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Theorem coprmprod 16537
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 3969 . . . . . . . . 9 (𝑥 = ∅ → (𝑥 ⊆ ℕ ↔ ∅ ⊆ ℕ))
213anbi1d 1440 . . . . . . . 8 (𝑥 = ∅ → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
3 raleq 3309 . . . . . . . 8 (𝑥 = ∅ → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ ∅ ((𝐹𝑚) gcd 𝑁) = 1))
4 difeq1 4075 . . . . . . . . . 10 (𝑥 = ∅ → (𝑥 ∖ {𝑚}) = (∅ ∖ {𝑚}))
54raleqdv 3313 . . . . . . . . 9 (𝑥 = ∅ → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
65raleqbi1dv 3307 . . . . . . . 8 (𝑥 = ∅ → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚 ∈ ∅ ∀𝑛 ∈ (∅ ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
72, 3, 63anbi123d 1436 . . . . . . 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 15792 . . . . . . . . 9 (𝑥 = ∅ → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚 ∈ ∅ (𝐹𝑚))
98oveq1d 7372 . . . . . . . 8 (𝑥 = ∅ → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁))
109eqeq1d 2738 . . . . . . 7 (𝑥 = ∅ → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1))
117, 10imbi12d 344 . . . . . 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 3969 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ⊆ ℕ ↔ 𝑦 ⊆ ℕ))
13123anbi1d 1440 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
14 raleq 3309 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1))
15 difeq1 4075 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∖ {𝑚}) = (𝑦 ∖ {𝑚}))
1615raleqdv 3313 . . . . . . . . 9 (𝑥 = 𝑦 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
1716raleqbi1dv 3307 . . . . . . . 8 (𝑥 = 𝑦 → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
1813, 14, 173anbi123d 1436 . . . . . . 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 15792 . . . . . . . . 9 (𝑥 = 𝑦 → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚𝑦 (𝐹𝑚))
2019oveq1d 7372 . . . . . . . 8 (𝑥 = 𝑦 → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
2120eqeq1d 2738 . . . . . . 7 (𝑥 = 𝑦 → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁) = 1))
2218, 21imbi12d 344 . . . . . 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 3969 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℕ ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ))
24233anbi1d 1440 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
25 raleq 3309 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1))
26 difeq1 4075 . . . . . . . . . 10 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∖ {𝑚}) = ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
2726raleqdv 3313 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
2827raleqbi1dv 3307 . . . . . . . 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 1436 . . . . . . 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 15792 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚))
3130oveq1d 7372 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁))
3231eqeq1d 2738 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = 1))
3329, 32imbi12d 344 . . . . . 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 3969 . . . . . . . . 9 (𝑥 = 𝑀 → (𝑥 ⊆ ℕ ↔ 𝑀 ⊆ ℕ))
35343anbi1d 1440 . . . . . . . 8 (𝑥 = 𝑀 → ((𝑥 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ↔ (𝑀 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
36 raleq 3309 . . . . . . . 8 (𝑥 = 𝑀 → (∀𝑚𝑥 ((𝐹𝑚) gcd 𝑁) = 1 ↔ ∀𝑚𝑀 ((𝐹𝑚) gcd 𝑁) = 1))
37 difeq1 4075 . . . . . . . . . 10 (𝑥 = 𝑀 → (𝑥 ∖ {𝑚}) = (𝑀 ∖ {𝑚}))
3837raleqdv 3313 . . . . . . . . 9 (𝑥 = 𝑀 → (∀𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
3938raleqbi1dv 3307 . . . . . . . 8 (𝑥 = 𝑀 → (∀𝑚𝑥𝑛 ∈ (𝑥 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 ↔ ∀𝑚𝑀𝑛 ∈ (𝑀 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
4035, 36, 393anbi123d 1436 . . . . . . 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 15792 . . . . . . . . 9 (𝑥 = 𝑀 → ∏𝑚𝑥 (𝐹𝑚) = ∏𝑚𝑀 (𝐹𝑚))
4241oveq1d 7372 . . . . . . . 8 (𝑥 = 𝑀 → (∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁))
4342eqeq1d 2738 . . . . . . 7 (𝑥 = 𝑀 → ((∏𝑚𝑥 (𝐹𝑚) gcd 𝑁) = 1 ↔ (∏𝑚𝑀 (𝐹𝑚) gcd 𝑁) = 1))
4440, 43imbi12d 344 . . . . . 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 15826 . . . . . . . . . . 11 𝑚 ∈ ∅ (𝐹𝑚) = 1
4645a1i 11 . . . . . . . . . 10 (𝑁 ∈ ℕ → ∏𝑚 ∈ ∅ (𝐹𝑚) = 1)
4746oveq1d 7372 . . . . . . . . 9 (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = (1 gcd 𝑁))
48 nnz 12520 . . . . . . . . . 10 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
49 1gcd 16414 . . . . . . . . . 10 (𝑁 ∈ ℤ → (1 gcd 𝑁) = 1)
5048, 49syl 17 . . . . . . . . 9 (𝑁 ∈ ℕ → (1 gcd 𝑁) = 1)
5147, 50eqtrd 2776 . . . . . . . 8 (𝑁 ∈ ℕ → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)
52513ad2ant2 1134 . . . . . . 7 ((∅ ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∏𝑚 ∈ ∅ (𝐹𝑚) gcd 𝑁) = 1)
53523ad2ant1 1133 . . . . . 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 1917 . . . . . . . . . . . . . . . 16 𝑚(((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦))
55 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑚(𝐹𝑧)
56 simprl 769 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ∈ Fin)
57 unss 4144 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ)
58 vex 3449 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
5958snss 4746 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ℕ ↔ {𝑧} ⊆ ℕ)
6059biimpri 227 . . . . . . . . . . . . . . . . . . . 20 ({𝑧} ⊆ ℕ → 𝑧 ∈ ℕ)
6160adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑧 ∈ ℕ)
6257, 61sylbir 234 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑧 ∈ ℕ)
63623ad2ant1 1133 . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ¬ 𝑧𝑦)
66 simpll3 1214 . . . . . . . . . . . . . . . . . 18 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → 𝐹:ℕ⟶ℕ)
67 simpl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑦 ⊆ ℕ)
6857, 67sylbir 234 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ)
69683ad2ant1 1133 . . . . . . . . . . . . . . . . . . . 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 3944 . . . . . . . . . . . . . . . . . 18 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → 𝑚 ∈ ℕ)
7266, 71ffvelcdmd 7036 . . . . . . . . . . . . . . . . 17 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → (𝐹𝑚) ∈ ℕ)
7372nncnd 12169 . . . . . . . . . . . . . . . 16 (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑚𝑦) → (𝐹𝑚) ∈ ℂ)
74 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑚 = 𝑧 → (𝐹𝑚) = (𝐹𝑧))
75 simpr 485 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝐹:ℕ⟶ℕ)
7662adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝑧 ∈ ℕ)
7775, 76ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℕ)
78773adant2 1131 . . . . . . . . . . . . . . . . . 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 12169 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℂ)
8154, 55, 56, 64, 65, 73, 74, 80fprodsplitsn 15872 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) = (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))
8281oveq1d 7372 . . . . . . . . . . . . . 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 15838 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ)
8483nnzd 12526 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ∏𝑚𝑦 (𝐹𝑚) ∈ ℤ)
8579nnzd 12526 . . . . . . . . . . . . . . . 16 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹𝑧) ∈ ℤ)
8684, 85zmulcld 12613 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) ∈ ℤ)
87483ad2ant2 1134 . . . . . . . . . . . . . . . 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)
8986, 88gcdcomd 16394 . . . . . . . . . . . . . 14 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9082, 89eqtrd 2776 . . . . . . . . . . . . 13 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))))
9190ex 413 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹𝑚) gcd 𝑁) = (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧)))))
92913ad2ant1 1133 . . . . . . . . . . 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)))))
9392com12 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)))))
9493adantr 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)))))
9594imp 407 . . . . . . . 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))))
96 simpl2 1192 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑁 ∈ ℕ)
9796, 83, 793jca 1128 . . . . . . . . . . . . . 14 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ))
9897ex 413 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ)))
99983ad2ant1 1133 . . . . . . . . . . . 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)))
10099com12 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)))
101100adantr 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)))
102101imp 407 . . . . . . . . 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))
10388, 84gcdcomd 16394 . . . . . . . . . . . . . . 15 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁))
104103ex 413 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = (∏𝑚𝑦 (𝐹𝑚) gcd 𝑁)))
1051043ad2ant1 1133 . . . . . . . . . . . . 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)))
106105com12 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)))
107106adantr 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)))
108107imp 407 . . . . . . . . . 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))
10968a1i 11 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ))
110 idd 24 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝑁 ∈ ℕ → 𝑁 ∈ ℕ))
111 idd 24 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝐹:ℕ⟶ℕ → 𝐹:ℕ⟶ℕ))
112109, 110, 1113anim123d 1443 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑦 ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
113 ssun1 4132 . . . . . . . . . . . . . 14 𝑦 ⊆ (𝑦 ∪ {𝑧})
114 ssralv 4010 . . . . . . . . . . . . . 14 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 → ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1))
115113, 114mp1i 13 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 → ∀𝑚𝑦 ((𝐹𝑚) gcd 𝑁) = 1))
116 ssralv 4010 . . . . . . . . . . . . . . 15 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑚𝑦𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
117113, 116mp1i 13 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑚𝑦𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
118113a1i 11 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑚𝑦) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
119118ssdifd 4100 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑚𝑦) → (𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
120 ssralv 4010 . . . . . . . . . . . . . . . 16 ((𝑦 ∖ {𝑚}) ⊆ ((𝑦 ∪ {𝑧}) ∖ {𝑚}) → (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
121119, 120syl 17 . . . . . . . . . . . . . . 15 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑚𝑦) → (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
122121ralimdva 3164 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑚𝑦𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
123117, 122syld 47 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1 → ∀𝑚𝑦𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹𝑚) gcd (𝐹𝑛)) = 1))
124112, 115, 1233anim123d 1443 . . . . . . . . . . . 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)))
125124imim1d 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)))
126125imp31 418 . . . . . . . . . 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)
127108, 126eqtrd 2776 . . . . . . . . 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)
128 rpmulgcd 16437 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ ∏𝑚𝑦 (𝐹𝑚) ∈ ℕ ∧ (𝐹𝑧) ∈ ℕ) ∧ (𝑁 gcd ∏𝑚𝑦 (𝐹𝑚)) = 1) → (𝑁 gcd (∏𝑚𝑦 (𝐹𝑚) · (𝐹𝑧))) = (𝑁 gcd (𝐹𝑧)))
129102, 127, 128syl2anc 584 . . . . . . . 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)))
130 vsnid 4623 . . . . . . . . . . . . . . 15 𝑧 ∈ {𝑧}
131130olci 864 . . . . . . . . . . . . . 14 (𝑧𝑦𝑧 ∈ {𝑧})
132 elun 4108 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧𝑦𝑧 ∈ {𝑧}))
133131, 132mpbir 230 . . . . . . . . . . . . 13 𝑧 ∈ (𝑦 ∪ {𝑧})
13474oveq1d 7372 . . . . . . . . . . . . . . 15 (𝑚 = 𝑧 → ((𝐹𝑚) gcd 𝑁) = ((𝐹𝑧) gcd 𝑁))
135134eqeq1d 2738 . . . . . . . . . . . . . 14 (𝑚 = 𝑧 → (((𝐹𝑚) gcd 𝑁) = 1 ↔ ((𝐹𝑧) gcd 𝑁) = 1))
136135rspcv 3577 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 → ((𝐹𝑧) gcd 𝑁) = 1))
137133, 136mp1i 13 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1 → ((𝐹𝑧) gcd 𝑁) = 1))
138137imp 407 . . . . . . . . . . 11 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1) → ((𝐹𝑧) gcd 𝑁) = 1)
13978nnzd 12526 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝐹𝑧) ∈ ℤ)
14087, 139gcdcomd 16394 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → (𝑁 gcd (𝐹𝑧)) = ((𝐹𝑧) gcd 𝑁))
141140eqeq1d 2738 . . . . . . . . . . . 12 (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑁 gcd (𝐹𝑧)) = 1 ↔ ((𝐹𝑧) gcd 𝑁) = 1))
142141adantr 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))
143138, 142mpbird 256 . . . . . . . . . 10 ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})((𝐹𝑚) gcd 𝑁) = 1) → (𝑁 gcd (𝐹𝑧)) = 1)
1441433adant3 1132 . . . . . . . . 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)
145144adantl 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)
14695, 129, 1453eqtrd 2780 . . . . . . 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)
147146exp31 420 . . . . . 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)))
14811, 22, 33, 44, 53, 147findcard2s 9109 . . . . 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))
1491483expd 1353 . . . 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))))
1501493expd 1353 . . 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))))))
1511503imp 1111 . 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))))
1521513imp 1111 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wral 3064  cdif 3907  cun 3908  wss 3910  c0 4282  {csn 4586  wf 6492  cfv 6496  (class class class)co 7357  Fincfn 8883  1c1 11052   · cmul 11056  cn 12153  cz 12499  cprod 15788   gcd cgcd 16374
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-prod 15789  df-dvds 16137  df-gcd 16375
This theorem is referenced by:  coprmproddvdslem  16538
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