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Theorem coprmproddvdslem 15376

Description: Lemma for coprmproddvds 15377: Induction step. (Contributed by AV, 19-Aug-2020.)

**Assertion**
Ref	Expression
coprmproddvdslem	⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)))

Distinct variable groups: 𝑚,𝐹,𝑛 𝑚,𝐾,𝑦,𝑧 𝑦,𝑛,𝑧 Allowed substitution hints: 𝐹(𝑦,𝑧) 𝐾(𝑛)

**Proof of Theorem coprmproddvdslem**
Step	Hyp	Ref	Expression
1		nfv 1843	. . . . 5 ⊢ Ⅎ𝑚((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
2		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑚(𝐹‘𝑧)
3		simpll 790	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝑦 ∈ Fin)
4		unss 3787	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ)
5		vex 3203	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
6	5	snss 4316	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ ↔ {𝑧} ⊆ ℕ)
7	6	biimpri 218	. . . . . . . . 9 ⊢ ({𝑧} ⊆ ℕ → 𝑧 ∈ ℕ)
8	7	adantl 482	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑧 ∈ ℕ)
9	4, 8	sylbir 225	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑧 ∈ ℕ)
10	9	adantr 481	. . . . . 6 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → 𝑧 ∈ ℕ)
11	10	adantl 482	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝑧 ∈ ℕ)
12		simplr 792	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → ¬ 𝑧 ∈ 𝑦)
13		simprrr 805	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝐹:ℕ⟶ℕ)
14	13	adantr 481	. . . . . . 7 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → 𝐹:ℕ⟶ℕ)
15		simpl 473	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑦 ⊆ ℕ)
16	4, 15	sylbir 225	. . . . . . . . . 10 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ)
17	16	adantr 481	. . . . . . . . 9 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → 𝑦 ⊆ ℕ)
18	17	adantl 482	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝑦 ⊆ ℕ)
19	18	sselda 3603	. . . . . . 7 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → 𝑚 ∈ ℕ)
20	14, 19	ffvelrnd 6360	. . . . . 6 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℕ)
21	20	nncnd 11036	. . . . 5 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℂ)
22		fveq2 6191	. . . . 5 ⊢ (𝑚 = 𝑧 → (𝐹‘𝑚) = (𝐹‘𝑧))
23	13, 11	ffvelrnd 6360	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝐹‘𝑧) ∈ ℕ)
24	23	nncnd 11036	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝐹‘𝑧) ∈ ℂ)
25	1, 2, 3, 11, 12, 21, 22, 24	fprodsplitsn 14720	. . . 4 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)))
26	25	ad2ant2r 783	. . 3 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) = (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)))
27		simprl 794	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
28		simprr 796	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → 𝐹:ℕ⟶ℕ)
29	28	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐹:ℕ⟶ℕ)
30	29	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → 𝐹:ℕ⟶ℕ)
31	17	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ⊆ ℕ)
32	31	sselda 3603	. . . . . . . . . . . . 13 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → 𝑚 ∈ ℕ)
33	30, 32	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℕ)
34	27, 33	fprodnncl 14685	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ)
35	34	ex 450	. . . . . . . . . 10 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ))
36	35	adantr 481	. . . . . . . . 9 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ))
37	36	com12 32	. . . . . . . 8 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ))
38	37	adantr 481	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ))
39	38	imp 445	. . . . . 6 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ)
40	39	nnzd 11481	. . . . 5 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℤ)
41	28, 10	ffvelrnd 6360	. . . . . . . 8 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → (𝐹‘𝑧) ∈ ℕ)
42	41	nnzd 11481	. . . . . . 7 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → (𝐹‘𝑧) ∈ ℤ)
43	42	adantr 481	. . . . . 6 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → (𝐹‘𝑧) ∈ ℤ)
44	43	adantl 482	. . . . 5 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → (𝐹‘𝑧) ∈ ℤ)
45		nnz 11399	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ)
46	45	adantr 481	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝐾 ∈ ℤ)
47	46	adantl 482	. . . . . . 7 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → 𝐾 ∈ ℤ)
48	47	adantr 481	. . . . . 6 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → 𝐾 ∈ ℤ)
49	48	adantl 482	. . . . 5 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → 𝐾 ∈ ℤ)
50	40, 44, 49	3jca 1242	. . . 4 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℤ ∧ (𝐹‘𝑧) ∈ ℤ ∧ 𝐾 ∈ ℤ))
51		simpl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑦 ∪ {𝑧}) ⊆ ℕ) → 𝐹:ℕ⟶ℕ)
52	9	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑦 ∪ {𝑧}) ⊆ ℕ) → 𝑧 ∈ ℕ)
53	51, 52	ffvelrnd 6360	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑦 ∪ {𝑧}) ⊆ ℕ) → (𝐹‘𝑧) ∈ ℕ)
54	53	ex 450	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ℕ⟶ℕ → ((𝑦 ∪ {𝑧}) ⊆ ℕ → (𝐹‘𝑧) ∈ ℕ))
55	54	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∪ {𝑧}) ⊆ ℕ → (𝐹‘𝑧) ∈ ℕ))
56	55	impcom 446	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → (𝐹‘𝑧) ∈ ℕ)
57	56	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝐹‘𝑧) ∈ ℕ)
58	3, 18, 57	3jca 1242	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ ℕ ∧ (𝐹‘𝑧) ∈ ℕ))
59	58	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ ℕ ∧ (𝐹‘𝑧) ∈ ℕ))
60	13	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → 𝐹:ℕ⟶ℕ)
61		ralunb 3794	. . . . . . . . . . . . 13 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ {𝑧}∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
62	61	simplbi 476	. . . . . . . . . . . 12 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
63		vsnid 4209	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ {𝑧}
64	63	olci 406	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧})
65		elun 3753	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
66	64, 65	mpbir 221	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
67	66	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
68		snssi 4339	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ 𝑦 → {𝑚} ⊆ 𝑦)
69	68	ssneld 3605	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ 𝑦 → (¬ 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑚}))
70	69	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑚 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑚}))
71	70	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝑚 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑚}))
72	71	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝑚 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑚}))
73	72	imp 445	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → ¬ 𝑧 ∈ {𝑚})
74	67, 73	eldifd 3585	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → 𝑧 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
75		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → (𝐹‘𝑛) = (𝐹‘𝑧))
76	75	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑧 → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = ((𝐹‘𝑚) gcd (𝐹‘𝑧)))
77	76	eqeq1d 2624	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑧 → (((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
78	77	rspcv 3305	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚}) → (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
79	74, 78	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
80	79	ralimdva 2962	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
81	62, 80	syl5 34	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
82	81	imp 445	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1)
83		raldifb 3750	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})(𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
84		ralunb 3794	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})(𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ (∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ∧ ∀𝑛 ∈ {𝑧} (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)))
85		raldifb 3750	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
86	85	biimpi 206	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
87	86	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ∧ ∀𝑛 ∈ {𝑧} (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)) → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
88	84, 87	sylbi 207	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})(𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
89	83, 88	sylbir 225	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
90	89	ralimi 2952	. . . . . . . . . . . . 13 ⊢ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
91	90	adantr 481	. . . . . . . . . . . 12 ⊢ ((∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ {𝑧}∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
92	61, 91	sylbi 207	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
93	92	adantl 482	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
94		coprmprod 15375	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝑦 ⊆ ℕ ∧ (𝐹‘𝑧) ∈ ℕ) ∧ 𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1) → (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
95	94	imp 445	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ Fin ∧ 𝑦 ⊆ ℕ ∧ (𝐹‘𝑧) ∈ ℕ) ∧ 𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ 𝑦 ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1) ∧ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1)
96	59, 60, 82, 93, 95	syl31anc 1329	. . . . . . . . 9 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1)
97	96	ex 450	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
98	97	adantrd 484	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → ((∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
99	98	expimpd 629	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
100	99	adantr 481	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
101	100	imp 445	. . . 4 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1)
102	84	simplbi 476	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})(𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
103	83, 102	sylbir 225	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
104	103	ralimi 2952	. . . . . . . 8 ⊢ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
105	104	adantr 481	. . . . . . 7 ⊢ ((∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ {𝑧}∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
106	61, 105	sylbi 207	. . . . . 6 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
107		ralunb 3794	. . . . . . 7 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾 ↔ (∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾 ∧ ∀𝑚 ∈ {𝑧} (𝐹‘𝑚) ∥ 𝐾))
108	107	simplbi 476	. . . . . 6 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾 → ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)
109	85	ralbii 2980	. . . . . . . 8 ⊢ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
110	109	anbi1i 731	. . . . . . 7 ⊢ ((∀𝑚 ∈ 𝑦 ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) ↔ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
111	17	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝑦 ⊆ ℕ)
112		simprrl 804	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝐾 ∈ ℕ)
113		simprrr 805	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝐹:ℕ⟶ℕ)
114	111, 112, 113	jca32 558	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
115		simplr 792	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
116		pm2.27 42	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ((((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
117	114, 115, 116	syl2anc 693	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → ((((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
118	117	exp31 630	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → ((((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))))
119	118	com24 95	. . . . . . . . 9 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → ((∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))))
120	119	imp 445	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → ((∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)))
121	120	imp 445	. . . . . . 7 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → ((∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
122	110, 121	syl5bi 232	. . . . . 6 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → ((∀𝑚 ∈ 𝑦 ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
123	106, 108, 122	syl2ani 688	. . . . 5 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → ((∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾))
124	123	impr 649	. . . 4 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)
125	22	breq1d 4663	. . . . . . . . 9 ⊢ (𝑚 = 𝑧 → ((𝐹‘𝑚) ∥ 𝐾 ↔ (𝐹‘𝑧) ∥ 𝐾))
126	125	rspcv 3305	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾 → (𝐹‘𝑧) ∥ 𝐾))
127	66, 126	ax-mp 5	. . . . . . 7 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾 → (𝐹‘𝑧) ∥ 𝐾)
128	127	adantl 482	. . . . . 6 ⊢ ((∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾) → (𝐹‘𝑧) ∥ 𝐾)
129	128	adantl 482	. . . . 5 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → (𝐹‘𝑧) ∥ 𝐾)
130	129	adantl 482	. . . 4 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → (𝐹‘𝑧) ∥ 𝐾)
131		coprmdvds2 15368	. . . . 5 ⊢ (((∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℤ ∧ (𝐹‘𝑧) ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1) → ((∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾 ∧ (𝐹‘𝑧) ∥ 𝐾) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)) ∥ 𝐾))
132	131	imp 445	. . . 4 ⊢ ((((∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℤ ∧ (𝐹‘𝑧) ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1) ∧ (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾 ∧ (𝐹‘𝑧) ∥ 𝐾)) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)) ∥ 𝐾)
133	50, 101, 124, 130, 132	syl22anc 1327	. . 3 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → (∏𝑚 ∈ 𝑦 (𝐹‘𝑚) · (𝐹‘𝑧)) ∥ 𝐾)
134	26, 133	eqbrtrd 4675	. 2 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) ∧ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾))) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)
135	134	exp31 630	1 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((((𝑦 ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾) → ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)) → ∏𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 ∀wral 2912 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 {csn 4177 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 1c1 9937 · cmul 9941 ℕcn 11020 ℤcz 11377 ∏cprod 14635 ∥ cdvds 14983 gcd cgcd 15216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-prod 14636 df-dvds 14984 df-gcd 15217

This theorem is referenced by: coprmproddvds 15377

W3C validator