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Theorem coprmprod 15306
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 3610 . . . . . . . . 9 (𝑥 = ∅ → (𝑥 ⊆ ℕ ↔ ∅ ⊆ ℕ))
213anbi1d 1400 . . . . . . . 8 (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
3 raleq 3130 . . . . . . . 8 (𝑥 = ∅ → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ ∅ ((𝐹𝑚) gcd 𝑁) = 1))
4 difeq1 3704 . . . . . . . . . 10 (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
54raleqdv 3136 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
65raleqbi1dv 3138 . . . . . . . 8 (𝑥 = ∅ → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
72, 3, 63anbi123d 1396 . . . . . . 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 14571 . . . . . . . . 9 (𝑥 = ∅ → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚 ∈ ∅ (𝐹𝑚))
98oveq1d 6625 . . . . . . . 8 (𝑥 = ∅ → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁))
109eqeq1d 2623 . . . . . . 7 (𝑥 = ∅ → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1))
117, 10imbi12d 334 . . . . . 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 3610 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ⊆ ℕ ↔ 𝑦 ⊆ ℕ))
13123anbi1d 1400 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
14 raleq 3130 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1))
15 difeq1 3704 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
1615raleqdv 3136 . . . . . . . . 9 (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
1716raleqbi1dv 3138 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
1813, 14, 173anbi123d 1396 . . . . . . 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 14571 . . . . . . . . 9 (𝑥 = 𝑦 → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚𝑦 (𝐹𝑚))
2019oveq1d 6625 . . . . . . . 8 (𝑥 = 𝑦 → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
2120eqeq1d 2623 . . . . . . 7 (𝑥 = 𝑦 → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1))
2218, 21imbi12d 334 . . . . . 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 3610 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℕ ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ))
24233anbi1d 1400 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
25 raleq 3130 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1))
26 difeq1 3704 . . . . . . . . . 10 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
2726raleqdv 3136 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
2827raleqbi1dv 3138 . . . . . . . 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 1396 . . . . . . 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 14571 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚))
3130oveq1d 6625 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁))
3231eqeq1d 2623 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = 1))
3329, 32imbi12d 334 . . . . . 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 3610 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑥 ⊆ ℕ ↔ 𝑀 ⊆ ℕ))
35343anbi1d 1400 . . . . . . . 8 (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
36 raleq 3130 . . . . . . . 8 (𝑥 = 𝑀 → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1))
37 difeq1 3704 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
3837raleqdv 3136 . . . . . . . . 9 (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
3938raleqbi1dv 3138 . . . . . . . 8 (𝑥 = 𝑀 → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
4035, 36, 393anbi123d 1396 . . . . . . 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 14571 . . . . . . . . 9 (𝑥 = 𝑀 → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚𝑀 (𝐹𝑚))
4241oveq1d 6625 . . . . . . . 8 (𝑥 = 𝑀 → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁))
4342eqeq1d 2623 . . . . . . 7 (𝑥 = 𝑀 → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))
4440, 43imbi12d 334 . . . . . 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 14605 . . . . . . . . . . 11 𝑚 ∈ ∅ (𝐹𝑚) = 1
4645a1i 11 . . . . . . . . . 10 (𝑁 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹𝑚) = 1)
4746oveq1d 6625 . . . . . . . . 9 (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = (1 gcd 𝑁))
48 nnz 11350 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
49 1gcd 15185 . . . . . . . . . 10 (𝑁 ∈ ℤ → (1 gcd 𝑁) = 1)
5048, 49syl 17 . . . . . . . . 9 (𝑁 ∈ ℕ → (1 gcd 𝑁) = 1)
5147, 50eqtrd 2655 . . . . . . . 8 (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)
52513ad2ant2 1081 . . . . . . 7 ((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)
53523ad2ant1 1080 . . . . . 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 1840 . . . . . . . . . . . . . . . 16 𝑚(((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦))
55 nfcv 2761 . . . . . . . . . . . . . . . 16 𝑚(𝐹𝑧)
56 simprl 793 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ∈ Fin)
57 unss 3770 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ)
58 vex 3192 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
5958snss 4291 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ℕ ↔ {𝑧} ⊆ ℕ)
6059biimpri 218 . . . . . . . . . . . . . . . . . . . 20 ({𝑧} ⊆ ℕ → 𝑧 ∈ ℕ)
6160adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑧 ∈ ℕ)
6257, 61sylbir 225 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑧 ∈ ℕ)
63623ad2ant1 1080 . . . . . . . . . . . . . . . . 17 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
6463adantr 481 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑧 ∈ ℕ)
65 simprr 795 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ¬ 𝑧𝑦)
66 simpll3 1100 . . . . . . . . . . . . . . . . . 18 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → 𝐹:ℕ⟶ℕ)
67 simpl 473 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑦 ⊆ ℕ)
6857, 67sylbir 225 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ)
69683ad2ant1 1080 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑦 ⊆ ℕ)
7069adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ⊆ ℕ)
7170sselda 3587 . . . . . . . . . . . . . . . . . 18 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → 𝑚 ∈ ℕ)
7266, 71ffvelrnd 6321 . . . . . . . . . . . . . . . . 17 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → (𝐹𝑚) ∈ ℕ)
7372nncnd 10987 . . . . . . . . . . . . . . . 16 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → (𝐹𝑚) ∈ ℂ)
74 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑧 → (𝐹𝑚) = (𝐹𝑧))
75 simpr 477 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝐹:ℕ⟶ℕ)
7662adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
7775, 76ffvelrnd 6321 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℕ)
78773adant2 1078 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℕ)
7978adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℕ)
8079nncnd 10987 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℂ)
8154, 55, 56, 64, 65, 73, 74, 80fprodsplitsn 14652 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) = (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))
8281oveq1d 6625 . . . . . . . . . . . . . 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 14617 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ)
8483nnzd 11432 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚𝑦 (𝐹𝑚) ∈ ℤ)
8579nnzd 11432 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℤ)
8684, 85zmulcld 11439 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) ∈ ℤ)
87483ad2ant2 1081 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑁 ∈ ℤ)
8887adantr 481 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑁 ∈ ℤ)
89 gcdcom 15166 . . . . . . . . . . . . . . 15 (((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9086, 88, 89syl2anc 692 . . . . . . . . . . . . . 14 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9182, 90eqtrd 2655 . . . . . . . . . . . . 13 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9291ex 450 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))))
93923ad2ant1 1080 . . . . . . . . . . 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 481 . . . . . . . . 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 445 . . . . . . . 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 1063 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑁 ∈ ℕ)
9897, 83, 793jca 1240 . . . . . . . . . . . . . 14 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ))
9998ex 450 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ)))
100993ad2ant1 1080 . . . . . . . . . . . 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 481 . . . . . . . . . 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 445 . . . . . . . . 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 15166 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℤ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℤ) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
10588, 84, 104syl2anc 692 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
106105ex 450 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁)))
1071063ad2ant1 1080 . . . . . . . . . . . . 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 481 . . . . . . . . . . 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 445 . . . . . . . . . 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 1403 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
115 ssun1 3759 . . . . . . . . . . . . . 14 𝑦 ⊆ (𝑦 ∪ {𝑧})
116 ssralv 3650 . . . . . . . . . . . . . 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 3650 . . . . . . . . . . . . . . 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 3729 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑚𝑦) → (𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
122 ssralv 3650 . . . . . . . . . . . . . . . 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 2957 . . . . . . . . . . . . . 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 1403 . . . . . . . . . . . 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 448 . . . . . . . . . 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 2655 . . . . . . . . 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 15206 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ) ∧ (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = 1) → (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))) = (𝑁 gcd (𝐹𝑧)))
131103, 129, 130syl2anc 692 . . . . . . . 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 4185 . . . . . . . . . . . . . . 15 𝑧 ∈ {𝑧}
133132olci 406 . . . . . . . . . . . . . 14 (𝑧𝑦𝑧 ∈ {𝑧})
134 elun 3736 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧𝑦𝑧 ∈ {𝑧}))
135133, 134mpbir 221 . . . . . . . . . . . . 13 𝑧 ∈ (𝑦 ∪ {𝑧})
13674oveq1d 6625 . . . . . . . . . . . . . . 15 (𝑚 = 𝑧 → ((𝐹𝑚) gcd 𝑁) = ((𝐹𝑧) gcd 𝑁))
137136eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑚 = 𝑧 → (((𝐹𝑚) gcd 𝑁) = 1 ↔ ((𝐹𝑧) gcd 𝑁) = 1))
138137rspcv 3294 . . . . . . . . . . . . 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 445 . . . . . . . . . . 11 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1) → ((𝐹𝑧) gcd 𝑁) = 1)
14178nnzd 11432 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℤ)
142 gcdcom 15166 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℤ ∧ (𝐹𝑧) ∈ ℤ) → (𝑁 gcd (𝐹𝑧)) = ((𝐹𝑧) gcd 𝑁))
14387, 141, 142syl2anc 692 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑁 gcd (𝐹𝑧)) = ((𝐹𝑧) gcd 𝑁))
144143eqeq1d 2623 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑁 gcd (𝐹𝑧)) = 1 ↔ ((𝐹𝑧) gcd 𝑁) = 1))
145144adantr 481 . . . . . . . . . . 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 247 . . . . . . . . . 10 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1) → (𝑁 gcd (𝐹𝑧)) = 1)
1471463adant3 1079 . . . . . . . . 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 482 . . . . . . . 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 2659 . . . . . . 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 629 . . . . . 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 8152 . . . . 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 1281 . . . 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 1281 . . 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 1254 . 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 1254 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 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cdif 3556  cun 3557  wss 3559  c0 3896  {csn 4153  wf 5848  cfv 5852  (class class class)co 6610  Fincfn 7906  1c1 9888   · cmul 9892  cn 10971  cz 11328  cprod 14567   gcd cgcd 15147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-z 11329  df-uz 11639  df-rp 11784  df-fz 12276  df-fzo 12414  df-fl 12540  df-mod 12616  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-prod 14568  df-dvds 14915  df-gcd 15148
This theorem is referenced by:  coprmproddvdslem  15307
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