Theorem prmgaplcm 17031

Description: Alternate proof of prmgap 17030: in contrast to prmgap 17030, where the gap starts at n! , the factorial of n, the gap starts at the least common multiple of all positive integers less than or equal to n. (Contributed by AV, 13-Aug-2020.) (Revised by AV, 27-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
prmgaplcm	⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

Distinct variable group:   𝑛,𝑝,𝑞,𝑧

Proof of Theorem prmgaplcm

Dummy variables 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ)
2		fzssz 13480	. . . . . . . 8 ⊢ (1...𝑥) ⊆ ℤ
3	2	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ⊆ ℤ)
4		fzfi 13934	. . . . . . . 8 ⊢ (1...𝑥) ∈ Fin
5	4	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1...𝑥) ∈ Fin)
6		0nelfz1 13497	. . . . . . . 8 ⊢ 0 ∉ (1...𝑥)
7	6	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → 0 ∉ (1...𝑥))
8		lcmfn0cl 16595	. . . . . . 7 ⊢ (((1...𝑥) ⊆ ℤ ∧ (1...𝑥) ∈ Fin ∧ 0 ∉ (1...𝑥)) → (lcm‘(1...𝑥)) ∈ ℕ)
9	3, 5, 7, 8	syl3anc 1374	. . . . . 6 ⊢ (𝑥 ∈ ℕ → (lcm‘(1...𝑥)) ∈ ℕ)
10	9	adantl 481	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (lcm‘(1...𝑥)) ∈ ℕ)
11		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) = (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))
12	10, 11	fmptd 7066	. . . 4 ⊢ (𝑛 ∈ ℕ → (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))):ℕ⟶ℕ)
13		nnex 12180	. . . . . 6 ⊢ ℕ ∈ V
14	13, 13	pm3.2i 470	. . . . 5 ⊢ (ℕ ∈ V ∧ ℕ ∈ V)
15		elmapg 8786	. . . . 5 ⊢ ((ℕ ∈ V ∧ ℕ ∈ V) → ((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) ∈ (ℕ ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))):ℕ⟶ℕ))
16	14, 15	mp1i 13	. . . 4 ⊢ (𝑛 ∈ ℕ → ((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) ∈ (ℕ ↑m ℕ) ↔ (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))):ℕ⟶ℕ))
17	12, 16	mpbird 257	. . 3 ⊢ (𝑛 ∈ ℕ → (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) ∈ (ℕ ↑m ℕ))
18		prmgaplcmlem2 17023	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < (((lcm‘(1...𝑛)) + 𝑖) gcd 𝑖))
19		eqidd 2737	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))) = (𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥))))
20		oveq2 7375	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → (1...𝑥) = (1...𝑛))
21	20	fveq2d 6844	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (lcm‘(1...𝑥)) = (lcm‘(1...𝑛)))
22	21	adantl 481	. . . . . . . 8 ⊢ (((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) ∧ 𝑥 = 𝑛) → (lcm‘(1...𝑥)) = (lcm‘(1...𝑛)))
23		simpl 482	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 𝑛 ∈ ℕ)
24		fzssz 13480	. . . . . . . . . 10 ⊢ (1...𝑛) ⊆ ℤ
25		fzfi 13934	. . . . . . . . . 10 ⊢ (1...𝑛) ∈ Fin
26	24, 25	pm3.2i 470	. . . . . . . . 9 ⊢ ((1...𝑛) ⊆ ℤ ∧ (1...𝑛) ∈ Fin)
27		lcmfcl 16597	. . . . . . . . 9 ⊢ (((1...𝑛) ⊆ ℤ ∧ (1...𝑛) ∈ Fin) → (lcm‘(1...𝑛)) ∈ ℕ0)
28	26, 27	mp1i 13	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (lcm‘(1...𝑛)) ∈ ℕ0)
29	19, 22, 23, 28	fvmptd 6955	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) = (lcm‘(1...𝑛)))
30	29	oveq1d 7382	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → (((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) = ((lcm‘(1...𝑛)) + 𝑖))
31	30	oveq1d 7382	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → ((((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) gcd 𝑖) = (((lcm‘(1...𝑛)) + 𝑖) gcd 𝑖))
32	18, 31	breqtrrd 5113	. . . 4 ⊢ ((𝑛 ∈ ℕ ∧ 𝑖 ∈ (2...𝑛)) → 1 < ((((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) gcd 𝑖))
33	32	ralrimiva 3129	. . 3 ⊢ (𝑛 ∈ ℕ → ∀𝑖 ∈ (2...𝑛)1 < ((((𝑥 ∈ ℕ ↦ (lcm‘(1...𝑥)))‘𝑛) + 𝑖) gcd 𝑖))
34	1, 17, 33	prmgaplem8 17029	. 2 ⊢ (𝑛 ∈ ℕ → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))
35	34	rgen 3053	1 ⊢ ∀𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞 − 𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∉ wnel 3036  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889   class class class wbr 5085   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773  Fincfn 8893  0cc0 11038  1c1 11039   + caddc 11041   < clt 11179   ≤ cle 11180   − cmin 11377  ℕcn 12174  2c2 12236  ℕ₀cn0 12437  ℤcz 12524  ...cfz 13461  ..^cfzo 13608   gcd cgcd 16463  lcmclcmf 16558  ℙcprime 16640

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-fac 14236  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-prod 15869  df-dvds 16222  df-gcd 16464  df-lcmf 16560  df-prm 16641

This theorem is referenced by: (None)