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Theorem prmefexple 16308

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 12913	. . . . . 6 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
3	1, 2	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℕ)
4	3	nnrpd 10112	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ⁺)
5		simp2 1029	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑁 ∈ ℤ)
6		reexplog 16101	. . . 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 10112	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ⁺)
10	9	reeflogd 16113	. . . 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 4144	. 2 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴↑𝑁) ≤ 𝐵 ↔ (exp‘(𝑁 · (log‘𝐴))) ≤ (exp‘(log‘𝐵))))
13	5	zred 9779	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑁 ∈ ℝ)
14	3	nnred 9326	. . . . . 6 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ)
15		prmuz2 12935	. . . . . . . 8 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ (ℤ_≥‘2))
16	1, 15	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ (ℤ_≥‘2))
17		eluz2gt1 10018	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 1 < 𝐴)
18	16, 17	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 1 < 𝐴)
19	14, 18	rplogcld 16118	. . . . 5 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (log‘𝐴) ∈ ℝ⁺)
20	19	rpred 10114	. . . 4 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (log‘𝐴) ∈ ℝ)
21	13, 20	remulcld 8362	. . 3 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝑁 · (log‘𝐴)) ∈ ℝ)
22	9	relogcld 16112	. . 3 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (log‘𝐵) ∈ ℝ)
23		efle 16002	. . 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 10164	. . 3 ⊢ ((𝐴 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝑁 · (log‘𝐴)) ≤ (log‘𝐵) ↔ 𝑁 ≤ ((log‘𝐵) / (log‘𝐴))))
26		zprmlogbap 16218	. . . . . 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 16187	. . . . . . . 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 4141	. . . . . . . 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 10737	. . . 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 4131 ‘cfv 5380 (class class class)co 6088 ℝcr 8184 1c1 8186 · cmul 8190 < clt 8366 ≤ cle 8367 # cap 8917 / cdiv 9010 ℕcn 9313 2c2 9364 ℤcz 9655 ℤ_≥cuz 9937 ℚcq 10035 ℝ⁺crp 10071 ⌊cfl 10720 ↑cexp 10996 expce 12434 ℙcprime 12910 logclog 16085 log_b clogb 16179

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305 ax-pre-suploc 8306 ax-addf 8307 ax-mulf 8308

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-2o 6692 df-oadd 6695 df-er 6811 df-map 6928 df-pm 6929 df-en 7027 df-dom 7028 df-fin 7029 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-q 10036 df-rp 10072 df-xneg 10191 df-xadd 10192 df-ioo 10311 df-ico 10313 df-icc 10314 df-fz 10429 df-fzo 10567 df-fl 10722 df-mod 10781 df-seqfrec 10906 df-exp 10997 df-fac 11186 df-bc 11208 df-ihash 11237 df-shft 11602 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-clim 12070 df-sumdc 12145 df-ef 12440 df-e 12441 df-dvds 12580 df-gcd 12756 df-prm 12911 df-rest 13654 df-topgen 13673 df-psmet 14989 df-xmet 14990 df-met 14991 df-bl 14992 df-mopn 14993 df-top 15224 df-topon 15237 df-bases 15269 df-ntr 15322 df-cn 15414 df-cnp 15415 df-tx 15479 df-cncf 15797 df-limced 15882 df-dvap 15883 df-relog 16087 df-rpcxp 16088 df-logb 16180

This theorem is used by: bposlem1 16311

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