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Theorem lgsdilem2 15361

Description: Lemma for lgsdi 15362. (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		mulrid 8040	. . 3 ⊢ (𝑘 ∈ ℂ → (𝑘 · 1) = 𝑘)
2	1	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℂ) → (𝑘 · 1) = 𝑘)
3		lgsdilem2.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		lgsdilem2.4	. . . 4 ⊢ (𝜑 → 𝑀 ≠ 0)
5		nnabscl 11282	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
6	3, 4, 5	syl2anc 411	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℕ)
7		nnuz 9654	. . 3 ⊢ ℕ = (ℤ_≥‘1)
8	6, 7	eleqtrdi 2289	. 2 ⊢ (𝜑 → (abs‘𝑀) ∈ (ℤ_≥‘1))
9	6	nnzd 9464	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℤ)
10		lgsdilem2.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
11	3, 10	zmulcld 9471	. . . . 5 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℤ)
12	3	zcnd 9466	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℂ)
13	10	zcnd 9466	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
14		0z 9354	. . . . . . . . 9 ⊢ 0 ∈ ℤ
15		zapne 9417	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 # 0 ↔ 𝑀 ≠ 0))
16	3, 14, 15	sylancl 413	. . . . . . . 8 ⊢ (𝜑 → (𝑀 # 0 ↔ 𝑀 ≠ 0))
17	4, 16	mpbird 167	. . . . . . 7 ⊢ (𝜑 → 𝑀 # 0)
18		lgsdilem2.5	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ≠ 0)
19		zapne 9417	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 # 0 ↔ 𝑁 ≠ 0))
20	10, 14, 19	sylancl 413	. . . . . . . 8 ⊢ (𝜑 → (𝑁 # 0 ↔ 𝑁 ≠ 0))
21	18, 20	mpbird 167	. . . . . . 7 ⊢ (𝜑 → 𝑁 # 0)
22	12, 13, 17, 21	mulap0d 8702	. . . . . 6 ⊢ (𝜑 → (𝑀 · 𝑁) # 0)
23		zapne 9417	. . . . . . 7 ⊢ (((𝑀 · 𝑁) ∈ ℤ ∧ 0 ∈ ℤ) → ((𝑀 · 𝑁) # 0 ↔ (𝑀 · 𝑁) ≠ 0))
24	11, 14, 23	sylancl 413	. . . . . 6 ⊢ (𝜑 → ((𝑀 · 𝑁) # 0 ↔ (𝑀 · 𝑁) ≠ 0))
25	22, 24	mpbid 147	. . . . 5 ⊢ (𝜑 → (𝑀 · 𝑁) ≠ 0)
26		nnabscl 11282	. . . . 5 ⊢ (((𝑀 · 𝑁) ∈ ℤ ∧ (𝑀 · 𝑁) ≠ 0) → (abs‘(𝑀 · 𝑁)) ∈ ℕ)
27	11, 25, 26	syl2anc 411	. . . 4 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ ℕ)
28	27	nnzd 9464	. . 3 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ ℤ)
29	12	abscld 11363	. . . . 5 ⊢ (𝜑 → (abs‘𝑀) ∈ ℝ)
30	13	abscld 11363	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) ∈ ℝ)
31	12	absge0d 11366	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘𝑀))
32		nnabscl 11282	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
33	10, 18, 32	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (abs‘𝑁) ∈ ℕ)
34	33	nnge1d 9050	. . . . 5 ⊢ (𝜑 → 1 ≤ (abs‘𝑁))
35	29, 30, 31, 34	lemulge11d 8981	. . . 4 ⊢ (𝜑 → (abs‘𝑀) ≤ ((abs‘𝑀) · (abs‘𝑁)))
36	12, 13	absmuld 11376	. . . 4 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
37	35, 36	breqtrrd 4062	. . 3 ⊢ (𝜑 → (abs‘𝑀) ≤ (abs‘(𝑀 · 𝑁)))
38		eluz2 9624	. . 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 1183	. 2 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ (ℤ_≥‘(abs‘𝑀)))
40		1zzd 9370	. . . . 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 15346	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝐹:ℕ⟶ℤ)
44	41, 3, 4, 43	syl3anc 1249	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℤ)
45	44	ffvelcdmda 5700	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℤ)
46		zmulcl 9396	. . . . . 6 ⊢ ((𝑘 ∈ ℤ ∧ 𝑣 ∈ ℤ) → (𝑘 · 𝑣) ∈ ℤ)
47	46	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (𝑘 · 𝑣) ∈ ℤ)
48	7, 40, 45, 47	seqf 10573	. . . 4 ⊢ (𝜑 → seq1( · , 𝐹):ℕ⟶ℤ)
49	48, 6	ffvelcdmd 5701	. . 3 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℤ)
50	49	zcnd 9466	. 2 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℂ)
51		eleq1w 2257	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ))
52		oveq2 5933	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐴 /_L 𝑛) = (𝐴 /_L 𝑘))
53		oveq1 5932	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 pCnt 𝑀) = (𝑘 pCnt 𝑀))
54	52, 53	oveq12d 5943	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)))
55	51, 54	ifbieq1d 3584	. . . 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 9022	. . . . 5 ⊢ (𝜑 → ((abs‘𝑀) + 1) ∈ ℕ)
57		elfzuz 10113	. . . . 5 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ (ℤ_≥‘((abs‘𝑀) + 1)))
58		eluznn 9691	. . . . 5 ⊢ ((((abs‘𝑀) + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘((abs‘𝑀) + 1))) → 𝑘 ∈ ℕ)
59	56, 57, 58	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → 𝑘 ∈ ℕ)
60	41	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝐴 ∈ ℤ)
61		prmz 12304	. . . . . . . 8 ⊢ (𝑘 ∈ ℙ → 𝑘 ∈ ℤ)
62	61	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑘 ∈ ℤ)
63		lgscl 15339	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝐴 /_L 𝑘) ∈ ℤ)
64	60, 62, 63	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝐴 /_L 𝑘) ∈ ℤ)
65		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑘 ∈ ℙ)
66	3	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑀 ∈ ℤ)
67	4	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑀 ≠ 0)
68		pczcl 12492	. . . . . . 7 ⊢ ((𝑘 ∈ ℙ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
69	65, 66, 67, 68	syl12anc 1247	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
70		zexpcl 10663	. . . . . 6 ⊢ (((𝐴 /_L 𝑘) ∈ ℤ ∧ (𝑘 pCnt 𝑀) ∈ ℕ₀) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) ∈ ℤ)
71	64, 69, 70	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) ∈ ℤ)
72		1zzd 9370	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ ¬ 𝑘 ∈ ℙ) → 1 ∈ ℤ)
73		prmdc 12323	. . . . . 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 3591	. . . 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 5658	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → (𝐹‘𝑘) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1))
77		zq 9717	. . . . . . . . . 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 12520	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℙ ∧ 𝑀 ∈ ℚ) → (𝑘 pCnt (abs‘𝑀)) = (𝑘 pCnt 𝑀))
80	65, 78, 79	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt (abs‘𝑀)) = (𝑘 pCnt 𝑀))
81		elfzle1 10119	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → ((abs‘𝑀) + 1) ≤ 𝑘)
82	81	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → ((abs‘𝑀) + 1) ≤ 𝑘)
83		elfzelz 10117	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ ℤ)
84		zltp1le 9397	. . . . . . . . . . . . . 14 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((abs‘𝑀) < 𝑘 ↔ ((abs‘𝑀) + 1) ≤ 𝑘))
85	9, 83, 84	syl2an 289	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → ((abs‘𝑀) < 𝑘 ↔ ((abs‘𝑀) + 1) ≤ 𝑘))
86	82, 85	mpbird 167	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → (abs‘𝑀) < 𝑘)
87		zltnle 9389	. . . . . . . . . . . . 13 ⊢ (((abs‘𝑀) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((abs‘𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ (abs‘𝑀)))
88	9, 83, 87	syl2an 289	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → ((abs‘𝑀) < 𝑘 ↔ ¬ 𝑘 ≤ (abs‘𝑀)))
89	86, 88	mpbid 147	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → ¬ 𝑘 ≤ (abs‘𝑀))
90	89	adantr 276	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ¬ 𝑘 ≤ (abs‘𝑀))
91	66, 67, 5	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (abs‘𝑀) ∈ ℕ)
92		dvdsle 12026	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ (abs‘𝑀) ∈ ℕ) → (𝑘 ∥ (abs‘𝑀) → 𝑘 ≤ (abs‘𝑀)))
93	62, 91, 92	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 ∥ (abs‘𝑀) → 𝑘 ≤ (abs‘𝑀)))
94	90, 93	mtod 664	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ¬ 𝑘 ∥ (abs‘𝑀))
95		pceq0 12516	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℙ ∧ (abs‘𝑀) ∈ ℕ) → ((𝑘 pCnt (abs‘𝑀)) = 0 ↔ ¬ 𝑘 ∥ (abs‘𝑀)))
96	65, 91, 95	syl2anc 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝑘 pCnt (abs‘𝑀)) = 0 ↔ ¬ 𝑘 ∥ (abs‘𝑀)))
97	94, 96	mpbird 167	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt (abs‘𝑀)) = 0)
98	80, 97	eqtr3d 2231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) = 0)
99	98	oveq2d 5941	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑0))
100	64	zcnd 9466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝐴 /_L 𝑘) ∈ ℂ)
101	100	exp0d 10776	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑0) = 1)
102	99, 101	eqtrd 2229	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) = 1)
103	102, 74	ifeq1dadc 3592	. . . 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 3596	. . . . 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 2229	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) = 1)
107	76, 106	eqtrd 2229	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → (𝐹‘𝑘) = 1)
108	44	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝐹:ℕ⟶ℤ)
109		elnnuz 9655	. . . . . 6 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
110	109	biimpri 133	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘1) → 𝑘 ∈ ℕ)
111	110	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝑘 ∈ ℕ)
112	108, 111	ffvelcdmd 5701	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℤ)
113	112	zcnd 9466	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℂ)
114		mulcl 8023	. . 3 ⊢ ((𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑘 · 𝑣) ∈ ℂ)
115	114	adantl 277	. 2 ⊢ ((𝜑 ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
116	2, 8, 39, 50, 107, 113, 115	seq3id2 10635	1 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) = (seq1( · , 𝐹)‘(abs‘(𝑀 · 𝑁))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 = wceq 1364 ∈ wcel 2167 ≠ wne 2367 ifcif 3562 class class class wbr 4034 ↦ cmpt 4095 ⟶wf 5255 ‘cfv 5259 (class class class)co 5925 ℂcc 7894 0cc0 7896 1c1 7897 + caddc 7899 · cmul 7901 < clt 8078 ≤ cle 8079 # cap 8625 ℕcn 9007 ℕ₀cn0 9266 ℤcz 9343 ℤ_≥cuz 9618 ℚcq 9710 ...cfz 10100 seqcseq 10556 ↑cexp 10647 abscabs 11179 ∥ cdvds 11969 ℙcprime 12300 pCnt cpc 12478 /_L clgs 15322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-2o 6484 df-oadd 6487 df-er 6601 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-5 9069 df-6 9070 df-7 9071 df-8 9072 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-fz 10101 df-fzo 10235 df-fl 10377 df-mod 10432 df-seqfrec 10557 df-exp 10648 df-ihash 10885 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-clim 11461 df-proddc 11733 df-dvds 11970 df-gcd 12146 df-prm 12301 df-phi 12404 df-pc 12479 df-lgs 15323

This theorem is referenced by: lgsdi 15362

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