Theorem nnmaxpwlemparts 12977

Description: Lemma for nnmaxpw 12978. Decomposing a number into parts. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 19-Aug-2026.)

Assertion

Ref	Expression
nnmaxpwlemparts	⊢ (𝐵 ∈ (ℤ≥‘2) → ((((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋)) ↔ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))))

Distinct variable groups:   𝑧,𝐴   𝑧,𝑌   𝑧,𝐵

Allowed substitution hint:   𝑋(𝑧)

Proof of Theorem nnmaxpwlemparts

Step	Hyp	Ref	Expression
1		simprr 537	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → 𝐴 = ((𝐵↑𝑌) · 𝑋))
2		eluz2nn 9982	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℕ)
3	2	adantr 276	. . . . . 6 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → 𝐵 ∈ ℕ)
4		simprlr 544	. . . . . 6 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → 𝑌 ∈ ℕ0)
5	3, 4	nnexpcld 11154	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → (𝐵↑𝑌) ∈ ℕ)
6		simplll 539	. . . . . 6 ⊢ ((((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋)) → 𝑋 ∈ ℕ)
7	6	adantl 277	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → 𝑋 ∈ ℕ)
8	5, 7	nnmulcld 9362	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → ((𝐵↑𝑌) · 𝑋) ∈ ℕ)
9	1, 8	eqeltrd 2315	. . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → 𝐴 ∈ ℕ)
10		simpl 109	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → 𝐵 ∈ (ℤ≥‘2))
11		simpllr 540	. . . . 5 ⊢ ((((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋)) → ¬ 𝐵 ∥ 𝑋)
12	11	adantl 277	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → ¬ 𝐵 ∥ 𝑋)
13	7, 10, 12, 4, 1	nnmaxpwlemxy 12973	. . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))
14	9, 13	jca 306	. 2 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋))) → (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))))
15		simprl 535	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝐴 ∈ ℕ)
16		simpl 109	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝐵 ∈ (ℤ≥‘2))
17		simprrl 545	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))))
18		simpr 110	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) ∧ 𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))))
19		nnmaxpwlemdvds 12974	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) ∥ 𝐴)
20		simpl 109	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐴 ∈ ℕ)
21	2	adantl 277	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → 𝐵 ∈ ℕ)
22		pwbdvdseu 12972	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → ∃!𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))
23		riotacl 6057	. . . . . . . . . . 11 ⊢ (∃!𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴) → (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)) ∈ ℕ0)
24	22, 23	syl 14	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)) ∈ ℕ0)
25	21, 24	nnexpcld 11154	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) ∈ ℕ)
26		nndivdvds 12588	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) ∈ ℕ) → ((𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) ∥ 𝐴 ↔ (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∈ ℕ))
27	20, 25, 26	syl2anc 415	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → ((𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) ∥ 𝐴 ↔ (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∈ ℕ))
28	19, 27	mpbid 147	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∈ ℕ)
29	28	adantr 276	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) ∧ 𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∈ ℕ)
30	18, 29	eqeltrd 2315	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) ∧ 𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝑋 ∈ ℕ)
31	15, 16, 17, 30	syl21anc 1277	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝑋 ∈ ℕ)
32		nnmaxpwlemnfac 12976	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘2)) → ¬ 𝐵 ∥ (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))))
33	15, 16, 32	syl2anc 415	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → ¬ 𝐵 ∥ (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))))
34		breq2 4135	. . . . . . 7 ⊢ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) → (𝐵 ∥ 𝑋 ↔ 𝐵 ∥ (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))))
35	34	notbid 677	. . . . . 6 ⊢ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) → (¬ 𝐵 ∥ 𝑋 ↔ ¬ 𝐵 ∥ (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))))
36	17, 35	syl 14	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → (¬ 𝐵 ∥ 𝑋 ↔ ¬ 𝐵 ∥ (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))))
37	33, 36	mpbird 167	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → ¬ 𝐵 ∥ 𝑋)
38	31, 37	jca 306	. . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → (𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋))
39		simprrr 546	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))
40	15, 16, 24	syl2anc 415	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)) ∈ ℕ0)
41	39, 40	eqeltrd 2315	. . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝑌 ∈ ℕ0)
42	39	oveq2d 6104	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → (𝐵↑𝑌) = (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))
43	42, 17	oveq12d 6106	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → ((𝐵↑𝑌) · 𝑋) = ((𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) · (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))))
44	15	nncnd 9327	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝐴 ∈ ℂ)
45	15, 16, 25	syl2anc 415	. . . . . 6 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) ∈ ℕ)
46	45	nncnd 9327	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) ∈ ℂ)
47	45	nnap0d 9359	. . . . 5 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) # 0)
48	44, 46, 47	divcanap2d 9130	. . . 4 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → ((𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))) · (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) = 𝐴)
49	43, 48	eqtr2d 2272	. . 3 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → 𝐴 = ((𝐵↑𝑌) · 𝑋))
50	38, 41, 49	jca31 309	. 2 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))) → (((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋)))
51	14, 50	impbida 604	1 ⊢ (𝐵 ∈ (ℤ≥‘2) → ((((𝑋 ∈ ℕ ∧ ¬ 𝐵 ∥ 𝑋) ∧ 𝑌 ∈ ℕ0) ∧ 𝐴 = ((𝐵↑𝑌) · 𝑋)) ↔ (𝐴 ∈ ℕ ∧ (𝑋 = (𝐴 / (𝐵↑(℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴)))) ∧ 𝑌 = (℩𝑧 ∈ ℕ0 ((𝐵↑𝑧) ∥ 𝐴 ∧ ¬ (𝐵↑(𝑧 + 1)) ∥ 𝐴))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∃!wreu 2530   class class class wbr 4131  ‘cfv 5380  ℩crio 6040  (class class class)co 6088  1c1 8186   + caddc 8188   · cmul 8190   / cdiv 9010  ℕcn 9313  2c2 9364  ℕ₀cn0 9574  ℤ_≥cuz 9937  ↑cexp 10996   ∥ cdvds 12579

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

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 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-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  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-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-dvds 12580

This theorem is used by:  nnmaxpw  12978