Theorem coprmproddvds 16539

Description: If a positive integer is divisible by each element of a set of pairwise coprime positive integers, then it is divisible by their product. (Contributed by AV, 19-Aug-2020.)

Assertion

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

Distinct variable groups:   𝑚,𝐹,𝑛   𝑚,𝐾   𝑚,𝑀,𝑛

Allowed substitution hint:   𝐾(𝑛)

Proof of Theorem coprmproddvds

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cleq1lem 14867	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ (∅ ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
2		difeq1 4075	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
3	2	raleqdv 3313	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
4	3	raleqbi1dv 3307	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
5		raleq 3309	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∀𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾))
6	4, 5	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = ∅ → ((∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾) ↔ (∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)))
7	1, 6	anbi12d 631	. . . . . 6 ⊢ (𝑥 = ∅ → (((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾)) ↔ ((∅ ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾))))
8		prodeq1 15792	. . . . . . 7 ⊢ (𝑥 = ∅ → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ ∅ (𝐹‘𝑚))
9	8	breq1d 5115	. . . . . 6 ⊢ (𝑥 = ∅ → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∏𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾))
10	7, 9	imbi12d 344	. . . . 5 ⊢ (𝑥 = ∅ → ((((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾) ↔ (((∅ ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)))
11		cleq1lem 14867	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ (𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
12		difeq1 4075	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
13	12	raleqdv 3313	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
14	13	raleqbi1dv 3307	. . . . . . . 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))
15		raleq 3309	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
16	14, 15	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾) ↔ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)))
17	11, 16	anbi12d 631	. . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾)) ↔ ((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))))
18		prodeq1 15792	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑦 (𝐹‘𝑚))
19	18	breq1d 5115	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
20	17, 19	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑦 → ((((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾) ↔ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)))
21		cleq1lem 14867	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
22		difeq1 4075	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
23	22	raleqdv 3313	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
24	23	raleqbi1dv 3307	. . . . . . . 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))
25		raleq 3309	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))
26	24, 25	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾) ↔ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)))
27	21, 26	anbi12d 631	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾)) ↔ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))))
28		prodeq1 15792	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚))
29	28	breq1d 5115	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))
30	27, 29	imbi12d 344	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾) ↔ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)))
31		cleq1lem 14867	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ (𝑀 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
32		difeq1 4075	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
33	32	raleqdv 3313	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
34	33	raleqbi1dv 3307	. . . . . . . 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))
35		raleq 3309	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾))
36	34, 35	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾) ↔ (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾)))
37	31, 36	anbi12d 631	. . . . . 6 ⊢ (𝑥 = 𝑀 → (((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾)) ↔ ((𝑀 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾))))
38		prodeq1 15792	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑀 (𝐹‘𝑚))
39	38	breq1d 5115	. . . . . 6 ⊢ (𝑥 = 𝑀 → (∏𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∏𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾))
40	37, 39	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑀 → ((((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾) ↔ (((𝑀 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾)))
41		prod0 15826	. . . . . . . 8 ⊢ ∏𝑚 ∈ ∅ (𝐹‘𝑚) = 1
42		nnz 12520	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ)
43		1dvds 16153	. . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → 1 ∥ 𝐾)
44	42, 43	syl 17	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → 1 ∥ 𝐾)
45	41, 44	eqbrtrid 5140	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)
46	45	adantr 481	. . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ∏𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)
47	46	ad2antlr 725	. . . . 5 ⊢ (((∅ ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)
48		coprmproddvdslem 16538	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)))
49	10, 20, 30, 40, 47, 48	findcard2s 9109	. . . 4 ⊢ (𝑀 ∈ Fin → (((𝑀 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾))
50	49	exp4c 433	. . 3 ⊢ (𝑀 ∈ Fin → (𝑀 ⊆ ℕ → ((𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾))))
51	50	impcom 408	. 2 ⊢ ((𝑀 ⊆ ℕ ∧ 𝑀 ∈ Fin) → ((𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾)))
52	51	3imp 1111	1 ⊢ (((𝑀 ⊆ ℕ ∧ 𝑀 ∈ Fin) ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (∀𝑚 ∈ 𝑀 ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ∖ cdif 3907   ∪ cun 3908   ⊆ wss 3910  ∅c0 4282  {csn 4586   class class class wbr 5105  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  1c1 11052  ℕcn 12153  ℤcz 12499  ∏cprod 15788   ∥ cdvds 16136   gcd cgcd 16374

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-prod 15789  df-dvds 16137  df-gcd 16375

This theorem is referenced by:  prmodvdslcmf  16919