Theorem zprmlogbaplem3 16136

Description: Lemma for zprmlogbap 16137. Decomposing a natural number into a power of a prime base and a factor not divisible by that prime. (Contributed by Jim Kingdon, 20-Aug-2026.)

Hypotheses

Ref	Expression
zprmlogbaplem3.j	⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑧}
zprmlogbaplem3.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((𝐵↑𝑦) · 𝑥))

Assertion

Ref	Expression
zprmlogbaplem3	⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ∃𝑚 ∈ ℕ ∃𝑎 ∈ ℕ0 (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚)))

Distinct variable groups:   𝐵,𝑎,𝑚   𝑥,𝐵,𝑦,𝑧   𝐹,𝑎,𝑚   𝑥,𝐹,𝑦,𝑧   𝑥,𝐽,𝑦   𝑋,𝑎,𝑚   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝐽(𝑧, 𝑚, 𝑎)

Proof of Theorem zprmlogbaplem3

Step	Hyp	Ref	Expression
1		prmuz2 12926	. . . . . . . . 9 ⊢ (𝐵 ∈ ℙ → 𝐵 ∈ (ℤ≥‘2))
2		zprmlogbaplem3.j	. . . . . . . . . 10 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 𝐵 ∥ 𝑧}
3		zprmlogbaplem3.f	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((𝐵↑𝑦) · 𝑥))
4	2, 3	nnmaxpw 12969	. . . . . . . . 9 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ)
5	1, 4	syl 14	. . . . . . . 8 ⊢ (𝐵 ∈ ℙ → 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ)
6		simpl 109	. . . . . . . 8 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → 𝑋 ∈ ℕ)
7		f1ocnvdm 5987	. . . . . . . 8 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝑋 ∈ ℕ) → (◡𝐹‘𝑋) ∈ (𝐽 × ℕ0))
8	5, 6, 7	syl2an2 602	. . . . . . 7 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (◡𝐹‘𝑋) ∈ (𝐽 × ℕ0))
9		elxp6 6403	. . . . . . 7 ⊢ ((◡𝐹‘𝑋) ∈ (𝐽 × ℕ0) ↔ ((◡𝐹‘𝑋) = ⟨(1st ‘(◡𝐹‘𝑋)), (2nd ‘(◡𝐹‘𝑋))⟩ ∧ ((1st ‘(◡𝐹‘𝑋)) ∈ 𝐽 ∧ (2nd ‘(◡𝐹‘𝑋)) ∈ ℕ0)))
10	8, 9	sylib 122	. . . . . 6 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ((◡𝐹‘𝑋) = ⟨(1st ‘(◡𝐹‘𝑋)), (2nd ‘(◡𝐹‘𝑋))⟩ ∧ ((1st ‘(◡𝐹‘𝑋)) ∈ 𝐽 ∧ (2nd ‘(◡𝐹‘𝑋)) ∈ ℕ0)))
11	10	simprd 114	. . . . 5 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ((1st ‘(◡𝐹‘𝑋)) ∈ 𝐽 ∧ (2nd ‘(◡𝐹‘𝑋)) ∈ ℕ0))
12	11	simpld 112	. . . 4 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (1st ‘(◡𝐹‘𝑋)) ∈ 𝐽)
13		breq2 4134	. . . . . 6 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝑋)) → (𝐵 ∥ 𝑧 ↔ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋))))
14	13	notbid 677	. . . . 5 ⊢ (𝑧 = (1st ‘(◡𝐹‘𝑋)) → (¬ 𝐵 ∥ 𝑧 ↔ ¬ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋))))
15	14, 2	elrab2 2985	. . . 4 ⊢ ((1st ‘(◡𝐹‘𝑋)) ∈ 𝐽 ↔ ((1st ‘(◡𝐹‘𝑋)) ∈ ℕ ∧ ¬ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋))))
16	12, 15	sylib 122	. . 3 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ((1st ‘(◡𝐹‘𝑋)) ∈ ℕ ∧ ¬ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋))))
17	16	simpld 112	. 2 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (1st ‘(◡𝐹‘𝑋)) ∈ ℕ)
18	11	simprd 114	. 2 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (2nd ‘(◡𝐹‘𝑋)) ∈ ℕ0)
19	16	simprd 114	. 2 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ¬ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋)))
20	10	simpld 112	. . . . 5 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (◡𝐹‘𝑋) = ⟨(1st ‘(◡𝐹‘𝑋)), (2nd ‘(◡𝐹‘𝑋))⟩)
21	20	fveq2d 5699	. . . 4 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (𝐹‘(◡𝐹‘𝑋)) = (𝐹‘⟨(1st ‘(◡𝐹‘𝑋)), (2nd ‘(◡𝐹‘𝑋))⟩))
22		df-ov 6088	. . . 4 ⊢ ((1st ‘(◡𝐹‘𝑋))𝐹(2nd ‘(◡𝐹‘𝑋))) = (𝐹‘⟨(1st ‘(◡𝐹‘𝑋)), (2nd ‘(◡𝐹‘𝑋))⟩)
23	21, 22	eqtr4di 2289	. . 3 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (𝐹‘(◡𝐹‘𝑋)) = ((1st ‘(◡𝐹‘𝑋))𝐹(2nd ‘(◡𝐹‘𝑋))))
24		f1ocnvfv2 5984	. . . 4 ⊢ ((𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ ∧ 𝑋 ∈ ℕ) → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
25	5, 6, 24	syl2an2 602	. . 3 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (𝐹‘(◡𝐹‘𝑋)) = 𝑋)
26		prmnn 12904	. . . . . 6 ⊢ (𝐵 ∈ ℙ → 𝐵 ∈ ℕ)
27		nnexpcl 11002	. . . . . 6 ⊢ ((𝐵 ∈ ℕ ∧ (2nd ‘(◡𝐹‘𝑋)) ∈ ℕ0) → (𝐵↑(2nd ‘(◡𝐹‘𝑋))) ∈ ℕ)
28	26, 18, 27	syl2an2 602	. . . . 5 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → (𝐵↑(2nd ‘(◡𝐹‘𝑋))) ∈ ℕ)
29	28, 17	nnmulcld 9355	. . . 4 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋))) ∈ ℕ)
30		oveq2 6093	. . . . 5 ⊢ (𝑥 = (1st ‘(◡𝐹‘𝑋)) → ((𝐵↑𝑦) · 𝑥) = ((𝐵↑𝑦) · (1st ‘(◡𝐹‘𝑋))))
31		oveq2 6093	. . . . . 6 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝑋)) → (𝐵↑𝑦) = (𝐵↑(2nd ‘(◡𝐹‘𝑋))))
32	31	oveq1d 6100	. . . . 5 ⊢ (𝑦 = (2nd ‘(◡𝐹‘𝑋)) → ((𝐵↑𝑦) · (1st ‘(◡𝐹‘𝑋))) = ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋))))
33	30, 32, 3	ovmpog 6223	. . . 4 ⊢ (((1st ‘(◡𝐹‘𝑋)) ∈ 𝐽 ∧ (2nd ‘(◡𝐹‘𝑋)) ∈ ℕ0 ∧ ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋))) ∈ ℕ) → ((1st ‘(◡𝐹‘𝑋))𝐹(2nd ‘(◡𝐹‘𝑋))) = ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋))))
34	12, 18, 29, 33	syl3anc 1278	. . 3 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ((1st ‘(◡𝐹‘𝑋))𝐹(2nd ‘(◡𝐹‘𝑋))) = ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋))))
35	23, 25, 34	3eqtr3d 2279	. 2 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → 𝑋 = ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋))))
36		breq2 4134	. . . . 5 ⊢ (𝑚 = (1st ‘(◡𝐹‘𝑋)) → (𝐵 ∥ 𝑚 ↔ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋))))
37	36	notbid 677	. . . 4 ⊢ (𝑚 = (1st ‘(◡𝐹‘𝑋)) → (¬ 𝐵 ∥ 𝑚 ↔ ¬ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋))))
38		oveq2 6093	. . . . 5 ⊢ (𝑚 = (1st ‘(◡𝐹‘𝑋)) → ((𝐵↑𝑎) · 𝑚) = ((𝐵↑𝑎) · (1st ‘(◡𝐹‘𝑋))))
39	38	eqeq2d 2250	. . . 4 ⊢ (𝑚 = (1st ‘(◡𝐹‘𝑋)) → (𝑋 = ((𝐵↑𝑎) · 𝑚) ↔ 𝑋 = ((𝐵↑𝑎) · (1st ‘(◡𝐹‘𝑋)))))
40	37, 39	anbi12d 477	. . 3 ⊢ (𝑚 = (1st ‘(◡𝐹‘𝑋)) → ((¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚)) ↔ (¬ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋)) ∧ 𝑋 = ((𝐵↑𝑎) · (1st ‘(◡𝐹‘𝑋))))))
41		oveq2 6093	. . . . . 6 ⊢ (𝑎 = (2nd ‘(◡𝐹‘𝑋)) → (𝐵↑𝑎) = (𝐵↑(2nd ‘(◡𝐹‘𝑋))))
42	41	oveq1d 6100	. . . . 5 ⊢ (𝑎 = (2nd ‘(◡𝐹‘𝑋)) → ((𝐵↑𝑎) · (1st ‘(◡𝐹‘𝑋))) = ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋))))
43	42	eqeq2d 2250	. . . 4 ⊢ (𝑎 = (2nd ‘(◡𝐹‘𝑋)) → (𝑋 = ((𝐵↑𝑎) · (1st ‘(◡𝐹‘𝑋))) ↔ 𝑋 = ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋)))))
44	43	anbi2d 468	. . 3 ⊢ (𝑎 = (2nd ‘(◡𝐹‘𝑋)) → ((¬ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋)) ∧ 𝑋 = ((𝐵↑𝑎) · (1st ‘(◡𝐹‘𝑋)))) ↔ (¬ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋)) ∧ 𝑋 = ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋))))))
45	40, 44	rspc2ev 2945	. 2 ⊢ (((1st ‘(◡𝐹‘𝑋)) ∈ ℕ ∧ (2nd ‘(◡𝐹‘𝑋)) ∈ ℕ0 ∧ (¬ 𝐵 ∥ (1st ‘(◡𝐹‘𝑋)) ∧ 𝑋 = ((𝐵↑(2nd ‘(◡𝐹‘𝑋))) · (1st ‘(◡𝐹‘𝑋))))) → ∃𝑚 ∈ ℕ ∃𝑎 ∈ ℕ0 (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚)))
46	17, 18, 19, 35, 45	syl112anc 1282	1 ⊢ ((𝑋 ∈ ℕ ∧ 𝐵 ∈ ℙ) → ∃𝑚 ∈ ℕ ∃𝑎 ∈ ℕ0 (¬ 𝐵 ∥ 𝑚 ∧ 𝑋 = ((𝐵↑𝑎) · 𝑚)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  {crab 2532  ⟨cop 3712   class class class wbr 4130   × cxp 4772  ^◡ccnv 4773  –1-1-onto→wf1o 5376  ‘cfv 5377  (class class class)co 6085   ∈ cmpo 6087  1^st c1st 6372  2^nd c2nd 6373   · cmul 8184  ℕcn 9306  2c2 9357  ℕ₀cn0 9567  ℤ_≥cuz 9930  ↑cexp 10988   ∥ cdvds 12570  ℙcprime 12901

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

This proof depends on definitions:  df-bi 117  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-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-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  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-fz 10422  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-dvds 12571  df-prm 12902

This theorem is used by:  zprmlogbap  16137