 13 ⊢ (3 ∥ ;31 ↔ 3 ∥ 4)
31	26, 30	mtbir 314	. . . . . . . . . . . 12 ⊢ ¬ 3 ∥ ;31
32		breq1 4812	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 318	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 883	. . . . . . . . . 10 ⊢ ((𝑛 = 2 ∨ 𝑛 = 3) → ¬ 𝑛 ∥ ;31)
35	18, 34	sylbi 208	. . . . . . . . 9 ⊢ (𝑛 ∈ {2, 3} → ¬ 𝑛 ∥ ;31)
36	35	adantr 472	. . . . . . . 8 ⊢ ((𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
37	16, 36	sylbi 208	. . . . . . 7 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
38		elin 3958	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4357	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2832	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 15687	. . . . . . . . . . . . 13 ⊢ ¬ 4 ∈ ℙ
42	41	pm2.21i 117	. . . . . . . . . . . 12 ⊢ (4 ∈ ℙ → ¬ 𝑛 ∥ ;31)
43	40, 42	syl6bi 244	. . . . . . . . . . 11 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
44		1nn 11287	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 11458	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 16047	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 4812	. . . . . . . . . . . . 13 ⊢ (𝑛 = 5 → (𝑛 ∥ ;31 ↔ 5 ∥ ;31))
48	46, 47	mtbiri 318	. . . . . . . . . . . 12 ⊢ (𝑛 = 5 → ¬ 𝑛 ∥ ;31)
49	48	a1d 25	. . . . . . . . . . 11 ⊢ (𝑛 = 5 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
50	43, 49	jaoi 883	. . . . . . . . . 10 ⊢ ((𝑛 = 4 ∨ 𝑛 = 5) → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
51	39, 50	sylbi 208	. . . . . . . . 9 ⊢ (𝑛 ∈ {4, 5} → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31))
52	51	imp 395	. . . . . . . 8 ⊢ ((𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ) → ¬ 𝑛 ∥ ;31)
53	38, 52	sylbi 208	. . . . . . 7 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) → ¬ 𝑛 ∥ ;31)
54	37, 53	jaoi 883	. . . . . 6 ⊢ ((𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
55	15, 54	sylbi 208	. . . . 5 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31)
56		indir 4040	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2862	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 11560	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 11361	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 11480	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 10414	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 11450	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 11776	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 11364	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 11463	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 11769	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 11550	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 10295	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 11392	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 10414	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 11777	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 462	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 42185	. . . . . . . . . 10 ⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) → ((;25 ≤ ;31 ∧ ;31 < ;36) ↔ (⌊‘(√‘;31)) = 5))
74	73	bicomd 214	. . . . . . . . 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 702	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 6853	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 11360	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 11648	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 11657	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1438	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 11352	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 10414	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 11456	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 10414	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 462	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 12540	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 702	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 12574	. . . . . . 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 11336	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 6853	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 12603	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 11421	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4427	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2791	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 6852	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 11338	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 6853	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 11658	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 12603	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 11424	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4427	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2787	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2791	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 3927	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2791	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 3972	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2862	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3069	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 15699	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 702	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∨ wo 873   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  ∀wral 3055   ∪ cun 3730   ∩ cin 3731  {cpr 4336   class class class wbr 4809  ‘cfv 6068  (class class class)co 6842  ℝcr 10188  0cc0 10189  1c1 10190   + caddc 10192   < clt 10328   ≤ cle 10329  2c2 11327  3c3 11328  4c4 11329  5c5 11330  6c6 11331  9c9 11334  ℤcz 11624  ;cdc 11740  ℤ_≥cuz 11886  ...cfz 12533  ⌊cfl 12799  √csqrt 14258   ∥ cdvds 15265  ℙcprime 15665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-rp 12029  df-fz 12534  df-fl 12801  df-seq 13009  df-exp 13068  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-dvds 15266  df-prm 15666

This theorem is referenced by:  m5prm  42189