Theorem coprmproddvds 16710

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 15031	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ (∅ ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
2		difeq1 4142	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
3	2	raleqdv 3334	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
4	3	raleqbi1dv 3346	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∀𝑚 ∈ 𝑥 ∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
5		raleq 3331	. . . . . . . 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 15955	. . . . . . 7 ⊢ (𝑥 = ∅ → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ ∅ (𝐹‘𝑚))
9	8	breq1d 5176	. . . . . 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 15031	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ (𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
12		difeq1 4142	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
13	12	raleqdv 3334	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
14	13	raleqbi1dv 3346	. . . . . . . 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 3331	. . . . . . . 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 15955	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑦 (𝐹‘𝑚))
19	18	breq1d 5176	. . . . . 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 15031	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
22		difeq1 4142	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
23	22	raleqdv 3334	. . . . . . . . 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 3346	. . . . . . . 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 3331	. . . . . . . 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 15955	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚))
29	28	breq1d 5176	. . . . . 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 15031	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ↔ (𝑀 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))))
32		difeq1 4142	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
33	32	raleqdv 3334	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
34	33	raleqbi1dv 3346	. . . . . . . 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 3331	. . . . . . . 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 15955	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ∏𝑚 ∈ 𝑥 (𝐹‘𝑚) = ∏𝑚 ∈ 𝑀 (𝐹‘𝑚))
39	38	breq1d 5176	. . . . . 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 15991	. . . . . . . 8 ⊢ ∏𝑚 ∈ ∅ (𝐹‘𝑚) = 1
42		nnz 12660	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ)
43		1dvds 16319	. . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → 1 ∥ 𝐾)
44	42, 43	syl 17	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → 1 ∥ 𝐾)
45	41, 44	eqbrtrid 5201	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)
46	45	adantr 480	. . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ∏𝑚 ∈ ∅ (𝐹‘𝑚) ∥ 𝐾)
47	46	ad2antlr 726	. . . . 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 16709	. . . . 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 9231	. . . 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 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 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∖ cdif 3973   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  {csn 4648   class class class wbr 5166  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  1c1 11185  ℕcn 12293  ℤcz 12639  ∏cprod 15951   ∥ cdvds 16302   gcd cgcd 16540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-fl 13843  df-mod 13921  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-prod 15952  df-dvds 16303  df-gcd 16541

This theorem is referenced by:  prmodvdslcmf  17094