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Theorem lgsdilem2 15764

Description: Lemma for lgsdi 15765. (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 8175	. . 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 11660	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
6	3, 4, 5	syl2anc 411	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℕ)
7		nnuz 9791	. . 3 ⊢ ℕ = (ℤ_≥‘1)
8	6, 7	eleqtrdi 2324	. 2 ⊢ (𝜑 → (abs‘𝑀) ∈ (ℤ_≥‘1))
9	6	nnzd 9600	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℤ)
10		lgsdilem2.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
11	3, 10	zmulcld 9607	. . . . 5 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℤ)
12	3	zcnd 9602	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℂ)
13	10	zcnd 9602	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
14		0z 9489	. . . . . . . . 9 ⊢ 0 ∈ ℤ
15		zapne 9553	. . . . . . . . 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 9553	. . . . . . . . 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 8837	. . . . . 6 ⊢ (𝜑 → (𝑀 · 𝑁) # 0)
23		zapne 9553	. . . . . . 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 11660	. . . . 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 9600	. . 3 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ ℤ)
29	12	abscld 11741	. . . . 5 ⊢ (𝜑 → (abs‘𝑀) ∈ ℝ)
30	13	abscld 11741	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) ∈ ℝ)
31	12	absge0d 11744	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘𝑀))
32		nnabscl 11660	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
33	10, 18, 32	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (abs‘𝑁) ∈ ℕ)
34	33	nnge1d 9185	. . . . 5 ⊢ (𝜑 → 1 ≤ (abs‘𝑁))
35	29, 30, 31, 34	lemulge11d 9116	. . . 4 ⊢ (𝜑 → (abs‘𝑀) ≤ ((abs‘𝑀) · (abs‘𝑁)))
36	12, 13	absmuld 11754	. . . 4 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
37	35, 36	breqtrrd 4116	. . 3 ⊢ (𝜑 → (abs‘𝑀) ≤ (abs‘(𝑀 · 𝑁)))
38		eluz2 9760	. . 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 1207	. 2 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ (ℤ_≥‘(abs‘𝑀)))
40		1zzd 9505	. . . . 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 15749	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝐹:ℕ⟶ℤ)
44	41, 3, 4, 43	syl3anc 1273	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℤ)
45	44	ffvelcdmda 5782	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℤ)
46		zmulcl 9532	. . . . . 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 10725	. . . 4 ⊢ (𝜑 → seq1( · , 𝐹):ℕ⟶ℤ)
49	48, 6	ffvelcdmd 5783	. . 3 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℤ)
50	49	zcnd 9602	. 2 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℂ)
51		eleq1w 2292	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ))
52		oveq2 6025	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐴 /_L 𝑛) = (𝐴 /_L 𝑘))
53		oveq1 6024	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 pCnt 𝑀) = (𝑘 pCnt 𝑀))
54	52, 53	oveq12d 6035	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)))
55	51, 54	ifbieq1d 3628	. . . 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 9157	. . . . 5 ⊢ (𝜑 → ((abs‘𝑀) + 1) ∈ ℕ)
57		elfzuz 10255	. . . . 5 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ (ℤ_≥‘((abs‘𝑀) + 1)))
58		eluznn 9833	. . . . 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 12682	. . . . . . . 8 ⊢ (𝑘 ∈ ℙ → 𝑘 ∈ ℤ)
62	61	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑘 ∈ ℤ)
63		lgscl 15742	. . . . . . 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 12870	. . . . . . 7 ⊢ ((𝑘 ∈ ℙ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
69	65, 66, 67, 68	syl12anc 1271	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
70		zexpcl 10815	. . . . . 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 9505	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ ¬ 𝑘 ∈ ℙ) → 1 ∈ ℤ)
73		prmdc 12701	. . . . . 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 3635	. . . 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 5740	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → (𝐹‘𝑘) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1))
77		zq 9859	. . . . . . . . . 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 12898	. . . . . . . . 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 10261	. . . . . . . . . . . . . 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 10259	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ ℤ)
84		zltp1le 9533	. . . . . . . . . . . . . 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 9524	. . . . . . . . . . . . 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 12404	. . . . . . . . . . 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 12894	. . . . . . . . . 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 6033	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑0))
100	64	zcnd 9602	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝐴 /_L 𝑘) ∈ ℂ)
101	100	exp0d 10928	. . . . . 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 3636	. . . 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 3641	. . . . 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 9792	. . . . . 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 5783	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℤ)
113	112	zcnd 9602	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℂ)
114		mulcl 8158	. . 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 10787	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 841 = wceq 1397 ∈ wcel 2202 ≠ wne 2402 ifcif 3605 class class class wbr 4088 ↦ cmpt 4150 ⟶wf 5322 ‘cfv 5326 (class class class)co 6017 ℂcc 8029 0cc0 8031 1c1 8032 + caddc 8034 · cmul 8036 < clt 8213 ≤ cle 8214 # cap 8760 ℕcn 9142 ℕ₀cn0 9401 ℤcz 9478 ℤ_≥cuz 9754 ℚcq 9852 ...cfz 10242 seqcseq 10708 ↑cexp 10799 abscabs 11557 ∥ cdvds 12347 ℙcprime 12678 pCnt cpc 12856 /_L clgs 15725

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-xor 1420 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-frec 6556 df-1o 6581 df-2o 6582 df-oadd 6585 df-er 6701 df-en 6909 df-dom 6910 df-fin 6911 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-5 9204 df-6 9205 df-7 9206 df-8 9207 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-fz 10243 df-fzo 10377 df-fl 10529 df-mod 10584 df-seqfrec 10709 df-exp 10800 df-ihash 11037 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-clim 11839 df-proddc 12111 df-dvds 12348 df-gcd 12524 df-prm 12679 df-phi 12782 df-pc 12857 df-lgs 15726

This theorem is referenced by: lgsdi 15765

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