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Theorem lgsdilem2 13731

Description: Lemma for lgsdi 13732. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Hypotheses**
Ref	Expression
lgsdilem2.1	⊢ (𝜑 → 𝐴 ∈ ℤ)
lgsdilem2.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
lgsdilem2.3	⊢ (𝜑 → 𝑁 ∈ ℤ)
lgsdilem2.4	⊢ (𝜑 → 𝑀 ≠ 0)
lgsdilem2.5	⊢ (𝜑 → 𝑁 ≠ 0)
lgsdilem2.6	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))

**Assertion**
Ref	Expression
lgsdilem2	⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) = (seq1( · , 𝐹)‘(abs‘(𝑀 · 𝑁))))

Distinct variable groups: 𝑛,𝑀 𝐴,𝑛 𝑛,𝑁 Allowed substitution hints: 𝜑(𝑛) 𝐹(𝑛)

Proof of Theorem lgsdilem2 Dummy variables 𝑘 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulid1 7917	. . 3 ⊢ (𝑘 ∈ ℂ → (𝑘 · 1) = 𝑘)
2	1	adantl 275	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℂ) → (𝑘 · 1) = 𝑘)
3		lgsdilem2.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		lgsdilem2.4	. . . 4 ⊢ (𝜑 → 𝑀 ≠ 0)
5		nnabscl 11064	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
6	3, 4, 5	syl2anc 409	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℕ)
7		nnuz 9522	. . 3 ⊢ ℕ = (ℤ_≥‘1)
8	6, 7	eleqtrdi 2263	. 2 ⊢ (𝜑 → (abs‘𝑀) ∈ (ℤ_≥‘1))
9	6	nnzd 9333	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℤ)
10		lgsdilem2.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
11	3, 10	zmulcld 9340	. . . . 5 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℤ)
12	3	zcnd 9335	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℂ)
13	10	zcnd 9335	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
14		0z 9223	. . . . . . . . 9 ⊢ 0 ∈ ℤ
15		zapne 9286	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 # 0 ↔ 𝑀 ≠ 0))
16	3, 14, 15	sylancl 411	. . . . . . . 8 ⊢ (𝜑 → (𝑀 # 0 ↔ 𝑀 ≠ 0))
17	4, 16	mpbird 166	. . . . . . 7 ⊢ (𝜑 → 𝑀 # 0)
18		lgsdilem2.5	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ≠ 0)
19		zapne 9286	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 # 0 ↔ 𝑁 ≠ 0))
20	10, 14, 19	sylancl 411	. . . . . . . 8 ⊢ (𝜑 → (𝑁 # 0 ↔ 𝑁 ≠ 0))
21	18, 20	mpbird 166	. . . . . . 7 ⊢ (𝜑 → 𝑁 # 0)
22	12, 13, 17, 21	mulap0d 8576	. . . . . 6 ⊢ (𝜑 → (𝑀 · 𝑁) # 0)
23		zapne 9286	. . . . . . 7 ⊢ (((𝑀 · 𝑁) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝑀 · 𝑁) # 0 ↔ (𝑀 · 𝑁) ≠ 0))
24	11, 14, 23	sylancl 411	. . . . . 6 ⊢ (𝜑 → ((𝑀 · 𝑁) # 0 ↔ (𝑀 · 𝑁) ≠ 0))
25	22, 24	mpbid 146	. . . . 5 ⊢ (𝜑 → (𝑀 · 𝑁) ≠ 0)
26		nnabscl 11064	. . . . 5 ⊢ (((𝑀 · 𝑁) ∈ ℤ ∧ (𝑀 · 𝑁) ≠ 0) → (abs‘(𝑀 · 𝑁)) ∈ ℕ)
27	11, 25, 26	syl2anc 409	. . . 4 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ ℕ)
28	27	nnzd 9333	. . 3 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ ℤ)
29	12	abscld 11145	. . . . 5 ⊢ (𝜑 → (abs‘𝑀) ∈ ℝ)
30	13	abscld 11145	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) ∈ ℝ)
31	12	absge0d 11148	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘𝑀))
32		nnabscl 11064	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
33	10, 18, 32	syl2anc 409	. . . . . 6 ⊢ (𝜑 → (abs‘𝑁) ∈ ℕ)
34	33	nnge1d 8921	. . . . 5 ⊢ (𝜑 → 1 ≤ (abs‘𝑁))
35	29, 30, 31, 34	lemulge11d 8853	. . . 4 ⊢ (𝜑 → (abs‘𝑀) ≤ ((abs‘𝑀) · (abs‘𝑁)))
36	12, 13	absmuld 11158	. . . 4 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
37	35, 36	breqtrrd 4017	. . 3 ⊢ (𝜑 → (abs‘𝑀) ≤ (abs‘(𝑀 · 𝑁)))
38		eluz2 9493	. . 3 ⊢ ((abs‘(𝑀 · 𝑁)) ∈ (ℤ_≥‘(abs‘𝑀)) ↔ ((abs‘𝑀) ∈ ℤ ∧ (abs‘(𝑀 · 𝑁)) ∈ ℤ ∧ (abs‘𝑀) ≤ (abs‘(𝑀 · 𝑁))))
39	9, 28, 37, 38	syl3anbrc 1176	. 2 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ (ℤ_≥‘(abs‘𝑀)))
40		1zzd 9239	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
41		lgsdilem2.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℤ)
42		lgsdilem2.6	. . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))
43	42	lgsfcl3 13716	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝐹:ℕ⟶ℤ)
44	41, 3, 4, 43	syl3anc 1233	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℤ)
45	44	ffvelrnda 5631	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℤ)
46		zmulcl 9265	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑣 ∈ ℤ) → (𝑘 · 𝑣) ∈ ℤ)
47	46	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝑘 · 𝑣) ∈ ℤ)
48	7, 40, 45, 47	seqf 10417	. . . 4 ⊢ (𝜑 → seq1( · , 𝐹):ℕ⟶ℤ)
49	48, 6	ffvelrnd 5632	. . 3 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℤ)
50	49	zcnd 9335	. 2 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℂ)
51		eleq1w 2231	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ))
52		oveq2 5861	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐴 /_L 𝑛) = (𝐴 /_L 𝑘))
53		oveq1 5860	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 pCnt 𝑀) = (𝑘 pCnt 𝑀))
54	52, 53	oveq12d 5871	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)))
55	51, 54	ifbieq1d 3548	. . . 4 ⊢ (𝑛 = 𝑘 → if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1))
56	6	peano2nnd 8893	. . . . 5 ⊢ (𝜑 → ((abs‘𝑀) + 1) ∈ ℕ)
57		elfzuz 9977	. . . . 5 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ (ℤ_≥‘((abs‘𝑀) + 1)))
58		eluznn 9559	. . . . 5 ⊢ ((((abs‘𝑀) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘((abs‘𝑀) + 1))) → 𝑘 ∈ ℕ)
59	56, 57, 58	syl2an 287	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → 𝑘 ∈ ℕ)
60	41	ad2antrr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝐴 ∈ ℤ)
61		prmz 12065	. . . . . . . 8 ⊢ (𝑘 ∈ ℙ → 𝑘 ∈ ℤ)
62	61	adantl 275	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑘 ∈ ℤ)
63		lgscl 13709	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝐴 /_L 𝑘) ∈ ℤ)
64	60, 62, 63	syl2anc 409	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝐴 /_L 𝑘) ∈ ℤ)
65		simpr 109	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑘 ∈ ℙ)
66	3	ad2antrr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑀 ∈ ℤ)
67	4	ad2antrr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑀 ≠ 0)
68		pczcl 12252	. . . . . . 7 ⊢ ((𝑘 ∈ ℙ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
69	65, 66, 67, 68	syl12anc 1231	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
70		zexpcl 10491	. . . . . 6 ⊢ (((𝐴 /_L 𝑘) ∈ ℤ ∧ (𝑘 pCnt 𝑀) ∈ ℕ₀) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) ∈ ℤ)
71	64, 69, 70	syl2anc 409	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) ∈ ℤ)
72		1zzd 9239	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ ¬ 𝑘 ∈ ℙ) → 1 ∈ ℤ)
73		prmdc 12084	. . . . . 6 ⊢ (𝑘 ∈ ℕ → DECID 𝑘 ∈ ℙ)
74	59, 73	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → DECID 𝑘 ∈ ℙ)
75	71, 72, 74	ifcldadc 3555	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) ∈ ℤ)
76	42, 55, 59, 75	fvmptd3 5589	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → (𝐹‘𝑘) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1))
77		zq 9585	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℚ)
78	66, 77	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑀 ∈ ℚ)
79		pcabs 12279	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℙ ∧ 𝑀 ∈ ℚ) → (𝑘 pCnt (abs‘𝑀)) = (𝑘 pCnt 𝑀))
80	65, 78, 79	syl2anc 409	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt (abs‘𝑀)) = (𝑘 pCnt 𝑀))
81		elfzle1 9983	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → ((abs‘𝑀) + 1) ≤ 𝑘)
82	81	adantl 275	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → ((abs‘𝑀) + 1) ≤ 𝑘)
83		elfzelz 9981	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ ℤ)
84		zltp1le 9266	. . . . . . . . . . . . . 14 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((abs‘𝑀) < 𝑘 ↔ ((abs‘𝑀) + 1) ≤ 𝑘))
85	9, 83, 84	syl2an 287	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → ((abs‘𝑀) < 𝑘 ↔ ((abs‘𝑀) + 1) ≤ 𝑘))
86	82, 85	mpbird 166	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → (abs‘𝑀) < 𝑘)
87		zltnle 9258	. . . . . . . . . . . . 13 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((abs‘𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ (abs‘𝑀)))
88	9, 83, 87	syl2an 287	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → ((abs‘𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ (abs‘𝑀)))
89	86, 88	mpbid 146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → ¬ 𝑘 ≤ (abs‘𝑀))
90	89	adantr 274	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ¬ 𝑘 ≤ (abs‘𝑀))
91	66, 67, 5	syl2anc 409	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (abs‘𝑀) ∈ ℕ)
92		dvdsle 11804	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ (abs‘𝑀) ∈ ℕ) → (𝑘 ∥ (abs‘𝑀) → 𝑘 ≤ (abs‘𝑀)))
93	62, 91, 92	syl2anc 409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 ∥ (abs‘𝑀) → 𝑘 ≤ (abs‘𝑀)))
94	90, 93	mtod 658	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ¬ 𝑘 ∥ (abs‘𝑀))
95		pceq0 12275	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℙ ∧ (abs‘𝑀) ∈ ℕ) → ((𝑘 pCnt (abs‘𝑀)) = 0 ↔ ¬ 𝑘 ∥ (abs‘𝑀)))
96	65, 91, 95	syl2anc 409	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝑘 pCnt (abs‘𝑀)) = 0 ↔ ¬ 𝑘 ∥ (abs‘𝑀)))
97	94, 96	mpbird 166	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt (abs‘𝑀)) = 0)
98	80, 97	eqtr3d 2205	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) = 0)
99	98	oveq2d 5869	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑0))
100	64	zcnd 9335	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝐴 /_L 𝑘) ∈ ℂ)
101	100	exp0d 10603	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑0) = 1)
102	99, 101	eqtrd 2203	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) = 1)
103	102, 74	ifeq1dadc 3556	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) = if(𝑘 ∈ ℙ, 1, 1))
104		ifiddc 3559	. . . . 5 ⊢ (DECID 𝑘 ∈ ℙ → if(𝑘 ∈ ℙ, 1, 1) = 1)
105	74, 104	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → if(𝑘 ∈ ℙ, 1, 1) = 1)
106	103, 105	eqtrd 2203	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) = 1)
107	76, 106	eqtrd 2203	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → (𝐹‘𝑘) = 1)
108	44	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝐹:ℕ⟶ℤ)
109		elnnuz 9523	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
110	109	biimpri 132	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘1) → 𝑘 ∈ ℕ)
111	110	adantl 275	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝑘 ∈ ℕ)
112	108, 111	ffvelrnd 5632	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℤ)
113	112	zcnd 9335	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℂ)
114		mulcl 7901	. . 3 ⊢ ((𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑘 · 𝑣) ∈ ℂ)
115	114	adantl 275	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
116	2, 8, 39, 50, 107, 113, 115	seq3id2 10465	1 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) = (seq1( · , 𝐹)‘(abs‘(𝑀 · 𝑁))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 829 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ifcif 3526 class class class wbr 3989 ↦ cmpt 4050 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 0cc0 7774 1c1 7775 + caddc 7777 · cmul 7779 < clt 7954 ≤ cle 7955 # cap 8500 ℕcn 8878 ℕ₀cn0 9135 ℤcz 9212 ℤ_≥cuz 9487 ℚcq 9578 ...cfz 9965 seqcseq 10401 ↑cexp 10475 abscabs 10961 ∥ cdvds 11749 ℙcprime 12061 pCnt cpc 12238 /_L clgs 13692

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-2o 6396 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-proddc 11514 df-dvds 11750 df-gcd 11898 df-prm 12062 df-phi 12165 df-pc 12239 df-lgs 13693

This theorem is referenced by: lgsdi 13732

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