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Theorem coprmprod 16600

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 3961	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ ℕ ↔ ∅ ⊆ ℕ))
2	1	3anbi1d 1443	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
3		raleq 3295	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ ∅ ((𝐹‘𝑚) gcd 𝑁) = 1))
4		difeq1 4073	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
5	4	raleqdv 3298	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
6	5	raleqbi1dv 3310	. . . . . . . 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 1439	. . . . . . 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 15842	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ ∅ (𝐹‘𝑚))
9	8	oveq1d 7383	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁))
10	9	eqeq1d 2739	. . . . . . 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 3961	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ ℕ ↔ 𝑦 ⊆ ℕ))
13	12	3anbi1d 1443	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
14		raleq 3295	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1))
15		difeq1 4073	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
16	15	raleqdv 3298	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
17	16	raleqbi1dv 3310	. . . . . . . 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 1439	. . . . . . 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 15842	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑦 (𝐹‘𝑚))
20	19	oveq1d 7383	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁))
21	20	eqeq1d 2739	. . . . . . 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 3961	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℕ ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ))
24	23	3anbi1d 1443	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
25		raleq 3295	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1))
26		difeq1 4073	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
27	26	raleqdv 3298	. . . . . . . . 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 3310	. . . . . . . 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 1439	. . . . . . 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 15842	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚))
31	30	oveq1d 7383	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁))
32	31	eqeq1d 2739	. . . . . . 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 3961	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑥 ⊆ ℕ ↔ 𝑀 ⊆ ℕ))
35	34	3anbi1d 1443	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
36		raleq 3295	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1))
37		difeq1 4073	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
38	37	raleqdv 3298	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
39	38	raleqbi1dv 3310	. . . . . . . 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 1439	. . . . . . 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 15842	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑀 (𝐹‘𝑚))
42	41	oveq1d 7383	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁))
43	42	eqeq1d 2739	. . . . . . 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 15878	. . . . . . . . . . 11 ⊢ ∏𝑚 ∈ ∅ (𝐹‘𝑚) = 1
46	45	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹‘𝑚) = 1)
47	46	oveq1d 7383	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = (1 gcd 𝑁))
48		nnz 12521	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
49		1gcd 16472	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (1 gcd 𝑁) = 1)
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1 gcd 𝑁) = 1)
51	47, 50	eqtrd 2772	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1)
52	51	3ad2ant2 1135	. . . . . . 7 ⊢ ((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1)
53	52	3ad2ant1 1134	. . . . . 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 1916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦))
55		nfcv 2899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝐹‘𝑧)
56		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
57		unss 4144	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ)
58		vex 3446	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑧 ∈ V
59	58	snss 4743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ℕ ↔ {𝑧} ⊆ ℕ)
60	59	biimpri 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑧} ⊆ ℕ → 𝑧 ∈ ℕ)
61	60	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑧 ∈ ℕ)
62	57, 61	sylbir 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑧 ∈ ℕ)
63	62	3ad2ant1 1134	. . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
66		simpll3 1216	. . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ)
69	68	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . 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 3935	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → 𝑚 ∈ ℕ)
72	66, 71	ffvelcdmd 7039	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℕ)
73	72	nncnd 12173	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℂ)
74		fveq2 6842	. . . . . . . . . . . . . . . 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	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹‘𝑧) ∈ ℕ)
78	77	3adant2 1132	. . . . . . . . . . . . . . . . . 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 12173	. . . . . . . . . . . . . . . 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 15924	. . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . 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 15890	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ)
84	83	nnzd 12526	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℤ)
85	79	nnzd 12526	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹‘𝑧) ∈ ℤ)
86	84, 85	zmulcld 12614	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)) ∈ ℤ)
87	48	3ad2ant2 1135	. . . . . . . . . . . . . . . 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 16453	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . 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 1134	. . . . . . . . . . 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 1194	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑁 ∈ ℕ)
97	96, 83, 79	3jca 1129	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . 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 16453	. . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . 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 1446	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
113		ssun1 4132	. . . . . . . . . . . . . 14 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
114		ssralv 4004	. . . . . . . . . . . . . 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 4004	. . . . . . . . . . . . . . 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 4099	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑚 ∈ 𝑦) → (𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
120		ssralv 4004	. . . . . . . . . . . . . . . 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 3150	. . . . . . . . . . . . . 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 1446	. . . . . . . . . . . 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 2772	. . . . . . . . 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 16496	. . . . . . . . 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 585	. . . . . . . 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 4622	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ {𝑧}
131	130	olci 867	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧})
132		elun 4107	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
133	131, 132	mpbir 231	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
134	74	oveq1d 7383	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑧 → ((𝐹‘𝑚) gcd 𝑁) = ((𝐹‘𝑧) gcd 𝑁))
135	134	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑧 → (((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ((𝐹‘𝑧) gcd 𝑁) = 1))
136	135	rspcv 3574	. . . . . . . . . . . . 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 12526	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹‘𝑧) ∈ ℤ)
140	87, 139	gcdcomd 16453	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑁 gcd (𝐹‘𝑧)) = ((𝐹‘𝑧) gcd 𝑁))
141	140	eqeq1d 2739	. . . . . . . . . . . 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 257	. . . . . . . . . 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 1133	. . . . . . . . 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 2776	. . . . . . 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 9102	. . . . 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 1355	. . . 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 1355	. . 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 1111	. 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 1111	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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∖ cdif 3900 ∪ cun 3901 ⊆ wss 3903 ∅c0 4287 {csn 4582 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 1c1 11039 · cmul 11043 ℕcn 12157 ℤcz 12500 ∏cprod 15838 gcd cgcd 16433

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-inf2 9562 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-se 5586 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-isom 6509 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-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-oi 9427 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-z 12501 df-uz 12764 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-clim 15423 df-prod 15839 df-dvds 16192 df-gcd 16434

This theorem is referenced by: coprmproddvdslem 16601

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