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Theorem coprmprod 15656
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 3785 . . . . . . . . 9 (𝑥 = ∅ → (𝑥 ⊆ ℕ ↔ ∅ ⊆ ℕ))
213anbi1d 1564 . . . . . . . 8 (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
3 raleq 3285 . . . . . . . 8 (𝑥 = ∅ → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ ∅ ((𝐹𝑚) gcd 𝑁) = 1))
4 difeq1 3882 . . . . . . . . . 10 (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
54raleqdv 3291 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
65raleqbi1dv 3293 . . . . . . . 8 (𝑥 = ∅ → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
72, 3, 63anbi123d 1560 . . . . . . 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 14923 . . . . . . . . 9 (𝑥 = ∅ → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚 ∈ ∅ (𝐹𝑚))
98oveq1d 6856 . . . . . . . 8 (𝑥 = ∅ → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁))
109eqeq1d 2766 . . . . . . 7 (𝑥 = ∅ → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1))
117, 10imbi12d 335 . . . . . 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 3785 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ⊆ ℕ ↔ 𝑦 ⊆ ℕ))
13123anbi1d 1564 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
14 raleq 3285 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1))
15 difeq1 3882 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
1615raleqdv 3291 . . . . . . . . 9 (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
1716raleqbi1dv 3293 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
1813, 14, 173anbi123d 1560 . . . . . . 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 14923 . . . . . . . . 9 (𝑥 = 𝑦 → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚𝑦 (𝐹𝑚))
2019oveq1d 6856 . . . . . . . 8 (𝑥 = 𝑦 → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
2120eqeq1d 2766 . . . . . . 7 (𝑥 = 𝑦 → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1))
2218, 21imbi12d 335 . . . . . 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 3785 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℕ ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ))
24233anbi1d 1564 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
25 raleq 3285 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1))
26 difeq1 3882 . . . . . . . . . 10 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
2726raleqdv 3291 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
2827raleqbi1dv 3293 . . . . . . . 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 1560 . . . . . . 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 14923 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚))
3130oveq1d 6856 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁))
3231eqeq1d 2766 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = 1))
3329, 32imbi12d 335 . . . . . 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 3785 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑥 ⊆ ℕ ↔ 𝑀 ⊆ ℕ))
35343anbi1d 1564 . . . . . . . 8 (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
36 raleq 3285 . . . . . . . 8 (𝑥 = 𝑀 → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1))
37 difeq1 3882 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
3837raleqdv 3291 . . . . . . . . 9 (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
3938raleqbi1dv 3293 . . . . . . . 8 (𝑥 = 𝑀 → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
4035, 36, 393anbi123d 1560 . . . . . . 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 14923 . . . . . . . . 9 (𝑥 = 𝑀 → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚𝑀 (𝐹𝑚))
4241oveq1d 6856 . . . . . . . 8 (𝑥 = 𝑀 → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁))
4342eqeq1d 2766 . . . . . . 7 (𝑥 = 𝑀 → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))
4440, 43imbi12d 335 . . . . . 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 14957 . . . . . . . . . . 11 𝑚 ∈ ∅ (𝐹𝑚) = 1
4645a1i 11 . . . . . . . . . 10 (𝑁 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹𝑚) = 1)
4746oveq1d 6856 . . . . . . . . 9 (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = (1 gcd 𝑁))
48 nnz 11645 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
49 1gcd 15536 . . . . . . . . . 10 (𝑁 ∈ ℤ → (1 gcd 𝑁) = 1)
5048, 49syl 17 . . . . . . . . 9 (𝑁 ∈ ℕ → (1 gcd 𝑁) = 1)
5147, 50eqtrd 2798 . . . . . . . 8 (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)
52513ad2ant2 1164 . . . . . . 7 ((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)
53523ad2ant1 1163 . . . . . 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 2009 . . . . . . . . . . . . . . . 16 𝑚(((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦))
55 nfcv 2906 . . . . . . . . . . . . . . . 16 𝑚(𝐹𝑧)
56 simprl 787 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ∈ Fin)
57 unss 3948 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ)
58 vex 3352 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
5958snss 4469 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ℕ ↔ {𝑧} ⊆ ℕ)
6059biimpri 219 . . . . . . . . . . . . . . . . . . . 20 ({𝑧} ⊆ ℕ → 𝑧 ∈ ℕ)
6160adantl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑧 ∈ ℕ)
6257, 61sylbir 226 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑧 ∈ ℕ)
63623ad2ant1 1163 . . . . . . . . . . . . . . . . 17 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
6463adantr 472 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑧 ∈ ℕ)
65 simprr 789 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ¬ 𝑧𝑦)
66 simpll3 1273 . . . . . . . . . . . . . . . . . 18 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → 𝐹:ℕ⟶ℕ)
67 simpl 474 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑦 ⊆ ℕ)
6857, 67sylbir 226 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ)
69683ad2ant1 1163 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑦 ⊆ ℕ)
7069adantr 472 . . . . . . . . . . . . . . . . . . 19 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ⊆ ℕ)
7170sselda 3760 . . . . . . . . . . . . . . . . . 18 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → 𝑚 ∈ ℕ)
7266, 71ffvelrnd 6549 . . . . . . . . . . . . . . . . 17 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → (𝐹𝑚) ∈ ℕ)
7372nncnd 11291 . . . . . . . . . . . . . . . 16 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → (𝐹𝑚) ∈ ℂ)
74 fveq2 6374 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑧 → (𝐹𝑚) = (𝐹𝑧))
75 simpr 477 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝐹:ℕ⟶ℕ)
7662adantr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
7775, 76ffvelrnd 6549 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℕ)
78773adant2 1161 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℕ)
7978adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℕ)
8079nncnd 11291 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℂ)
8154, 55, 56, 64, 65, 73, 74, 80fprodsplitsn 15003 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) = (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))
8281oveq1d 6856 . . . . . . . . . . . . . 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 14969 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ)
8483nnzd 11727 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚𝑦 (𝐹𝑚) ∈ ℤ)
8579nnzd 11727 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℤ)
8684, 85zmulcld 11734 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) ∈ ℤ)
87483ad2ant2 1164 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑁 ∈ ℤ)
8887adantr 472 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑁 ∈ ℤ)
89 gcdcom 15517 . . . . . . . . . . . . . . 15 (((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9086, 88, 89syl2anc 579 . . . . . . . . . . . . . 14 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9182, 90eqtrd 2798 . . . . . . . . . . . . 13 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9291ex 401 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))))
93923ad2ant1 1163 . . . . . . . . . . 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 472 . . . . . . . . 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 395 . . . . . . . 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 1244 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑁 ∈ ℕ)
9897, 83, 793jca 1158 . . . . . . . . . . . . . 14 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ))
9998ex 401 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ)))
100993ad2ant1 1163 . . . . . . . . . . . 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 472 . . . . . . . . . 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 395 . . . . . . . . 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 15517 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℤ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℤ) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
10588, 84, 104syl2anc 579 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
106105ex 401 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁)))
1071063ad2ant1 1163 . . . . . . . . . . . . 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 472 . . . . . . . . . . 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 395 . . . . . . . . . 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 1567 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
115 ssun1 3937 . . . . . . . . . . . . . 14 𝑦 ⊆ (𝑦 ∪ {𝑧})
116 ssralv 3825 . . . . . . . . . . . . . 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 3825 . . . . . . . . . . . . . . 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 3907 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑚𝑦) → (𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
122 ssralv 3825 . . . . . . . . . . . . . . . 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 3108 . . . . . . . . . . . . . 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 1567 . . . . . . . . . . . 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 408 . . . . . . . . . 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 2798 . . . . . . . . 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 15557 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ) ∧ (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = 1) → (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))) = (𝑁 gcd (𝐹𝑧)))
131103, 129, 130syl2anc 579 . . . . . . . 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 4366 . . . . . . . . . . . . . . 15 𝑧 ∈ {𝑧}
133132olci 892 . . . . . . . . . . . . . 14 (𝑧𝑦𝑧 ∈ {𝑧})
134 elun 3914 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧𝑦𝑧 ∈ {𝑧}))
135133, 134mpbir 222 . . . . . . . . . . . . 13 𝑧 ∈ (𝑦 ∪ {𝑧})
13674oveq1d 6856 . . . . . . . . . . . . . . 15 (𝑚 = 𝑧 → ((𝐹𝑚) gcd 𝑁) = ((𝐹𝑧) gcd 𝑁))
137136eqeq1d 2766 . . . . . . . . . . . . . 14 (𝑚 = 𝑧 → (((𝐹𝑚) gcd 𝑁) = 1 ↔ ((𝐹𝑧) gcd 𝑁) = 1))
138137rspcv 3456 . . . . . . . . . . . . 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 395 . . . . . . . . . . 11 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1) → ((𝐹𝑧) gcd 𝑁) = 1)
14178nnzd 11727 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℤ)
142 gcdcom 15517 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℤ ∧ (𝐹𝑧) ∈ ℤ) → (𝑁 gcd (𝐹𝑧)) = ((𝐹𝑧) gcd 𝑁))
14387, 141, 142syl2anc 579 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑁 gcd (𝐹𝑧)) = ((𝐹𝑧) gcd 𝑁))
144143eqeq1d 2766 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑁 gcd (𝐹𝑧)) = 1 ↔ ((𝐹𝑧) gcd 𝑁) = 1))
145144adantr 472 . . . . . . . . . . 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 248 . . . . . . . . . 10 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1) → (𝑁 gcd (𝐹𝑧)) = 1)
1471463adant3 1162 . . . . . . . . 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 473 . . . . . . . 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 2802 . . . . . . 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 410 . . . . . 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 8407 . . . . 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 1462 . . . 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 1462 . . 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 1137 . 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 1137 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wral 3054  cdif 3728  cun 3729  wss 3731  c0 4078  {csn 4333  wf 6063  cfv 6067  (class class class)co 6841  Fincfn 8159  1c1 10189   · cmul 10193  cn 11273  cz 11623  cprod 14919   gcd cgcd 15498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-inf2 8752  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265  ax-pre-sup 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-se 5236  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-isom 6076  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-oadd 7767  df-er 7946  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-sup 8554  df-inf 8555  df-oi 8621  df-card 9015  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-div 10938  df-nn 11274  df-2 11334  df-3 11335  df-n0 11538  df-z 11624  df-uz 11886  df-rp 12028  df-fz 12533  df-fzo 12673  df-fl 12800  df-mod 12876  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14125  df-re 14126  df-im 14127  df-sqrt 14261  df-abs 14262  df-clim 14505  df-prod 14920  df-dvds 15267  df-gcd 15499
This theorem is referenced by:  coprmproddvdslem  15657
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