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Theorem coprmprod 16718

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 3970	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ ℕ ↔ ∅ ⊆ ℕ))
2	1	3anbi1d 1466	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
3		raleq 3326	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ ∅ ((𝐹‘𝑚) gcd 𝑁) = 1))
4		difeq1 4082	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
5	4	raleqdv 3329	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
6	5	raleqbi1dv 3339	. . . . . . . 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 1462	. . . . . . 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 15960	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ ∅ (𝐹‘𝑚))
9	8	oveq1d 7426	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁))
10	9	eqeq1d 2771	. . . . . . 7 ⊢ (𝑥 = ∅ → ((∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1))
11	7, 10	imbi12d 347	. . . . . 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 3970	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ ℕ ↔ 𝑦 ⊆ ℕ))
13	12	3anbi1d 1466	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
14		raleq 3326	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1))
15		difeq1 4082	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
16	15	raleqdv 3329	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
17	16	raleqbi1dv 3339	. . . . . . . 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 1462	. . . . . . 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 15960	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑦 (𝐹‘𝑚))
20	19	oveq1d 7426	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁))
21	20	eqeq1d 2771	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd 𝑁) = 1))
22	18, 21	imbi12d 347	. . . . . 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 3970	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℕ ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ))
24	23	3anbi1d 1466	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
25		raleq 3326	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1))
26		difeq1 4082	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
27	26	raleqdv 3329	. . . . . . . . 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 3339	. . . . . . . 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 1462	. . . . . . 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 15960	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚))
31	30	oveq1d 7426	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁))
32	31	eqeq1d 2771	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) gcd 𝑁) = 1))
33	29, 32	imbi12d 347	. . . . . 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 3970	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝑥 ⊆ ℕ ↔ 𝑀 ⊆ ℕ))
35	34	3anbi1d 1466	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
36		raleq 3326	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∀𝑚 ∈ 𝑥 ((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ 𝑀 ((𝐹‘𝑚) gcd 𝑁) = 1))
37		difeq1 4082	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
38	37	raleqdv 3329	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
39	38	raleqbi1dv 3339	. . . . . . . 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 1462	. . . . . . 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 15960	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑀 (𝐹‘𝑚))
42	41	oveq1d 7426	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁))
43	42	eqeq1d 2771	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((∏𝑚 ∈ 𝑥 (𝐹‘𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ 𝑀 (𝐹‘𝑚) gcd 𝑁) = 1))
44	40, 43	imbi12d 347	. . . . . 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 15996	. . . . . . . . . . 11 ⊢ ∏𝑚 ∈ ∅ (𝐹‘𝑚) = 1
46	45	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹‘𝑚) = 1)
47	46	oveq1d 7426	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = (1 gcd 𝑁))
48		nnz 12611	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
49		1gcd 16590	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (1 gcd 𝑁) = 1)
50	48, 49	syl 18	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (1 gcd 𝑁) = 1)
51	47, 50	eqtrd 2804	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1)
52	51	3ad2ant2 1150	. . . . . . 7 ⊢ ((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∏𝑚 ∈ ∅ (𝐹‘𝑚) gcd 𝑁) = 1)
53	52	3ad2ant1 1149	. . . . . 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 1941	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦))
55		nfcv 2931	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝐹‘𝑧)
56		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
57		unss 4151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ)
58		vex 3467	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
59	58	snss 4755	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ℕ ↔ {𝑧} ⊆ ℕ)
60	59	bilanri 511	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑧 ∈ ℕ)
61	57, 60	sylbir 238	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑧 ∈ ℕ)
62	61	3ad2ant1 1149	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
63	62	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ ℕ)
64		simprr 784	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
65		simpll3 1231	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → 𝐹:ℕ⟶ℕ)
66		simpl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑦 ⊆ ℕ)
67	57, 66	sylbir 238	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ)
68	67	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑦 ⊆ ℕ)
69	68	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ⊆ ℕ)
70	69	sselda 3945	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → 𝑚 ∈ ℕ)
71	65, 70	ffvelcdmd 7081	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℕ)
72	71	nncnd 12248	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℂ)
73		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑧 → (𝐹‘𝑚) = (𝐹‘𝑧))
74		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝐹:ℕ⟶ℕ)
75	61	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
76	74, 75	ffvelcdmd 7081	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹‘𝑧) ∈ ℕ)
77	76	3adant2 1147	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹‘𝑧) ∈ ℕ)
78	77	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹‘𝑧) ∈ ℕ)
79	78	nncnd 12248	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹‘𝑧) ∈ ℂ)
80	54, 55, 56, 63, 64, 72, 73, 79	fprodsplitsn 16042	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)))
81	80	oveq1d 7426	. . . . . . . . . . . . . 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))
82	56, 71	fprodnncl 16008	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ)
83	82	nnzd 12616	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℤ)
84	78	nnzd 12616	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹‘𝑧) ∈ ℤ)
85	83, 84	zmulcld 12705	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)) ∈ ℤ)
86	48	3ad2ant2 1150	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑁 ∈ ℤ)
87	86	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑁 ∈ ℤ)
88	85, 87	gcdcomd 16571	. . . . . . . . . . . . . 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))))
89	81, 88	eqtrd 2804	. . . . . . . . . . . . 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))))
90	89	ex 417	. . . . . . . . . . . 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)))))
91	90	3ad2ant1 1149	. . . . . . . . . . 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)))))
92	91	com12 33	. . . . . . . . . 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)))))
93	92	adantr 485	. . . . . . . . 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)))))
94	93	imp 411	. . . . . . . 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))))
95		simpl2 1209	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑁 ∈ ℕ)
96	95, 82, 78	3jca 1144	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑁 ∈ ℕ ∧ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ ∧ (𝐹‘𝑧) ∈ ℕ))
97	96	ex 417	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝑁 ∈ ℕ ∧ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ ∧ (𝐹‘𝑧) ∈ ℕ)))
98	97	3ad2ant1 1149	. . . . . . . . . . . 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)))
99	98	com12 33	. . . . . . . . . . 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)))
100	99	adantr 485	. . . . . . . . . 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)))
101	100	imp 411	. . . . . . . . 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))
102	87, 83	gcdcomd 16571	. . . . . . . . . . . . . . 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))
103	102	ex 417	. . . . . . . . . . . . . 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)))
104	103	3ad2ant1 1149	. . . . . . . . . . . . 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)))
105	104	com12 33	. . . . . . . . . . . 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)))
106	105	adantr 485	. . . . . . . . . . 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)))
107	106	imp 411	. . . . . . . . . 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))
108	67	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ))
109		idd 25	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝑁 ∈ ℕ → 𝑁 ∈ ℕ))
110		idd 25	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐹:ℕ⟶ℕ → 𝐹:ℕ⟶ℕ))
111	108, 109, 110	3anim123d 1469	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
112		ssun1 4139	. . . . . . . . . . . . . 14 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
113		ssralv 4014	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 → ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1))
114	112, 113	mp1i 14	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 → ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd 𝑁) = 1))
115		ssralv 4014	. . . . . . . . . . . . . . 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))
116	112, 115	mp1i 14	. . . . . . . . . . . . . 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))
117	112	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑚 ∈ 𝑦) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
118	117	ssdifd 4107	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ 𝑚 ∈ 𝑦) → (𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
119		ssralv 4014	. . . . . . . . . . . . . . . 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))
120	118, 119	syl 18	. . . . . . . . . . . . . . 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))
121	120	ralimdva 3183	. . . . . . . . . . . . . 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))
122	116, 121	syld 48	. . . . . . . . . . . . 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))
123	111, 114, 122	3anim123d 1469	. . . . . . . . . . . 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)))
124	123	imim1d 83	. . . . . . . . . . 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)))
125	124	imp31 422	. . . . . . . . . 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)
126	107, 125	eqtrd 2804	. . . . . . . . 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)
127		rpmulgcd 16614	. . . . . . . . 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)))
128	101, 126, 127	syl2anc 595	. . . . . . . 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)))
129		vsnid 4634	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ {𝑧}
130	129	olci 879	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧})
131		elun 4115	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
132	130, 131	mpbir 234	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
133	73	oveq1d 7426	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑧 → ((𝐹‘𝑚) gcd 𝑁) = ((𝐹‘𝑧) gcd 𝑁))
134	133	eqeq1d 2771	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑧 → (((𝐹‘𝑚) gcd 𝑁) = 1 ↔ ((𝐹‘𝑧) gcd 𝑁) = 1))
135	134	rspcv 3586	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 → ((𝐹‘𝑧) gcd 𝑁) = 1))
136	132, 135	mp1i 14	. . . . . . . . . . . 12 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1 → ((𝐹‘𝑧) gcd 𝑁) = 1))
137	136	imp 411	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1) → ((𝐹‘𝑧) gcd 𝑁) = 1)
138	77	nnzd 12616	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹‘𝑧) ∈ ℤ)
139	86, 138	gcdcomd 16571	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑁 gcd (𝐹‘𝑧)) = ((𝐹‘𝑧) gcd 𝑁))
140	139	eqeq1d 2771	. . . . . . . . . . . 12 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑁 gcd (𝐹‘𝑧)) = 1 ↔ ((𝐹‘𝑧) gcd 𝑁) = 1))
141	140	adantr 485	. . . . . . . . . . 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))
142	137, 141	mpbird 260	. . . . . . . . . 10 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹‘𝑚) gcd 𝑁) = 1) → (𝑁 gcd (𝐹‘𝑧)) = 1)
143	142	3adant3 1148	. . . . . . . . 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)
144	143	adantl 486	. . . . . . . 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)
145	94, 128, 144	3eqtrd 2808	. . . . . . 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)
146	145	exp31 424	. . . . . 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)))
147	11, 22, 33, 44, 53, 146	findcard2s 9149	. . . . 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))
148	147	3expd 1370	. . . 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))))
149	148	3expd 1370	. . 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))))))
150	149	3imp 1126	. 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))))
151	150	3imp 1126	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∖ cdif 3910 ∪ cun 3911 ⊆ wss 3913 ∅c0 4294 {csn 4594 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 Fincfn 8942 1c1 11100 · cmul 11104 ℕcn 12232 ℤcz 12590 ∏cprod 15956 gcd cgcd 16551

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-n0 12504 df-z 12591 df-uz 12862 df-rp 13016 df-fz 13535 df-fzo 13682 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-prod 15957 df-dvds 16310 df-gcd 16552

This theorem is referenced by: coprmproddvdslem 16719

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