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Theorem coprmprod 16294

Description: The product of the elements of a sequence of pairwise coprime positive integers is coprime to a positive integer which is coprime to all integers of the sequence. (Contributed by AV, 18-Aug-2020.)

**Assertion**
Ref	Expression
coprmprod	⊢ (((𝑀 ∈ Fin ∧ 𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1) → (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1))

Distinct variable groups: 𝑚,𝐹 𝑚,𝑀,𝑛 𝑚,𝑁,𝑛 Allowed substitution hint: 𝐹(𝑛)

Proof of Theorem coprmprod Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3942	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ ℕ ↔ ∅ ⊆ ℕ))
2	1	3anbi1d 1438	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
3		raleq 3333	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ ∅ ((𝐹‘𝑚) gcd 𝑁) = 1))
4		difeq1 4046	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
5	4	raleqdv 3339	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
6	5	raleqbi1dv 3331	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
7	2, 3, 6	3anbi123d 1434	. . . . . . 7 ⊢ (𝑥 = ∅ → (((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ ((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ ∅ ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)))
8		prodeq1 15547	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ ∅ (𝐹‘𝑚))
9	8	oveq1d 7270	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁))
10	9	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = ∅ → ((∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1))
11	7, 10	imbi12d 344	. . . . . 6 ⊢ (𝑥 = ∅ → ((((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1) ↔ (((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ ∅ ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1)))
12		sseq1 3942	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ ℕ ↔ 𝑦 ⊆ ℕ))
13	12	3anbi1d 1438	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
14		raleq 3333	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1))
15		difeq1 4046	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
16	15	raleqdv 3339	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
17	16	raleqbi1dv 3331	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
18	13, 14, 17	3anbi123d 1434	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ ((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)))
19		prodeq1 15547	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑦 (𝐹‘𝑚))
20	19	oveq1d 7270	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁))
21	20	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1))
22	18, 21	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1) ↔ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)))
23		sseq1 3942	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℕ ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ))
24	23	3anbi1d 1438	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
25		raleq 3333	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1))
26		difeq1 4046	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
27	26	raleqdv 3339	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
28	27	raleqbi1dv 3331	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
29	24, 25, 28	3anbi123d 1434	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)))
30		prodeq1 15547	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚))
31	30	oveq1d 7270	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁))
32	31	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = 1))
33	29, 32	imbi12d 344	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1) ↔ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = 1)))
34		sseq1 3942	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑥 ⊆ ℕ ↔ 𝑀 ⊆ ℕ))
35	34	3anbi1d 1438	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
36		raleq 3333	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1))
37		difeq1 4046	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
38	37	raleqdv 3339	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
39	38	raleqbi1dv 3331	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
40	35, 36, 39	3anbi123d 1434	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ ((𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)))
41		prodeq1 15547	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑀 (𝐹‘𝑚))
42	41	oveq1d 7270	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁))
43	42	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1))
44	40, 43	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑀 → ((((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1) ↔ (((𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1)))
45		prod0 15581	. . . . . . . . . . 11 ⊢ ∏𝑚 ∈ ∅ (𝐹‘𝑚) = 1
46	45	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹‘𝑚) = 1)
47	46	oveq1d 7270	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = (1 gcd 𝑁))
48		nnz 12272	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
49		1gcd 16169	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (1 gcd 𝑁) = 1)
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1 gcd 𝑁) = 1)
51	47, 50	eqtrd 2778	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1)
52	51	3ad2ant2 1132	. . . . . . 7 ⊢ ((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1)
53	52	3ad2ant1 1131	. . . . . 6 ⊢ (((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ ∅ ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1)
54		nfv 1918	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦))
55		nfcv 2906	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝐹‘𝑧)
56		simprl 767	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
57		unss 4114	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ)
58		vex 3426	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑧 ∈ V
59	58	snss 4716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ℕ ↔ {𝑧} ⊆ ℕ)
60	59	biimpri 227	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑧} ⊆ ℕ → 𝑧 ∈ ℕ)
61	60	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑧 ∈ ℕ)
62	57, 61	sylbir 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑧 ∈ ℕ)
63	62	3ad2ant1 1131	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
64	63	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ ℕ)
65		simprr 769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
66		simpll3 1212	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → 𝐹:ℕ⟶ℕ)
67		simpl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑦 ⊆ ℕ)
68	57, 67	sylbir 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ)
69	68	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑦 ⊆ ℕ)
70	69	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ⊆ ℕ)
71	70	sselda 3917	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → 𝑚 ∈ ℕ)
72	66, 71	ffvelrnd 6944	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℕ)
73	72	nncnd 11919	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℂ)
74		fveq2 6756	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑧 → (𝐹‘𝑚) = (𝐹‘𝑧))
75		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝐹:ℕ⟶ℕ)
76	62	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
77	75, 76	ffvelrnd 6944	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹‘𝑧) ∈ ℕ)
78	77	3adant2 1129	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹‘𝑧) ∈ ℕ)
79	78	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹‘𝑧) ∈ ℕ)
80	79	nncnd 11919	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹‘𝑧) ∈ ℂ)
81	54, 55, 56, 64, 65, 73, 74, 80	fprodsplitsn 15627	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)))
82	81	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = ((∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)) gcd 𝑁))
83	56, 72	fprodnncl 15593	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ)
84	83	nnzd 12354	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℤ)
85	79	nnzd 12354	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹‘𝑧) ∈ ℤ)
86	84, 85	zmulcld 12361	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)) ∈ ℤ)
87	48	3ad2ant2 1132	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑁 ∈ ℤ)
88	87	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑁 ∈ ℤ)
89	86, 88	gcdcomd 16149	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)) gcd 𝑁) = (𝑁 gcd (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧))))
90	82, 89	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧))))
91	90	ex 412	. . . . . . . . . . . 12 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)))))
92	91	3ad2ant1 1131	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)))))
93	92	com12 32	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)))))
94	93	adantr 480	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)))))
95	94	imp 406	. . . . . . . 8 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧))))
96		simpl2 1190	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑁 ∈ ℕ)
97	96, 83, 79	3jca 1126	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑁 ∈ ℕ ∧ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ ∧ (𝐹‘𝑧) ∈ ℕ))
98	97	ex 412	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝑁 ∈ ℕ ∧ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ ∧ (𝐹‘𝑧) ∈ ℕ)))
99	98	3ad2ant1 1131	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝑁 ∈ ℕ ∧ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ ∧ (𝐹‘𝑧) ∈ ℕ)))
100	99	com12 32	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (𝑁 ∈ ℕ ∧ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ ∧ (𝐹‘𝑧) ∈ ℕ)))
101	100	adantr 480	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (𝑁 ∈ ℕ ∧ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ ∧ (𝐹‘𝑧) ∈ ℕ)))
102	101	imp 406	. . . . . . . . 9 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)) → (𝑁 ∈ ℕ ∧ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ ∧ (𝐹‘𝑧) ∈ ℕ))
103	88, 84	gcdcomd 16149	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑁 gcd ∏𝑚 ∈ 𝑦 (𝐹‘𝑚)) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁))
104	103	ex 412	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝑁 gcd ∏𝑚 ∈ 𝑦 (𝐹‘𝑚)) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁)))
105	104	3ad2ant1 1131	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝑁 gcd ∏𝑚 ∈ 𝑦 (𝐹‘𝑚)) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁)))
106	105	com12 32	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (𝑁 gcd ∏𝑚 ∈ 𝑦 (𝐹‘𝑚)) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁)))
107	106	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (𝑁 gcd ∏𝑚 ∈ 𝑦 (𝐹‘𝑚)) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁)))
108	107	imp 406	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)) → (𝑁 gcd ∏𝑚 ∈ 𝑦 (𝐹‘𝑚)) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁))
109	68	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ))
110		idd 24	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝑁 ∈ ℕ → 𝑁 ∈ ℕ))
111		idd 24	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐹:ℕ⟶ℕ → 𝐹:ℕ⟶ℕ))
112	109, 110, 111	3anim123d 1441	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
113		ssun1 4102	. . . . . . . . . . . . . 14 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
114		ssralv 3983	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 → ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1))
115	113, 114	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 → ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1))
116		ssralv 3983	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
117	113, 116	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
118	113	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑚 ∈ 𝑦) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
119	118	ssdifd 4071	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑚 ∈ 𝑦) → (𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
120		ssralv 3983	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}) → (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
121	119, 120	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑚 ∈ 𝑦) → (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
122	121	ralimdva 3102	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
123	117, 122	syld 47	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
124	112, 115, 123	3anim123d 1441	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)))
125	124	imim1d 82	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)))
126	125	imp31 417	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)
127	108, 126	eqtrd 2778	. . . . . . . . 9 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)) → (𝑁 gcd ∏𝑚 ∈ 𝑦 (𝐹‘𝑚)) = 1)
128		rpmulgcd 16194	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ ∧ (𝐹‘𝑧) ∈ ℕ) ∧ (𝑁 gcd ∏𝑚 ∈ 𝑦 (𝐹‘𝑚)) = 1) → (𝑁 gcd (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧))) = (𝑁 gcd (𝐹‘𝑧)))
129	102, 127, 128	syl2anc 583	. . . . . . . 8 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)) → (𝑁 gcd (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧))) = (𝑁 gcd (𝐹‘𝑧)))
130		vsnid 4595	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ {𝑧}
131	130	olci 862	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧})
132		elun 4079	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
133	131, 132	mpbir 230	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
134	74	oveq1d 7270	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑧 → ((𝐹‘𝑚) gcd 𝑁) = ((𝐹‘𝑧) gcd 𝑁))
135	134	eqeq1d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑧 → (((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ((𝐹‘𝑧) gcd 𝑁) = 1))
136	135	rspcv 3547	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 → ((𝐹‘𝑧) gcd 𝑁) = 1))
137	133, 136	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 → ((𝐹‘𝑧) gcd 𝑁) = 1))
138	137	imp 406	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1) → ((𝐹‘𝑧) gcd 𝑁) = 1)
139	78	nnzd 12354	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹‘𝑧) ∈ ℤ)
140	87, 139	gcdcomd 16149	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑁 gcd (𝐹‘𝑧)) = ((𝐹‘𝑧) gcd 𝑁))
141	140	eqeq1d 2740	. . . . . . . . . . . 12 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑁 gcd (𝐹‘𝑧)) = 1 ↔ ((𝐹‘𝑧) gcd 𝑁) = 1))
142	141	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1) → ((𝑁 gcd (𝐹‘𝑧)) = 1 ↔ ((𝐹‘𝑧) gcd 𝑁) = 1))
143	138, 142	mpbird 256	. . . . . . . . . 10 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1) → (𝑁 gcd (𝐹‘𝑧)) = 1)
144	143	3adant3 1130	. . . . . . . . 9 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (𝑁 gcd (𝐹‘𝑧)) = 1)
145	144	adantl 481	. . . . . . . 8 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)) → (𝑁 gcd (𝐹‘𝑧)) = 1)
146	95, 129, 145	3eqtrd 2782	. . . . . . 7 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = 1)
147	146	exp31 419	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = 1)))
148	11, 22, 33, 44, 53, 147	findcard2s 8910	. . . . 5 ⊢ (𝑀 ∈ Fin → (((𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1))
149	148	3expd 1351	. . . 4 ⊢ (𝑀 ∈ Fin → ((𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1 → (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1))))
150	149	3expd 1351	. . 3 ⊢ (𝑀 ∈ Fin → (𝑀 ⊆ ℕ → (𝑁 ∈ ℕ → (𝐹:ℕ⟶ℕ → (∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1 → (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1))))))
151	150	3imp 1109	. 2 ⊢ ((𝑀 ∈ Fin ∧ 𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ) → (𝐹:ℕ⟶ℕ → (∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1 → (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1))))
152	151	3imp 1109	1 ⊢ (((𝑀 ∈ Fin ∧ 𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1) → (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 ∅c0 4253 {csn 4558 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 1c1 10803 · cmul 10807 ℕcn 11903 ℤcz 12249 ∏cprod 15543 gcd cgcd 16129

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-prod 15544 df-dvds 15892 df-gcd 16130

This theorem is referenced by: coprmproddvdslem 16295

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