lgsdi - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lgsdi		Structured version Visualization version GIF version

Theorem lgsdi 27313

Description: The Legendre symbol is completely multiplicative in its right argument. Generalization of theorem 9.9(b) in [ApostolNT] p. 188 (which assumes that 𝑀 and 𝑁 are odd positive integers). (Contributed by Mario Carneiro, 5-Feb-2015.)

**Assertion**
Ref	Expression
lgsdi	⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝐴 /_L (𝑀 · 𝑁)) = ((𝐴 /_L 𝑀) · (𝐴 /_L 𝑁)))

Proof of Theorem lgsdi Dummy variables 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anrot 1100	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ))
2		lgsdilem 27303	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → if((𝐴 < 0 ∧ (𝑀 · 𝑁) < 0), -1, 1) = (if((𝐴 < 0 ∧ 𝑀 < 0), -1, 1) · if((𝐴 < 0 ∧ 𝑁 < 0), -1, 1)))
3	1, 2	sylanb 582	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → if((𝐴 < 0 ∧ (𝑀 · 𝑁) < 0), -1, 1) = (if((𝐴 < 0 ∧ 𝑀 < 0), -1, 1) · if((𝐴 < 0 ∧ 𝑁 < 0), -1, 1)))
4		ancom 460	. . . . 5 ⊢ (((𝑀 · 𝑁) < 0 ∧ 𝐴 < 0) ↔ (𝐴 < 0 ∧ (𝑀 · 𝑁) < 0))
5		ifbi 4504	. . . . 5 ⊢ ((((𝑀 · 𝑁) < 0 ∧ 𝐴 < 0) ↔ (𝐴 < 0 ∧ (𝑀 · 𝑁) < 0)) → if(((𝑀 · 𝑁) < 0 ∧ 𝐴 < 0), -1, 1) = if((𝐴 < 0 ∧ (𝑀 · 𝑁) < 0), -1, 1))
6	4, 5	ax-mp 5	. . . 4 ⊢ if(((𝑀 · 𝑁) < 0 ∧ 𝐴 < 0), -1, 1) = if((𝐴 < 0 ∧ (𝑀 · 𝑁) < 0), -1, 1)
7		ancom 460	. . . . . 6 ⊢ ((𝑀 < 0 ∧ 𝐴 < 0) ↔ (𝐴 < 0 ∧ 𝑀 < 0))
8		ifbi 4504	. . . . . 6 ⊢ (((𝑀 < 0 ∧ 𝐴 < 0) ↔ (𝐴 < 0 ∧ 𝑀 < 0)) → if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) = if((𝐴 < 0 ∧ 𝑀 < 0), -1, 1))
9	7, 8	ax-mp 5	. . . . 5 ⊢ if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) = if((𝐴 < 0 ∧ 𝑀 < 0), -1, 1)
10		ancom 460	. . . . . 6 ⊢ ((𝑁 < 0 ∧ 𝐴 < 0) ↔ (𝐴 < 0 ∧ 𝑁 < 0))
11		ifbi 4504	. . . . . 6 ⊢ (((𝑁 < 0 ∧ 𝐴 < 0) ↔ (𝐴 < 0 ∧ 𝑁 < 0)) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) = if((𝐴 < 0 ∧ 𝑁 < 0), -1, 1))
12	10, 11	ax-mp 5	. . . . 5 ⊢ if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) = if((𝐴 < 0 ∧ 𝑁 < 0), -1, 1)
13	9, 12	oveq12i 7380	. . . 4 ⊢ (if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) = (if((𝐴 < 0 ∧ 𝑀 < 0), -1, 1) · if((𝐴 < 0 ∧ 𝑁 < 0), -1, 1))
14	3, 6, 13	3eqtr4g 2797	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → if(((𝑀 · 𝑁) < 0 ∧ 𝐴 < 0), -1, 1) = (if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)))
15		simpl2 1194	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → 𝑀 ∈ ℤ)
16		simpl3 1195	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℤ)
17	15, 16	zmulcld 12614	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝑀 · 𝑁) ∈ ℤ)
18	15	zcnd 12609	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → 𝑀 ∈ ℂ)
19	16	zcnd 12609	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℂ)
20		simprl 771	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → 𝑀 ≠ 0)
21		simprr 773	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → 𝑁 ≠ 0)
22	18, 19, 20, 21	mulne0d 11801	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝑀 · 𝑁) ≠ 0)
23		nnabscl 15261	. . . . . . 7 ⊢ (((𝑀 · 𝑁) ∈ ℤ ∧ (𝑀 · 𝑁) ≠ 0) → (abs‘(𝑀 · 𝑁)) ∈ ℕ)
24	17, 22, 23	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (abs‘(𝑀 · 𝑁)) ∈ ℕ)
25		nnuz 12802	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
26	24, 25	eleqtrdi 2847	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (abs‘(𝑀 · 𝑁)) ∈ (ℤ_≥‘1))
27		simpl1 1193	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → 𝐴 ∈ ℤ)
28		eqid 2737	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))
29	28	lgsfcl3 27297	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)):ℕ⟶ℤ)
30	27, 15, 20, 29	syl3anc 1374	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)):ℕ⟶ℤ)
31		elfznn 13481	. . . . . . 7 ⊢ (𝑘 ∈ (1...(abs‘(𝑀 · 𝑁))) → 𝑘 ∈ ℕ)
32		ffvelcdm 7035	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)):ℕ⟶ℤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))‘𝑘) ∈ ℤ)
33	30, 31, 32	syl2an 597	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))‘𝑘) ∈ ℤ)
34	33	zcnd 12609	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))‘𝑘) ∈ ℂ)
35		eqid 2737	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))
36	35	lgsfcl3 27297	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)):ℕ⟶ℤ)
37	27, 16, 21, 36	syl3anc 1374	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)):ℕ⟶ℤ)
38		ffvelcdm 7035	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)):ℕ⟶ℤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑘) ∈ ℤ)
39	37, 31, 38	syl2an 597	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑘) ∈ ℤ)
40	39	zcnd 12609	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑘) ∈ ℂ)
41		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑘 ∈ ℙ)
42	15	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑀 ∈ ℤ)
43	20	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑀 ≠ 0)
44	16	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑁 ∈ ℤ)
45	21	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑁 ≠ 0)
46		pcmul 16791	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℙ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑘 pCnt (𝑀 · 𝑁)) = ((𝑘 pCnt 𝑀) + (𝑘 pCnt 𝑁)))
47	41, 42, 43, 44, 45, 46	syl122anc 1382	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt (𝑀 · 𝑁)) = ((𝑘 pCnt 𝑀) + (𝑘 pCnt 𝑁)))
48	47	oveq2d 7384	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))) = ((𝐴 /_L 𝑘)↑((𝑘 pCnt 𝑀) + (𝑘 pCnt 𝑁))))
49	27	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝐴 ∈ ℤ)
50		prmz 16614	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℙ → 𝑘 ∈ ℤ)
51	50	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → 𝑘 ∈ ℤ)
52		lgscl 27290	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝐴 /_L 𝑘) ∈ ℤ)
53	49, 51, 52	syl2anc 585	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝐴 /_L 𝑘) ∈ ℤ)
54	53	zcnd 12609	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝐴 /_L 𝑘) ∈ ℂ)
55		pczcl 16788	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑘 pCnt 𝑁) ∈ ℕ₀)
56	41, 44, 45, 55	syl12anc 837	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑁) ∈ ℕ₀)
57		pczcl 16788	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℙ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
58	41, 42, 43, 57	syl12anc 837	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (𝑘 pCnt 𝑀) ∈ ℕ₀)
59	54, 56, 58	expaddd 14083	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑((𝑘 pCnt 𝑀) + (𝑘 pCnt 𝑁))) = (((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) · ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁))))
60	48, 59	eqtrd 2772	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))) = (((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) · ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁))))
61		iftrue 4487	. . . . . . . . 9 ⊢ (𝑘 ∈ ℙ → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))))
62	61	adantl 481	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))))
63		iftrue 4487	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℙ → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)))
64		iftrue 4487	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℙ → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)))
65	63, 64	oveq12d 7386	. . . . . . . . 9 ⊢ (𝑘 ∈ ℙ → (if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) · if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1)) = (((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) · ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁))))
66	65	adantl 481	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) · if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1)) = (((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) · ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁))))
67	60, 62, 66	3eqtr4rd 2783	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ 𝑘 ∈ ℙ) → (if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) · if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1)) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1))
68		1t1e1 12314	. . . . . . . . 9 ⊢ (1 · 1) = 1
69		iffalse 4490	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ ℙ → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) = 1)
70		iffalse 4490	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ ℙ → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1) = 1)
71	69, 70	oveq12d 7386	. . . . . . . . 9 ⊢ (¬ 𝑘 ∈ ℙ → (if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) · if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1)) = (1 · 1))
72		iffalse 4490	. . . . . . . . 9 ⊢ (¬ 𝑘 ∈ ℙ → if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1) = 1)
73	68, 71, 72	3eqtr4a 2798	. . . . . . . 8 ⊢ (¬ 𝑘 ∈ ℙ → (if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) · if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1)) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1))
74	73	adantl 481	. . . . . . 7 ⊢ (((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) ∧ ¬ 𝑘 ∈ ℙ) → (if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) · if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1)) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1))
75	67, 74	pm2.61dan 813	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) → (if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) · if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1)) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1))
76	31	adantl 481	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) → 𝑘 ∈ ℕ)
77		eleq1w 2820	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ))
78		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝐴 /_L 𝑛) = (𝐴 /_L 𝑘))
79		oveq1 7375	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑛 pCnt 𝑀) = (𝑘 pCnt 𝑀))
80	78, 79	oveq12d 7386	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)))
81	77, 80	ifbieq1d 4506	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1))
82		ovex 7401	. . . . . . . . . 10 ⊢ ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)) ∈ V
83		1ex 11140	. . . . . . . . . 10 ⊢ 1 ∈ V
84	82, 83	ifex 4532	. . . . . . . . 9 ⊢ if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) ∈ V
85	81, 28, 84	fvmpt 6949	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))‘𝑘) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1))
86		oveq1 7375	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑛 pCnt 𝑁) = (𝑘 pCnt 𝑁))
87	78, 86	oveq12d 7386	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)))
88	77, 87	ifbieq1d 4506	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1))
89		ovex 7401	. . . . . . . . . 10 ⊢ ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)) ∈ V
90	89, 83	ifex 4532	. . . . . . . . 9 ⊢ if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1) ∈ V
91	88, 35, 90	fvmpt 6949	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑘) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1))
92	85, 91	oveq12d 7386	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))‘𝑘) · ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑘)) = (if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) · if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1)))
93	76, 92	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) → (((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))‘𝑘) · ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑘)) = (if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑀)), 1) · if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt 𝑁)), 1)))
94		oveq1 7375	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (𝑛 pCnt (𝑀 · 𝑁)) = (𝑘 pCnt (𝑀 · 𝑁)))
95	78, 94	oveq12d 7386	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))) = ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))))
96	77, 95	ifbieq1d 4506	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1))
97		eqid 2737	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1))
98		ovex 7401	. . . . . . . . 9 ⊢ ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))) ∈ V
99	98, 83	ifex 4532	. . . . . . . 8 ⊢ if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1) ∈ V
100	96, 97, 99	fvmpt 6949	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1))‘𝑘) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1))
101	76, 100	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1))‘𝑘) = if(𝑘 ∈ ℙ, ((𝐴 /_L 𝑘)↑(𝑘 pCnt (𝑀 · 𝑁))), 1))
102	75, 93, 101	3eqtr4rd 2783	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘(𝑀 · 𝑁)))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1))‘𝑘) = (((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))‘𝑘) · ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑘)))
103	26, 34, 40, 102	prodfmul 15825	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1)))‘(abs‘(𝑀 · 𝑁))) = ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘(𝑀 · 𝑁))) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘(𝑀 · 𝑁)))))
104	27, 15, 16, 20, 21, 28	lgsdilem2 27312	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀)) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘(𝑀 · 𝑁))))
105	27, 16, 15, 21, 20, 35	lgsdilem2 27312	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘(𝑁 · 𝑀))))
106	18, 19	mulcomd 11165	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
107	106	fveq2d 6846	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (abs‘(𝑀 · 𝑁)) = (abs‘(𝑁 · 𝑀)))
108	107	fveq2d 6846	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘(𝑀 · 𝑁))) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘(𝑁 · 𝑀))))
109	105, 108	eqtr4d 2775	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘(𝑀 · 𝑁))))
110	104, 109	oveq12d 7386	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁))) = ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘(𝑀 · 𝑁))) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘(𝑀 · 𝑁)))))
111	103, 110	eqtr4d 2775	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1)))‘(abs‘(𝑀 · 𝑁))) = ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁))))
112	14, 111	oveq12d 7386	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (if(((𝑀 · 𝑁) < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1)))‘(abs‘(𝑀 · 𝑁)))) = ((if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) · ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)))))
113	97	lgsval4 27296	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ ∧ (𝑀 · 𝑁) ≠ 0) → (𝐴 /_L (𝑀 · 𝑁)) = (if(((𝑀 · 𝑁) < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1)))‘(abs‘(𝑀 · 𝑁)))))
114	27, 17, 22, 113	syl3anc 1374	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝐴 /_L (𝑀 · 𝑁)) = (if(((𝑀 · 𝑁) < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt (𝑀 · 𝑁))), 1)))‘(abs‘(𝑀 · 𝑁)))))
115	28	lgsval4 27296	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (𝐴 /_L 𝑀) = (if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀))))
116	27, 15, 20, 115	syl3anc 1374	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝐴 /_L 𝑀) = (if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀))))
117	35	lgsval4 27296	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /_L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁))))
118	27, 16, 21, 117	syl3anc 1374	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝐴 /_L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁))))
119	116, 118	oveq12d 7386	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → ((𝐴 /_L 𝑀) · (𝐴 /_L 𝑁)) = ((if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀))) · (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)))))
120		neg1cn 12142	. . . . . 6 ⊢ -1 ∈ ℂ
121		ax-1cn 11096	. . . . . 6 ⊢ 1 ∈ ℂ
122	120, 121	ifcli 4529	. . . . 5 ⊢ if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) ∈ ℂ
123	122	a1i 11	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) ∈ ℂ)
124		nnabscl 15261	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑀 ≠ 0) → (abs‘𝑀) ∈ ℕ)
125	15, 20, 124	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (abs‘𝑀) ∈ ℕ)
126	125, 25	eleqtrdi 2847	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (abs‘𝑀) ∈ (ℤ_≥‘1))
127		elfznn 13481	. . . . . . 7 ⊢ (𝑘 ∈ (1...(abs‘𝑀)) → 𝑘 ∈ ℕ)
128	30, 127, 32	syl2an 597	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘𝑀))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))‘𝑘) ∈ ℤ)
129	128	zcnd 12609	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘𝑀))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1))‘𝑘) ∈ ℂ)
130		mulcl 11122	. . . . . 6 ⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ)
131	130	adantl 481	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ)
132	126, 129, 131	seqcl 13957	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀)) ∈ ℂ)
133	120, 121	ifcli 4529	. . . . 5 ⊢ if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) ∈ ℂ
134	133	a1i 11	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) ∈ ℂ)
135		nnabscl 15261	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
136	16, 21, 135	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (abs‘𝑁) ∈ ℕ)
137	136, 25	eleqtrdi 2847	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (abs‘𝑁) ∈ (ℤ_≥‘1))
138		elfznn 13481	. . . . . . 7 ⊢ (𝑘 ∈ (1...(abs‘𝑁)) → 𝑘 ∈ ℕ)
139	37, 138, 38	syl2an 597	. . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘𝑁))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑘) ∈ ℤ)
140	139	zcnd 12609	. . . . 5 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) ∧ 𝑘 ∈ (1...(abs‘𝑁))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑘) ∈ ℂ)
141	137, 140, 131	seqcl 13957	. . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)) ∈ ℂ)
142	123, 132, 134, 141	mul4d 11357	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → ((if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀))) · (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)))) = ((if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) · ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)))))
143	119, 142	eqtrd 2772	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → ((𝐴 /_L 𝑀) · (𝐴 /_L 𝑁)) = ((if((𝑀 < 0 ∧ 𝐴 < 0), -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) · ((seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑀)), 1)))‘(abs‘𝑀)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)))))
144	112, 114, 143	3eqtr4d 2782	1 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝐴 /_L (𝑀 · 𝑁)) = ((𝐴 /_L 𝑀) · (𝐴 /_L 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ifcif 4481 class class class wbr 5100 ↦ cmpt 5181 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 -cneg 11377 ℕcn 12157 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ...cfz 13435 seqcseq 13936 ↑cexp 13996 abscabs 15169 ℙcprime 16610 pCnt cpc 16776 /_L clgs 27273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-dvds 16192 df-gcd 16434 df-prm 16611 df-phi 16705 df-pc 16777 df-lgs 27274

This theorem is referenced by: lgssq2 27317 lgsdinn0 27324 lgsquad2lem1 27363

W3C validator