prmefexple - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > prmefexple		GIF version

Theorem prmefexple 16206

Description: Convert a bound on a power of a prime to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Jim Kingdon, 21-Aug-2026.)

**Assertion**
Ref	Expression
prmefexple	⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴↑𝑁) ≤ 𝐵 ↔ 𝑁 ≤ (⌊‘((log‘𝐵) / (log‘𝐴)))))

Proof of Theorem prmefexple Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1028	. . . . . 6 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℙ)
2		prmnn 12904	. . . . . 6 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
3	1, 2	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℕ)
4	3	nnrpd 10105	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ⁺)
5		simp2 1029	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑁 ∈ ℤ)
6		reexplog 16023	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
7	4, 5, 6	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴↑𝑁) = (exp‘(𝑁 · (log‘𝐴))))
8		simp3 1030	. . . . . 6 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
9	8	nnrpd 10105	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ⁺)
10	9	reeflogd 16035	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (exp‘(log‘𝐵)) = 𝐵)
11	10	eqcomd 2244	. . 3 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 = (exp‘(log‘𝐵)))
12	7, 11	breq12d 4143	. 2 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴↑𝑁) ≤ 𝐵 ↔ (exp‘(𝑁 · (log‘𝐴))) ≤ (exp‘(log‘𝐵))))
13	5	zred 9772	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑁 ∈ ℝ)
14	3	nnred 9319	. . . . . 6 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ)
15		prmuz2 12926	. . . . . . . 8 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ (ℤ_≥‘2))
16	1, 15	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ (ℤ_≥‘2))
17		eluz2gt1 10011	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 1 < 𝐴)
18	16, 17	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 1 < 𝐴)
19	14, 18	rplogcld 16040	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (log‘𝐴) ∈ ℝ⁺)
20	19	rpred 10107	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (log‘𝐴) ∈ ℝ)
21	13, 20	remulcld 8356	. . 3 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝑁 · (log‘𝐴)) ∈ ℝ)
22	9	relogcld 16034	. . 3 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (log‘𝐵) ∈ ℝ)
23		efle 15926	. . 3 ⊢ (((𝑁 · (log‘𝐴)) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ) → ((𝑁 · (log‘𝐴)) ≤ (log‘𝐵) ↔ (exp‘(𝑁 · (log‘𝐴))) ≤ (exp‘(log‘𝐵))))
24	21, 22, 23	syl2anc 415	. 2 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝑁 · (log‘𝐴)) ≤ (log‘𝐵) ↔ (exp‘(𝑁 · (log‘𝐴))) ≤ (exp‘(log‘𝐵))))
25	13, 22, 19	lemuldivd 10157	. . 3 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝑁 · (log‘𝐴)) ≤ (log‘𝐵) ↔ 𝑁 ≤ ((log‘𝐵) / (log‘𝐴))))
26		zprmlogbap 16137	. . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℙ) → ((𝐴 log_b 𝐵) ∈ ℚ ∨ ((𝐴 log_b 𝐵) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐴 log_b 𝐵) # 𝑞)))
27	8, 1, 26	syl2anc 415	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 log_b 𝐵) ∈ ℚ ∨ ((𝐴 log_b 𝐵) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐴 log_b 𝐵) # 𝑞)))
28		relogbval 16106	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝐵 ∈ ℝ⁺) → (𝐴 log_b 𝐵) = ((log‘𝐵) / (log‘𝐴)))
29	16, 9, 28	syl2anc 415	. . . . . . 7 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 log_b 𝐵) = ((log‘𝐵) / (log‘𝐴)))
30	29	eleq1d 2307	. . . . . 6 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 log_b 𝐵) ∈ ℚ ↔ ((log‘𝐵) / (log‘𝐴)) ∈ ℚ))
31	29	eleq1d 2307	. . . . . . 7 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 log_b 𝐵) ∈ ℝ ↔ ((log‘𝐵) / (log‘𝐴)) ∈ ℝ))
32	29	breq1d 4140	. . . . . . . 8 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 log_b 𝐵) # 𝑞 ↔ ((log‘𝐵) / (log‘𝐴)) # 𝑞))
33	32	ralbidv 2550	. . . . . . 7 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (∀𝑞 ∈ ℚ (𝐴 log_b 𝐵) # 𝑞 ↔ ∀𝑞 ∈ ℚ ((log‘𝐵) / (log‘𝐴)) # 𝑞))
34	31, 33	anbi12d 477	. . . . . 6 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (((𝐴 log_b 𝐵) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐴 log_b 𝐵) # 𝑞) ↔ (((log‘𝐵) / (log‘𝐴)) ∈ ℝ ∧ ∀𝑞 ∈ ℚ ((log‘𝐵) / (log‘𝐴)) # 𝑞)))
35	30, 34	orbi12d 805	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (((𝐴 log_b 𝐵) ∈ ℚ ∨ ((𝐴 log_b 𝐵) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐴 log_b 𝐵) # 𝑞)) ↔ (((log‘𝐵) / (log‘𝐴)) ∈ ℚ ∨ (((log‘𝐵) / (log‘𝐴)) ∈ ℝ ∧ ∀𝑞 ∈ ℚ ((log‘𝐵) / (log‘𝐴)) # 𝑞))))
36	27, 35	mpbid 147	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (((log‘𝐵) / (log‘𝐴)) ∈ ℚ ∨ (((log‘𝐵) / (log‘𝐴)) ∈ ℝ ∧ ∀𝑞 ∈ ℚ ((log‘𝐵) / (log‘𝐴)) # 𝑞)))
37		flapge 10730	. . . 4 ⊢ (((((log‘𝐵) / (log‘𝐴)) ∈ ℚ ∨ (((log‘𝐵) / (log‘𝐴)) ∈ ℝ ∧ ∀𝑞 ∈ ℚ ((log‘𝐵) / (log‘𝐴)) # 𝑞)) ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ ((log‘𝐵) / (log‘𝐴)) ↔ 𝑁 ≤ (⌊‘((log‘𝐵) / (log‘𝐴)))))
38	36, 5, 37	syl2anc 415	. . 3 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝑁 ≤ ((log‘𝐵) / (log‘𝐴)) ↔ 𝑁 ≤ (⌊‘((log‘𝐵) / (log‘𝐴)))))
39	25, 38	bitrd 188	. 2 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝑁 · (log‘𝐴)) ≤ (log‘𝐵) ↔ 𝑁 ≤ (⌊‘((log‘𝐵) / (log‘𝐴)))))
40	12, 24, 39	3bitr2d 216	1 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴↑𝑁) ≤ 𝐵 ↔ 𝑁 ≤ (⌊‘((log‘𝐵) / (log‘𝐴)))))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∀wral 2528 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℝcr 8178 1c1 8180 · cmul 8184 < clt 8360 ≤ cle 8361 # cap 8911 / cdiv 9004 ℕcn 9306 2c2 9357 ℤcz 9648 ℤ_≥cuz 9930 ℚcq 10028 ℝ⁺crp 10064 ⌊cfl 10713 ↑cexp 10988 expce 12425 ℙcprime 12901 logclog 16007 log_b clogb 16098

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299 ax-pre-suploc 8300 ax-addf 8301 ax-mulf 8302

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-map 6924 df-pm 6925 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8500 df-neg 8501 df-reap 8905 df-ap 8912 df-div 9005 df-inn 9307 df-2 9365 df-3 9366 df-4 9367 df-n0 9568 df-z 9649 df-uz 9931 df-q 10029 df-rp 10065 df-xneg 10184 df-xadd 10185 df-ioo 10304 df-ico 10306 df-icc 10307 df-fz 10422 df-fzo 10560 df-fl 10715 df-mod 10773 df-seqfrec 10898 df-exp 10989 df-fac 11178 df-bc 11200 df-ihash 11229 df-shft 11594 df-cj 11621 df-re 11622 df-im 11623 df-rsqrt 11778 df-abs 11779 df-clim 12061 df-sumdc 12136 df-ef 12431 df-e 12432 df-dvds 12571 df-gcd 12747 df-prm 12902 df-rest 13644 df-topgen 13663 df-psmet 14929 df-xmet 14930 df-met 14931 df-bl 14932 df-mopn 14933 df-top 15148 df-topon 15161 df-bases 15193 df-ntr 15246 df-cn 15338 df-cnp 15339 df-tx 15403 df-cncf 15721 df-limced 15806 df-dvap 15807 df-relog 16009 df-rpcxp 16010 df-logb 16099

This theorem is used by: bposlem1 16209

W3C validator