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Theorem zprmlogbap 16179

Description: The logarithm of a natural number to a prime base is either rational or irrational.

The proof decomposes 𝑋 into 𝑚 ∈ ℕ and 𝑎 ∈ ℕ₀ such that 𝑋 = ((𝐵↑𝑎) · 𝑚) (using nnmaxpw 12972). If 𝑚 = 1 the logarithm is 𝑎, which is rational. If 1 < 𝑚 then we can apply logbgcd1irrap 16167 to show that the logarithm is irrational. (Contributed by Jim Kingdon and Taylor Barrella, 20-Aug-2026.)

**Assertion**
Ref	Expression
zprmlogbap	⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ((𝐵 log_b 𝑋) ∈ ℚ ∨ ((𝐵 log_b 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b 𝑋) # 𝑞)))

Distinct variable groups: 𝐵,𝑞 𝑋,𝑞

Proof of Theorem zprmlogbap Dummy variables 𝑎 𝑚 𝑡 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4134	. . . . 5 ⊢ (𝑡 = 𝑧 → (𝐵 ∥ 𝑡 ↔ 𝐵 ∥ 𝑧))
2	1	notbid 677	. . . 4 ⊢ (𝑡 = 𝑧 → (¬ 𝐵 ∥ 𝑡 ↔ ¬ 𝐵 ∥ 𝑧))
3	2	cbvrabv 2820	. . 3 ⊢ {𝑡 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑡} = {𝑧 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑧}
4		oveq2 6093	. . . 4 ⊢ (𝑢 = 𝑥 → ((𝐵↑𝑣) · 𝑢) = ((𝐵↑𝑣) · 𝑥))
5		oveq2 6093	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝐵↑𝑣) = (𝐵↑𝑦))
6	5	oveq1d 6100	. . . 4 ⊢ (𝑣 = 𝑦 → ((𝐵↑𝑣) · 𝑥) = ((𝐵↑𝑦) · 𝑥))
7	4, 6	cbvmpov 6168	. . 3 ⊢ (𝑢 ∈ {𝑡 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑡}, 𝑣 ∈ ℕ₀ ↦ ((𝐵↑𝑣) · 𝑢)) = (𝑥 ∈ {𝑡 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑡}, 𝑦 ∈ ℕ₀ ↦ ((𝐵↑𝑦) · 𝑥))
8	3, 7	zprmlogbaplem3 16178	. 2 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ∃𝑚 ∈ ℕ ∃𝑎 ∈ ℕ₀ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚)))
9		simpllr 540	. . . . . 6 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → 𝐵 ∈ ℙ)
10		simplrl 541	. . . . . 6 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → 𝑚 ∈ ℕ)
11		simprl 535	. . . . . 6 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → ¬ 𝐵 ∥ 𝑚)
12		simplrr 542	. . . . . 6 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → 𝑎 ∈ ℕ₀)
13		eqid 2238	. . . . . 6 ⊢ ((𝐵↑𝑎) · 𝑚) = ((𝐵↑𝑎) · 𝑚)
14	9, 10, 11, 12, 13	zprmlogbaplem2 16177	. . . . 5 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → ((𝐵 log_b ((𝐵↑𝑎) · 𝑚)) ∈ ℚ ∨ ((𝐵 log_b ((𝐵↑𝑎) · 𝑚)) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b ((𝐵↑𝑎) · 𝑚)) # 𝑞)))
15		simprr 537	. . . . . . . 8 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → 𝑋 = ((𝐵↑𝑎) · 𝑚))
16	15	oveq2d 6101	. . . . . . 7 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → (𝐵 log_b 𝑋) = (𝐵 log_b ((𝐵↑𝑎) · 𝑚)))
17	16	eleq1d 2307	. . . . . 6 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → ((𝐵 log_b 𝑋) ∈ ℚ ↔ (𝐵 log_b ((𝐵↑𝑎) · 𝑚)) ∈ ℚ))
18	16	eleq1d 2307	. . . . . . 7 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → ((𝐵 log_b 𝑋) ∈ ℝ ↔ (𝐵 log_b ((𝐵↑𝑎) · 𝑚)) ∈ ℝ))
19	16	breq1d 4140	. . . . . . . 8 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → ((𝐵 log_b 𝑋) # 𝑞 ↔ (𝐵 log_b ((𝐵↑𝑎) · 𝑚)) # 𝑞))
20	19	ralbidv 2550	. . . . . . 7 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → (∀𝑞 ∈ ℚ (𝐵 log_b 𝑋) # 𝑞 ↔ ∀𝑞 ∈ ℚ (𝐵 log_b ((𝐵↑𝑎) · 𝑚)) # 𝑞))
21	18, 20	anbi12d 477	. . . . . 6 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → (((𝐵 log_b 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b 𝑋) # 𝑞) ↔ ((𝐵 log_b ((𝐵↑𝑎) · 𝑚)) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b ((𝐵↑𝑎) · 𝑚)) # 𝑞)))
22	17, 21	orbi12d 805	. . . . 5 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → (((𝐵 log_b 𝑋) ∈ ℚ ∨ ((𝐵 log_b 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b 𝑋) # 𝑞)) ↔ ((𝐵 log_b ((𝐵↑𝑎) · 𝑚)) ∈ ℚ ∨ ((𝐵 log_b ((𝐵↑𝑎) · 𝑚)) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b ((𝐵↑𝑎) · 𝑚)) # 𝑞))))
23	14, 22	mpbird 167	. . . 4 ⊢ ((((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) ∧ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚))) → ((𝐵 log_b 𝑋) ∈ ℚ ∨ ((𝐵 log_b 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b 𝑋) # 𝑞)))
24	23	ex 115	. . 3 ⊢ (((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) ∧ (𝑚 ∈ ℕ ∧ 𝑎 ∈ ℕ₀)) → ((¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚)) → ((𝐵 log_b 𝑋) ∈ ℚ ∨ ((𝐵 log_b 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b 𝑋) # 𝑞))))
25	24	rexlimdvva 2676	. 2 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (∃𝑚 ∈ ℕ ∃𝑎 ∈ ℕ₀ (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚)) → ((𝐵 log_b 𝑋) ∈ ℚ ∨ ((𝐵 log_b 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b 𝑋) # 𝑞))))
26	8, 25	mpd 13	1 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ((𝐵 log_b 𝑋) ∈ ℚ ∨ ((𝐵 log_b 𝑋) ∈ ℝ ∧ ∀𝑞 ∈ ℚ (𝐵 log_b 𝑋) # 𝑞)))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 720 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 {crab 2532 class class class wbr 4130 (class class class)co 6085 ∈ cmpo 6087 ℝcr 8179 · cmul 8185 # cap 8912 ℕcn 9307 ℕ₀cn0 9568 ℚcq 10029 ↑cexp 10990 ∥ cdvds 12573 ℙcprime 12904 log_b clogb 16140

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 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300 ax-pre-suploc 8301 ax-addf 8302 ax-mulf 8303

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 7325 df-inf 7326 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-n0 9569 df-z 9650 df-uz 9932 df-q 10030 df-rp 10066 df-xneg 10185 df-xadd 10186 df-ioo 10305 df-ico 10307 df-icc 10308 df-fz 10423 df-fzo 10561 df-fl 10716 df-mod 10775 df-seqfrec 10900 df-exp 10991 df-fac 11180 df-bc 11202 df-ihash 11231 df-shft 11596 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-clim 12064 df-sumdc 12139 df-ef 12434 df-e 12435 df-dvds 12574 df-gcd 12750 df-prm 12905 df-rest 13648 df-topgen 13667 df-psmet 14964 df-xmet 14965 df-met 14966 df-bl 14967 df-mopn 14968 df-top 15190 df-topon 15203 df-bases 15235 df-ntr 15288 df-cn 15380 df-cnp 15381 df-tx 15445 df-cncf 15763 df-limced 15848 df-dvap 15849 df-relog 16051 df-rpcxp 16052 df-logb 16141

This theorem is used by: prmefexple 16269 bposlem1 16272

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