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Theorem coprmprod 15982
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 ∧ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1) → (∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → (∏𝑚𝑀 (𝐹𝑚) 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.
StepHypRef Expression
1 sseq1 3968 . . . . . . . . 9 (𝑥 = ∅ → (𝑥 ⊆ ℕ ↔ ∅ ⊆ ℕ))
213anbi1d 1437 . . . . . . . 8 (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
3 raleq 3390 . . . . . . . 8 (𝑥 = ∅ → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ ∅ ((𝐹𝑚) gcd 𝑁) = 1))
4 difeq1 4068 . . . . . . . . . 10 (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
54raleqdv 3396 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
65raleqbi1dv 3388 . . . . . . . 8 (𝑥 = ∅ → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
72, 3, 63anbi123d 1433 . . . . . . 7 (𝑥 = ∅ → (((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) ↔ ((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ ∅ ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)))
8 prodeq1 15242 . . . . . . . . 9 (𝑥 = ∅ → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚 ∈ ∅ (𝐹𝑚))
98oveq1d 7145 . . . . . . . 8 (𝑥 = ∅ → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁))
109eqeq1d 2823 . . . . . . 7 (𝑥 = ∅ → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1))
117, 10imbi12d 348 . . . . . 6 (𝑥 = ∅ → ((((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1) ↔ (((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ ∅ ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)))
12 sseq1 3968 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ⊆ ℕ ↔ 𝑦 ⊆ ℕ))
13123anbi1d 1437 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
14 raleq 3390 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1))
15 difeq1 4068 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
1615raleqdv 3396 . . . . . . . . 9 (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
1716raleqbi1dv 3388 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
1813, 14, 173anbi123d 1433 . . . . . . 7 (𝑥 = 𝑦 → (((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) ↔ ((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)))
19 prodeq1 15242 . . . . . . . . 9 (𝑥 = 𝑦 → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚𝑦 (𝐹𝑚))
2019oveq1d 7145 . . . . . . . 8 (𝑥 = 𝑦 → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
2120eqeq1d 2823 . . . . . . 7 (𝑥 = 𝑦 → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1))
2218, 21imbi12d 348 . . . . . 6 (𝑥 = 𝑦 → ((((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1) ↔ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)))
23 sseq1 3968 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℕ ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ))
24233anbi1d 1437 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
25 raleq 3390 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1))
26 difeq1 4068 . . . . . . . . . 10 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
2726raleqdv 3396 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
2827raleqbi1dv 3388 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
2924, 25, 283anbi123d 1433 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) ↔ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)))
30 prodeq1 15242 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚))
3130oveq1d 7145 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁))
3231eqeq1d 2823 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = 1))
3329, 32imbi12d 348 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1) ↔ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = 1)))
34 sseq1 3968 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑥 ⊆ ℕ ↔ 𝑀 ⊆ ℕ))
35343anbi1d 1437 . . . . . . . 8 (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
36 raleq 3390 . . . . . . . 8 (𝑥 = 𝑀 → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1))
37 difeq1 4068 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
3837raleqdv 3396 . . . . . . . . 9 (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
3938raleqbi1dv 3388 . . . . . . . 8 (𝑥 = 𝑀 → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
4035, 36, 393anbi123d 1433 . . . . . . 7 (𝑥 = 𝑀 → (((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) ↔ ((𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)))
41 prodeq1 15242 . . . . . . . . 9 (𝑥 = 𝑀 → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚𝑀 (𝐹𝑚))
4241oveq1d 7145 . . . . . . . 8 (𝑥 = 𝑀 → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁))
4342eqeq1d 2823 . . . . . . 7 (𝑥 = 𝑀 → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))
4440, 43imbi12d 348 . . . . . 6 (𝑥 = 𝑀 → ((((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1) ↔ (((𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1)))
45 prod0 15276 . . . . . . . . . . 11 𝑚 ∈ ∅ (𝐹𝑚) = 1
4645a1i 11 . . . . . . . . . 10 (𝑁 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹𝑚) = 1)
4746oveq1d 7145 . . . . . . . . 9 (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = (1 gcd 𝑁))
48 nnz 11982 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
49 1gcd 15858 . . . . . . . . . 10 (𝑁 ∈ ℤ → (1 gcd 𝑁) = 1)
5048, 49syl 17 . . . . . . . . 9 (𝑁 ∈ ℕ → (1 gcd 𝑁) = 1)
5147, 50eqtrd 2856 . . . . . . . 8 (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)
52513ad2ant2 1131 . . . . . . 7 ((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)
53523ad2ant1 1130 . . . . . 6 (((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ ∅ ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)
54 nfv 1916 . . . . . . . . . . . . . . . 16 𝑚(((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦))
55 nfcv 2974 . . . . . . . . . . . . . . . 16 𝑚(𝐹𝑧)
56 simprl 770 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ∈ Fin)
57 unss 4136 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ)
58 vex 3474 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
5958snss 4691 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ℕ ↔ {𝑧} ⊆ ℕ)
6059biimpri 231 . . . . . . . . . . . . . . . . . . . 20 ({𝑧} ⊆ ℕ → 𝑧 ∈ ℕ)
6160adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑧 ∈ ℕ)
6257, 61sylbir 238 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑧 ∈ ℕ)
63623ad2ant1 1130 . . . . . . . . . . . . . . . . 17 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
6463adantr 484 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑧 ∈ ℕ)
65 simprr 772 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ¬ 𝑧𝑦)
66 simpll3 1211 . . . . . . . . . . . . . . . . . 18 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → 𝐹:ℕ⟶ℕ)
67 simpl 486 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑦 ⊆ ℕ)
6857, 67sylbir 238 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ)
69683ad2ant1 1130 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑦 ⊆ ℕ)
7069adantr 484 . . . . . . . . . . . . . . . . . . 19 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ⊆ ℕ)
7170sselda 3943 . . . . . . . . . . . . . . . . . 18 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → 𝑚 ∈ ℕ)
7266, 71ffvelrnd 6825 . . . . . . . . . . . . . . . . 17 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → (𝐹𝑚) ∈ ℕ)
7372nncnd 11631 . . . . . . . . . . . . . . . 16 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → (𝐹𝑚) ∈ ℂ)
74 fveq2 6643 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑧 → (𝐹𝑚) = (𝐹𝑧))
75 simpr 488 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝐹:ℕ⟶ℕ)
7662adantr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
7775, 76ffvelrnd 6825 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℕ)
78773adant2 1128 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℕ)
7978adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℕ)
8079nncnd 11631 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℂ)
8154, 55, 56, 64, 65, 73, 74, 80fprodsplitsn 15322 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) = (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))
8281oveq1d 7145 . . . . . . . . . . . . . 14 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = ((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) gcd 𝑁))
8356, 72fprodnncl 15288 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ)
8483nnzd 12064 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚𝑦 (𝐹𝑚) ∈ ℤ)
8579nnzd 12064 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℤ)
8684, 85zmulcld 12071 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) ∈ ℤ)
87483ad2ant2 1131 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑁 ∈ ℤ)
8887adantr 484 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑁 ∈ ℤ)
89 gcdcom 15839 . . . . . . . . . . . . . . 15 (((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9086, 88, 89syl2anc 587 . . . . . . . . . . . . . 14 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9182, 90eqtrd 2856 . . . . . . . . . . . . 13 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9291ex 416 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))))
93923ad2ant1 1130 . . . . . . . . . . 11 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))))
9493com12 32 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))))
9594adantr 484 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))))
9695imp 410 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
97 simpl2 1189 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑁 ∈ ℕ)
9897, 83, 793jca 1125 . . . . . . . . . . . . . 14 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ))
9998ex 416 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ)))
100993ad2ant1 1130 . . . . . . . . . . . 12 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ)))
101100com12 32 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ)))
102101adantr 484 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ)))
103102imp 410 . . . . . . . . 9 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ))
104 gcdcom 15839 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℤ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℤ) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
10588, 84, 104syl2anc 587 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
106105ex 416 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁)))
1071063ad2ant1 1130 . . . . . . . . . . . . 13 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁)))
108107com12 32 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁)))
109108adantr 484 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁)))
110109imp 410 . . . . . . . . . 10 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
11168a1i 11 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ))
112 idd 24 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 ∈ ℕ → 𝑁 ∈ ℕ))
113 idd 24 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝐹:ℕ⟶ℕ → 𝐹:ℕ⟶ℕ))
114111, 112, 1133anim123d 1440 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
115 ssun1 4124 . . . . . . . . . . . . . 14 𝑦 ⊆ (𝑦 ∪ {𝑧})
116 ssralv 4009 . . . . . . . . . . . . . 14 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 → ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1))
117115, 116mp1i 13 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 → ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1))
118 ssralv 4009 . . . . . . . . . . . . . . 15 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑚𝑦𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
119115, 118mp1i 13 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑚𝑦𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
120115a1i 11 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑚𝑦) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
121120ssdifd 4093 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑚𝑦) → (𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
122 ssralv 4009 . . . . . . . . . . . . . . . 16 ((𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}) → (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
123121, 122syl 17 . . . . . . . . . . . . . . 15 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑚𝑦) → (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
124123ralimdva 3165 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑚𝑦𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
125119, 124syld 47 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
126114, 117, 1253anim123d 1440 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → ((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)))
127126imim1d 82 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)))
128127imp31 421 . . . . . . . . . 10 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)
129110, 128eqtrd 2856 . . . . . . . . 9 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = 1)
130 rpmulgcd 15883 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ) ∧ (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = 1) → (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))) = (𝑁 gcd (𝐹𝑧)))
131103, 129, 130syl2anc 587 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)) → (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))) = (𝑁 gcd (𝐹𝑧)))
132 vsnid 4575 . . . . . . . . . . . . . . 15 𝑧 ∈ {𝑧}
133132olci 863 . . . . . . . . . . . . . 14 (𝑧𝑦𝑧 ∈ {𝑧})
134 elun 4101 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧𝑦𝑧 ∈ {𝑧}))
135133, 134mpbir 234 . . . . . . . . . . . . 13 𝑧 ∈ (𝑦 ∪ {𝑧})
13674oveq1d 7145 . . . . . . . . . . . . . . 15 (𝑚 = 𝑧 → ((𝐹𝑚) gcd 𝑁) = ((𝐹𝑧) gcd 𝑁))
137136eqeq1d 2823 . . . . . . . . . . . . . 14 (𝑚 = 𝑧 → (((𝐹𝑚) gcd 𝑁) = 1 ↔ ((𝐹𝑧) gcd 𝑁) = 1))
138137rspcv 3595 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 → ((𝐹𝑧) gcd 𝑁) = 1))
139135, 138mp1i 13 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 → ((𝐹𝑧) gcd 𝑁) = 1))
140139imp 410 . . . . . . . . . . 11 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1) → ((𝐹𝑧) gcd 𝑁) = 1)
14178nnzd 12064 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℤ)
142 gcdcom 15839 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℤ ∧ (𝐹𝑧) ∈ ℤ) → (𝑁 gcd (𝐹𝑧)) = ((𝐹𝑧) gcd 𝑁))
14387, 141, 142syl2anc 587 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑁 gcd (𝐹𝑧)) = ((𝐹𝑧) gcd 𝑁))
144143eqeq1d 2823 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑁 gcd (𝐹𝑧)) = 1 ↔ ((𝐹𝑧) gcd 𝑁) = 1))
145144adantr 484 . . . . . . . . . . 11 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1) → ((𝑁 gcd (𝐹𝑧)) = 1 ↔ ((𝐹𝑧) gcd 𝑁) = 1))
146140, 145mpbird 260 . . . . . . . . . 10 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1) → (𝑁 gcd (𝐹𝑧)) = 1)
1471463adant3 1129 . . . . . . . . 9 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (𝑁 gcd (𝐹𝑧)) = 1)
148147adantl 485 . . . . . . . 8 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)) → (𝑁 gcd (𝐹𝑧)) = 1)
14996, 131, 1483eqtrd 2860 . . . . . . 7 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = 1)
150149exp31 423 . . . . . 6 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((((𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = 1)))
15111, 22, 33, 44, 53, 150findcard2s 8735 . . . . 5 (𝑀 ∈ Fin → (((𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1 ∧ ∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1) → (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))
1521513expd 1350 . . . 4 (𝑀 ∈ Fin → ((𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1 → (∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))))
1531523expd 1350 . . 3 (𝑀 ∈ Fin → (𝑀 ⊆ ℕ → (𝑁 ∈ ℕ → (𝐹:ℕ⟶ℕ → (∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1 → (∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))))))
1541533imp 1108 . 2 ((𝑀 ∈ Fin ∧ 𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ) → (𝐹:ℕ⟶ℕ → (∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1 → (∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))))
1551543imp 1108 1 (((𝑀 ∈ Fin ∧ 𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝐹:ℕ⟶ℕ ∧ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1) → (∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2115  wral 3126  cdif 3907  cun 3908  wss 3910  c0 4266  {csn 4540  wf 6324  cfv 6328  (class class class)co 7130  Fincfn 8484  1c1 10515   · cmul 10519  cn 11615  cz 11959  cprod 15238   gcd cgcd 15820
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-sup 8882  df-inf 8883  df-oi 8950  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-3 11679  df-n0 11876  df-z 11960  df-uz 12222  df-rp 12368  df-fz 12876  df-fzo 13017  df-fl 13145  df-mod 13221  df-seq 13353  df-exp 13414  df-hash 13675  df-cj 14437  df-re 14438  df-im 14439  df-sqrt 14573  df-abs 14574  df-clim 14824  df-prod 15239  df-dvds 15587  df-gcd 15821
This theorem is referenced by:  coprmproddvdslem  15983
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