Theorem 31prm 47724

Description: 31 is a prime number. In contrast to 37prm 17036, the proof of this theorem is not based on the "blanket" prmlem2 17035, but on isprm7 16623. 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 17036 (1810 characters compared with 1213 for 37prm 17036). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17036). (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 12512	. . 3 ⊢ 2 ∈ ℤ
2		3nn0 12408	. . . . 5 ⊢ 3 ∈ ℕ0
3		1nn0 12406	. . . . 5 ⊢ 1 ∈ ℕ0
4	2, 3	deccl 12611	. . . 4 ⊢ ;31 ∈ ℕ0
5	4	nn0zi 12505	. . 3 ⊢ ;31 ∈ ℤ
6		3nn 12213	. . . 4 ⊢ 3 ∈ ℕ
7		2nn0 12407	. . . 4 ⊢ 2 ∈ ℕ0
8		2re 12208	. . . . 5 ⊢ 2 ∈ ℝ
9		9re 12233	. . . . 5 ⊢ 9 ∈ ℝ
10		2lt9 12334	. . . . 5 ⊢ 2 < 9
11	8, 9, 10	ltleii 11245	. . . 4 ⊢ 2 ≤ 9
12	6, 3, 7, 11	declei 12632	. . 3 ⊢ 2 ≤ ;31
13		eluz2 12746	. . 3 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31))
14	1, 5, 12, 13	mpbir3an 1342	. 2 ⊢ ;31 ∈ (ℤ≥‘2)
15		elun 4102	. . . . . 6 ⊢ (𝑛 ∈ (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩ ℙ)))
16		elin 3914	. . . . . . . 8 ⊢ (𝑛 ∈ ({2, 3} ∩ ℙ) ↔ (𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ))
17		vex 3441	. . . . . . . . . . 11 ⊢ 𝑛 ∈ V
18	17	elpr 4602	. . . . . . . . . 10 ⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3))
19		0nn0 12405	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ0
20		2cn 12209	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
21	20	mul02i 11311	. . . . . . . . . . . . 13 ⊢ (0 · 2) = 0
22		1e0p1 12638	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
23	2, 19, 21, 22	dec2dvds 16979	. . . . . . . . . . . 12 ⊢ ¬ 2 ∥ ;31
24		breq1 5098	. . . . . . . . . . . 12 ⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31))
25	23, 24	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31)
26		3ndvds4 47722	. . . . . . . . . . . . 13 ⊢ ¬ 3 ∥ 4
27	2, 3	3dvdsdec 16247	. . . . . . . . . . . . . 14 ⊢ (3 ∥ ;31 ↔ 3 ∥ (3 + 1))
28		3p1e4 12274	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
29	28	breq2i 5103	. . . . . . . . . . . . . 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 5098	. . . . . . . . . . . 12 ⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31))
33	31, 32	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31)
34	25, 33	jaoi 857	. . . . . . . . . 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 3914	. . . . . . . 8 ⊢ (𝑛 ∈ ({4, 5} ∩ ℙ) ↔ (𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ))
39	17	elpr 4602	. . . . . . . . . 10 ⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5))
40		eleq1 2821	. . . . . . . . . . . 12 ⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈ ℙ))
41		4nprm 16610	. . . . . . . . . . . . 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 12145	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
45		1lt5 12309	. . . . . . . . . . . . . 14 ⊢ 1 < 5
46	2, 44, 45	dec5dvds 16980	. . . . . . . . . . . . 13 ⊢ ¬ 5 ∥ ;31
47		breq1 5098	. . . . . . . . . . . . 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 857	. . . . . . . . . 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 857	. . . . . 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 4235	. . . . 5 ⊢ (({2, 3} ∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩ ℙ))
57	55, 56	eleq2s 2851	. . . 4 ⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5}) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
58		5nn0 12410	. . . . . . . . 9 ⊢ 5 ∈ ℕ0
59		5re 12221	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
60		5lt9 12331	. . . . . . . . . 10 ⊢ 5 < 9
61	59, 9, 60	ltleii 11245	. . . . . . . . 9 ⊢ 5 ≤ 9
62		2lt3 12301	. . . . . . . . 9 ⊢ 2 < 3
63	7, 2, 58, 3, 61, 62	decleh 12631	. . . . . . . 8 ⊢ ;25 ≤ ;31
64		6nn 12223	. . . . . . . . 9 ⊢ 6 ∈ ℕ
65		1lt6 12314	. . . . . . . . 9 ⊢ 1 < 6
66	2, 3, 64, 65	declt 12624	. . . . . . . 8 ⊢ ;31 < ;36
67	4	nn0rei 12401	. . . . . . . . . 10 ⊢ ;31 ∈ ℝ
68		0re 11123	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
69		9pos 12247	. . . . . . . . . . . 12 ⊢ 0 < 9
70	68, 9, 69	ltleii 11245	. . . . . . . . . . 11 ⊢ 0 ≤ 9
71	6, 3, 19, 70	declei 12632	. . . . . . . . . 10 ⊢ 0 ≤ ;31
72	67, 71	pm3.2i 470	. . . . . . . . 9 ⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31)
73		flsqrt5 47721	. . . . . . . . . 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 711	. . . . . . 7 ⊢ (⌊‘(√‘;31)) = 5
77	76	oveq2i 7365	. . . . . 6 ⊢ (2...(⌊‘(√‘;31))) = (2...5)
78		5nn 12220	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
79	78	nnzi 12504	. . . . . . . . 9 ⊢ 5 ∈ ℤ
80		3z 12513	. . . . . . . . 9 ⊢ 3 ∈ ℤ
81	1, 79, 80	3pm3.2i 1340	. . . . . . . 8 ⊢ (2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ)
82		3re 12214	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
83	8, 82, 62	ltleii 11245	. . . . . . . . 9 ⊢ 2 ≤ 3
84		3lt5 12307	. . . . . . . . . 10 ⊢ 3 < 5
85	82, 59, 84	ltleii 11245	. . . . . . . . 9 ⊢ 3 ≤ 5
86	83, 85	pm3.2i 470	. . . . . . . 8 ⊢ (2 ≤ 3 ∧ 3 ≤ 5)
87		elfz2 13418	. . . . . . . 8 ⊢ (3 ∈ (2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5)))
88	81, 86, 87	mpbir2an 711	. . . . . . 7 ⊢ 3 ∈ (2...5)
89		fzsplit 13454	. . . . . . 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 12198	. . . . . . . . 9 ⊢ 3 = (2 + 1)
92	91	oveq2i 7365	. . . . . . . 8 ⊢ (2...3) = (2...(2 + 1))
93		fzpr 13483	. . . . . . . . 9 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)})
94	1, 93	ax-mp 5	. . . . . . . 8 ⊢ (2...(2 + 1)) = {2, (2 + 1)}
95		2p1e3 12271	. . . . . . . . 9 ⊢ (2 + 1) = 3
96	95	preq2i 4691	. . . . . . . 8 ⊢ {2, (2 + 1)} = {2, 3}
97	92, 94, 96	3eqtri 2760	. . . . . . 7 ⊢ (2...3) = {2, 3}
98	28	oveq1i 7364	. . . . . . . 8 ⊢ ((3 + 1)...5) = (4...5)
99		df-5 12200	. . . . . . . . 9 ⊢ 5 = (4 + 1)
100	99	oveq2i 7365	. . . . . . . 8 ⊢ (4...5) = (4...(4 + 1))
101		4z 12514	. . . . . . . . . 10 ⊢ 4 ∈ ℤ
102		fzpr 13483	. . . . . . . . . 10 ⊢ (4 ∈ ℤ → (4...(4 + 1)) = {4, (4 + 1)})
103	101, 102	ax-mp 5	. . . . . . . . 9 ⊢ (4...(4 + 1)) = {4, (4 + 1)}
104		4p1e5 12275	. . . . . . . . . 10 ⊢ (4 + 1) = 5
105	104	preq2i 4691	. . . . . . . . 9 ⊢ {4, (4 + 1)} = {4, 5}
106	103, 105	eqtri 2756	. . . . . . . 8 ⊢ (4...(4 + 1)) = {4, 5}
107	98, 100, 106	3eqtri 2760	. . . . . . 7 ⊢ ((3 + 1)...5) = {4, 5}
108	97, 107	uneq12i 4115	. . . . . 6 ⊢ ((2...3) ∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5})
109	77, 90, 108	3eqtri 2760	. . . . 5 ⊢ (2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5})
110	109	ineq1i 4165	. . . 4 ⊢ ((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩ ℙ)
111	57, 110	eleq2s 2851	. . 3 ⊢ (𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31)
112	111	rgen 3050	. 2 ⊢ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31
113		isprm7 16623	. 2 ⊢ (;31 ∈ ℙ ↔ (;31 ∈ (ℤ≥‘2) ∧ ∀𝑛 ∈ ((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31))
114	14, 112, 113	mpbir2an 711	1 ⊢ ;31 ∈ ℙ

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ∪ cun 3896   ∩ cin 3897  {cpr 4579   class class class wbr 5095  ‘cfv 6488  (class class class)co 7354  ℝcr 11014  0cc0 11015  1c1 11016   + caddc 11018   < clt 11155   ≤ cle 11156  2c2 12189  3c3 12190  4c4 12191  5c5 12192  6c6 12193  9c9 12196  ℤcz 12477  ;cdc 12596  ℤ_≥cuz 12740  ...cfz 13411  ⌊cfl 13698  √csqrt 15144   ∥ cdvds 16167  ℙcprime 16586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-cnex 11071  ax-resscn 11072  ax-1cn 11073  ax-icn 11074  ax-addcl 11075  ax-addrcl 11076  ax-mulcl 11077  ax-mulrcl 11078  ax-mulcom 11079  ax-addass 11080  ax-mulass 11081  ax-distr 11082  ax-i2m1 11083  ax-1ne0 11084  ax-1rid 11085  ax-rnegex 11086  ax-rrecex 11087  ax-cnre 11088  ax-pre-lttri 11089  ax-pre-lttrn 11090  ax-pre-ltadd 11091  ax-pre-mulgt0 11092  ax-pre-sup 11093

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7805  df-1st 7929  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-1o 8393  df-2o 8394  df-er 8630  df-en 8878  df-dom 8879  df-sdom 8880  df-fin 8881  df-sup 9335  df-inf 9336  df-pnf 11157  df-mnf 11158  df-xr 11159  df-ltxr 11160  df-le 11161  df-sub 11355  df-neg 11356  df-div 11784  df-nn 12135  df-2 12197  df-3 12198  df-4 12199  df-5 12200  df-6 12201  df-7 12202  df-8 12203  df-9 12204  df-n0 12391  df-z 12478  df-dec 12597  df-uz 12741  df-rp 12895  df-fz 13412  df-fl 13700  df-seq 13913  df-exp 13973  df-cj 15010  df-re 15011  df-im 15012  df-sqrt 15146  df-abs 15147  df-dvds 16168  df-prm 16587

This theorem is referenced by:  m5prm  47725