Theorem coprmproddvds 16571

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 14886	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ (∅ ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
2		difeq1 4069	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
3	2	raleqdv 3292	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
4	3	raleqbi1dv 3304	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
5		raleq 3289	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∀𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾))
6	4, 5	anbi12d 632	. . . . . . 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 632	. . . . . 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 15811	. . . . . . 7 ⊢ (𝑥 = ∅ → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ ∅ (𝐹‘𝑚))
9	8	breq1d 5101	. . . . . 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 14886	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ (𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
12		difeq1 4069	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
13	12	raleqdv 3292	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
14	13	raleqbi1dv 3304	. . . . . . . 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 3289	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
16	14, 15	anbi12d 632	. . . . . . 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 632	. . . . . 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 15811	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑦 (𝐹‘𝑚))
19	18	breq1d 5101	. . . . . 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 14886	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
22		difeq1 4069	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
23	22	raleqdv 3292	. . . . . . . . 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 3304	. . . . . . . 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 3289	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))
26	24, 25	anbi12d 632	. . . . . . 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 632	. . . . . 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 15811	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚))
29	28	breq1d 5101	. . . . . 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 14886	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ (𝑀 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
32		difeq1 4069	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
33	32	raleqdv 3292	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
34	33	raleqbi1dv 3304	. . . . . . . 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 3289	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∀𝑚 ∈ 𝑥 (𝐹‘𝑚) ∥ 𝐾 ↔ ∀𝑚 ∈ 𝑀 (𝐹‘𝑚) ∥ 𝐾))
36	34, 35	anbi12d 632	. . . . . . 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 632	. . . . . 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 15811	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑀 (𝐹‘𝑚))
39	38	breq1d 5101	. . . . . 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 15847	. . . . . . . 8 ⊢ ∏𝑚 ∈ ∅ (𝐹‘𝑚) = 1
42		nnz 12486	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ)
43		1dvds 16178	. . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → 1 ∥ 𝐾)
44	42, 43	syl 17	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → 1 ∥ 𝐾)
45	41, 44	eqbrtrid 5126	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)
46	45	adantr 480	. . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ∏𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)
47	46	ad2antlr 727	. . . . 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 16570	. . . . 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 9075	. . . 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 432	. . 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 407	. 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 1110	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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  ∅c0 4283  {csn 4576   class class class wbr 5091  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Fincfn 8869  1c1 11004  ℕcn 12122  ℤcz 12465  ∏cprod 15807   ∥ cdvds 16160   gcd cgcd 16402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-pre-sup 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-oi 9396  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-div 11772  df-nn 12123  df-2 12185  df-3 12186  df-n0 12379  df-z 12466  df-uz 12730  df-rp 12888  df-fz 13405  df-fzo 13552  df-fl 13693  df-mod 13771  df-seq 13906  df-exp 13966  df-hash 14235  df-cj 15003  df-re 15004  df-im 15005  df-sqrt 15139  df-abs 15140  df-clim 15392  df-prod 15808  df-dvds 16161  df-gcd 16403

This theorem is referenced by:  prmodvdslcmf  16956