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Theorem lgsdilem2 15838

Description: Lemma for lgsdi 15839. (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 8219	. . 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 11723	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
6	3, 4, 5	syl2anc 411	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℕ)
7		nnuz 9836	. . 3 ⊢ ℕ = (ℤ_≥‘1)
8	6, 7	eleqtrdi 2324	. 2 ⊢ (𝜑 → (abs‘𝑀) ∈ (ℤ_≥‘1))
9	6	nnzd 9645	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℤ)
10		lgsdilem2.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
11	3, 10	zmulcld 9652	. . . . 5 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℤ)
12	3	zcnd 9647	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℂ)
13	10	zcnd 9647	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
14		0z 9534	. . . . . . . . 9 ⊢ 0 ∈ ℤ
15		zapne 9598	. . . . . . . . 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 9598	. . . . . . . . 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 8880	. . . . . 6 ⊢ (𝜑 → (𝑀 · 𝑁) # 0)
23		zapne 9598	. . . . . . 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 11723	. . . . 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 9645	. . 3 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ ℤ)
29	12	abscld 11804	. . . . 5 ⊢ (𝜑 → (abs‘𝑀) ∈ ℝ)
30	13	abscld 11804	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) ∈ ℝ)
31	12	absge0d 11807	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘𝑀))
32		nnabscl 11723	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
33	10, 18, 32	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (abs‘𝑁) ∈ ℕ)
34	33	nnge1d 9228	. . . . 5 ⊢ (𝜑 → 1 ≤ (abs‘𝑁))
35	29, 30, 31, 34	lemulge11d 9159	. . . 4 ⊢ (𝜑 → (abs‘𝑀) ≤ ((abs‘𝑀) · (abs‘𝑁)))
36	12, 13	absmuld 11817	. . . 4 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
37	35, 36	breqtrrd 4121	. . 3 ⊢ (𝜑 → (abs‘𝑀) ≤ (abs‘(𝑀 · 𝑁)))
38		eluz2 9805	. . 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 1208	. 2 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ (ℤ_≥‘(abs‘𝑀)))
40		1zzd 9550	. . . . 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 15823	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝐹:ℕ⟶ℤ)
44	41, 3, 4, 43	syl3anc 1274	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℤ)
45	44	ffvelcdmda 5790	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℤ)
46		zmulcl 9577	. . . . . 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 10772	. . . 4 ⊢ (𝜑 → seq1( · , 𝐹):ℕ⟶ℤ)
49	48, 6	ffvelcdmd 5791	. . 3 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℤ)
50	49	zcnd 9647	. 2 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℂ)
51		eleq1w 2292	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ))
52		oveq2 6036	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐴 /_L 𝑛) = (𝐴 /_L 𝑘))
53		oveq1 6035	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 pCnt 𝑀) = (𝑘 pCnt 𝑀))
54	52, 53	oveq12d 6046	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)))
55	51, 54	ifbieq1d 3632	. . . 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 9200	. . . . 5 ⊢ (𝜑 → ((abs‘𝑀) + 1) ∈ ℕ)
57		elfzuz 10301	. . . . 5 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ (ℤ_≥‘((abs‘𝑀) + 1)))
58		eluznn 9878	. . . . 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 12746	. . . . . . . 8 ⊢ (𝑘 ∈ ℙ → 𝑘 ∈ ℤ)
62	61	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑘 ∈ ℤ)
63		lgscl 15816	. . . . . . 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 12934	. . . . . . 7 ⊢ ((𝑘 ∈ ℙ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
69	65, 66, 67, 68	syl12anc 1272	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
70		zexpcl 10862	. . . . . 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 9550	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ ¬ 𝑘 ∈ ℙ) → 1 ∈ ℤ)
73		prmdc 12765	. . . . . 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 3639	. . . 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 5749	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → (𝐹‘𝑘) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1))
77		zq 9904	. . . . . . . . . 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 12962	. . . . . . . . 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 10307	. . . . . . . . . . . . . 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 10305	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ ℤ)
84		zltp1le 9578	. . . . . . . . . . . . . 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 9569	. . . . . . . . . . . . 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 12468	. . . . . . . . . . 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 669	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ¬ 𝑘 ∥ (abs‘𝑀))
95		pceq0 12958	. . . . . . . . . 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 2266	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) = 0)
99	98	oveq2d 6044	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑0))
100	64	zcnd 9647	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝐴 /_L 𝑘) ∈ ℂ)
101	100	exp0d 10975	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑0) = 1)
102	99, 101	eqtrd 2264	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) = 1)
103	102, 74	ifeq1dadc 3640	. . . 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 3645	. . . . 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 2264	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) = 1)
107	76, 106	eqtrd 2264	. 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 9837	. . . . . 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 5791	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℤ)
113	112	zcnd 9647	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℂ)
114		mulcl 8202	. . 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 10834	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 842 = wceq 1398 ∈ wcel 2202 ≠ wne 2403 ifcif 3607 class class class wbr 4093 ↦ cmpt 4155 ⟶wf 5329 ‘cfv 5333 (class class class)co 6028 ℂcc 8073 0cc0 8075 1c1 8076 + caddc 8078 · cmul 8080 < clt 8256 ≤ cle 8257 # cap 8803 ℕcn 9185 ℕ₀cn0 9444 ℤcz 9523 ℤ_≥cuz 9799 ℚcq 9897 ...cfz 10288 seqcseq 10755 ↑cexp 10846 abscabs 11620 ∥ cdvds 12411 ℙcprime 12742 pCnt cpc 12920 /_L clgs 15799

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-2o 6626 df-oadd 6629 df-er 6745 df-en 6953 df-dom 6954 df-fin 6955 df-sup 7226 df-inf 7227 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-5 9247 df-6 9248 df-7 9249 df-8 9250 df-n0 9445 df-z 9524 df-uz 9800 df-q 9898 df-rp 9933 df-fz 10289 df-fzo 10423 df-fl 10576 df-mod 10631 df-seqfrec 10756 df-exp 10847 df-ihash 11084 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622 df-clim 11902 df-proddc 12175 df-dvds 12412 df-gcd 12588 df-prm 12743 df-phi 12846 df-pc 12921 df-lgs 15800

This theorem is referenced by: lgsdi 15839

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