lgsdilem2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > lgsdilem2		GIF version

Theorem lgsdilem2 15513

Description: Lemma for lgsdi 15514. (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 8069	. . 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 11411	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
6	3, 4, 5	syl2anc 411	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℕ)
7		nnuz 9684	. . 3 ⊢ ℕ = (ℤ_≥‘1)
8	6, 7	eleqtrdi 2298	. 2 ⊢ (𝜑 → (abs‘𝑀) ∈ (ℤ_≥‘1))
9	6	nnzd 9494	. . 3 ⊢ (𝜑 → (abs‘𝑀) ∈ ℤ)
10		lgsdilem2.3	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
11	3, 10	zmulcld 9501	. . . . 5 ⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℤ)
12	3	zcnd 9496	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℂ)
13	10	zcnd 9496	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
14		0z 9383	. . . . . . . . 9 ⊢ 0 ∈ ℤ
15		zapne 9447	. . . . . . . . 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 9447	. . . . . . . . 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 8731	. . . . . 6 ⊢ (𝜑 → (𝑀 · 𝑁) # 0)
23		zapne 9447	. . . . . . 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 11411	. . . . 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 9494	. . 3 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ ℤ)
29	12	abscld 11492	. . . . 5 ⊢ (𝜑 → (abs‘𝑀) ∈ ℝ)
30	13	abscld 11492	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) ∈ ℝ)
31	12	absge0d 11495	. . . . 5 ⊢ (𝜑 → 0 ≤ (abs‘𝑀))
32		nnabscl 11411	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
33	10, 18, 32	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (abs‘𝑁) ∈ ℕ)
34	33	nnge1d 9079	. . . . 5 ⊢ (𝜑 → 1 ≤ (abs‘𝑁))
35	29, 30, 31, 34	lemulge11d 9010	. . . 4 ⊢ (𝜑 → (abs‘𝑀) ≤ ((abs‘𝑀) · (abs‘𝑁)))
36	12, 13	absmuld 11505	. . . 4 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) = ((abs‘𝑀) · (abs‘𝑁)))
37	35, 36	breqtrrd 4072	. . 3 ⊢ (𝜑 → (abs‘𝑀) ≤ (abs‘(𝑀 · 𝑁)))
38		eluz2 9654	. . 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 1184	. 2 ⊢ (𝜑 → (abs‘(𝑀 · 𝑁)) ∈ (ℤ_≥‘(abs‘𝑀)))
40		1zzd 9399	. . . . 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 15498	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → 𝐹:ℕ⟶ℤ)
44	41, 3, 4, 43	syl3anc 1250	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ℤ)
45	44	ffvelcdmda 5715	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℤ)
46		zmulcl 9426	. . . . . 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 10609	. . . 4 ⊢ (𝜑 → seq1( · , 𝐹):ℕ⟶ℤ)
49	48, 6	ffvelcdmd 5716	. . 3 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℤ)
50	49	zcnd 9496	. 2 ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) ∈ ℂ)
51		eleq1w 2266	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ))
52		oveq2 5952	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐴 /_L 𝑛) = (𝐴 /_L 𝑘))
53		oveq1 5951	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 pCnt 𝑀) = (𝑘 pCnt 𝑀))
54	52, 53	oveq12d 5962	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)))
55	51, 54	ifbieq1d 3593	. . . 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 9051	. . . . 5 ⊢ (𝜑 → ((abs‘𝑀) + 1) ∈ ℕ)
57		elfzuz 10143	. . . . 5 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ (ℤ_≥‘((abs‘𝑀) + 1)))
58		eluznn 9721	. . . . 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 12433	. . . . . . . 8 ⊢ (𝑘 ∈ ℙ → 𝑘 ∈ ℤ)
62	61	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑘 ∈ ℤ)
63		lgscl 15491	. . . . . . 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 12621	. . . . . . 7 ⊢ ((𝑘 ∈ ℙ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
69	65, 66, 67, 68	syl12anc 1248	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
70		zexpcl 10699	. . . . . 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 9399	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ ¬ 𝑘 ∈ ℙ) → 1 ∈ ℤ)
73		prmdc 12452	. . . . . 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 3600	. . . 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 5673	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → (𝐹‘𝑘) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1))
77		zq 9747	. . . . . . . . . 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 12649	. . . . . . . . 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 10149	. . . . . . . . . . . . . 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 10147	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ ℤ)
84		zltp1le 9427	. . . . . . . . . . . . . 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 9418	. . . . . . . . . . . . 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 12155	. . . . . . . . . . 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 665	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ¬ 𝑘 ∥ (abs‘𝑀))
95		pceq0 12645	. . . . . . . . . 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 2240	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) = 0)
99	98	oveq2d 5960	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑0))
100	64	zcnd 9496	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝐴 /_L 𝑘) ∈ ℂ)
101	100	exp0d 10812	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑0) = 1)
102	99, 101	eqtrd 2238	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) = 1)
103	102, 74	ifeq1dadc 3601	. . . 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 3606	. . . . 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 2238	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (((abs‘𝑀) + 1)...(abs‘(𝑀 · 𝑁)))) → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) = 1)
107	76, 106	eqtrd 2238	. 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 9685	. . . . . 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 5716	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℤ)
113	112	zcnd 9496	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐹‘𝑘) ∈ ℂ)
114		mulcl 8052	. . 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 10671	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 836 = wceq 1373 ∈ wcel 2176 ≠ wne 2376 ifcif 3571 class class class wbr 4044 ↦ cmpt 4105 ⟶wf 5267 ‘cfv 5271 (class class class)co 5944 ℂcc 7923 0cc0 7925 1c1 7926 + caddc 7928 · cmul 7930 < clt 8107 ≤ cle 8108 # cap 8654 ℕcn 9036 ℕ₀cn0 9295 ℤcz 9372 ℤ_≥cuz 9648 ℚcq 9740 ...cfz 10130 seqcseq 10592 ↑cexp 10683 abscabs 11308 ∥ cdvds 12098 ℙcprime 12429 pCnt cpc 12607 /_L clgs 15474

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044 ax-caucvg 8045

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-xor 1396 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-isom 5280 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-irdg 6456 df-frec 6477 df-1o 6502 df-2o 6503 df-oadd 6506 df-er 6620 df-en 6828 df-dom 6829 df-fin 6830 df-sup 7086 df-inf 7087 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099 df-7 9100 df-8 9101 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-fz 10131 df-fzo 10265 df-fl 10413 df-mod 10468 df-seqfrec 10593 df-exp 10684 df-ihash 10921 df-cj 11153 df-re 11154 df-im 11155 df-rsqrt 11309 df-abs 11310 df-clim 11590 df-proddc 11862 df-dvds 12099 df-gcd 12275 df-prm 12430 df-phi 12533 df-pc 12608 df-lgs 15475

This theorem is referenced by: lgsdi 15514

W3C validator