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Theorem coprmproddvdslem 16640

Description: Lemma for coprmproddvds 16641: 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 1909	. . . . 5 ⊢ Ⅎ𝑚((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)))
2		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑚(𝐹‘𝑧)
3		simpll 765	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝑦 ∈ Fin)
4		unss 4186	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℕ)
5		vex 3477	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
6	5	snss 4794	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ ↔ {𝑧} ⊆ ℕ)
7	6	biimpri 227	. . . . . . . . 9 ⊢ ({𝑧} ⊆ ℕ → 𝑧 ∈ ℕ)
8	7	adantl 480	. . . . . . . 8 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑧 ∈ ℕ)
9	4, 8	sylbir 234	. . . . . . 7 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑧 ∈ ℕ)
10	9	adantr 479	. . . . . 6 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → 𝑧 ∈ ℕ)
11	10	adantl 480	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝑧 ∈ ℕ)
12		simplr 767	. . . . 5 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → ¬ 𝑧 ∈ 𝑦)
13		simprrr 780	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝐹:ℕ⟶ℕ)
14	13	adantr 479	. . . . . . 7 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → 𝐹:ℕ⟶ℕ)
15		simpl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ ℕ ∧ {𝑧} ⊆ ℕ) → 𝑦 ⊆ ℕ)
16	4, 15	sylbir 234	. . . . . . . . . 10 ⊢ ((𝑦 ∪ {𝑧}) ⊆ ℕ → 𝑦 ⊆ ℕ)
17	16	adantr 479	. . . . . . . . 9 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → 𝑦 ⊆ ℕ)
18	17	adantl 480	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → 𝑦 ⊆ ℕ)
19	18	sselda 3982	. . . . . . 7 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → 𝑚 ∈ ℕ)
20	14, 19	ffvelcdmd 7100	. . . . . 6 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℕ)
21	20	nncnd 12266	. . . . 5 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℂ)
22		fveq2 6902	. . . . 5 ⊢ (𝑚 = 𝑧 → (𝐹‘𝑚) = (𝐹‘𝑧))
23	13, 11	ffvelcdmd 7100	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝐹‘𝑧) ∈ ℕ)
24	23	nncnd 12266	. . . . 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 15973	. . . 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 745	. . 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 769	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
28		simprr 771	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → 𝐹:ℕ⟶ℕ)
29	28	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐹:ℕ⟶ℕ)
30	29	adantr 479	. . . . . . . . . . . . 13 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → 𝐹:ℕ⟶ℕ)
31	17	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ⊆ ℕ)
32	31	sselda 3982	. . . . . . . . . . . . 13 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → 𝑚 ∈ ℕ)
33	30, 32	ffvelcdmd 7100	. . . . . . . . . . . 12 ⊢ (((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑚 ∈ 𝑦) → (𝐹‘𝑚) ∈ ℕ)
34	27, 33	fprodnncl 15939	. . . . . . . . . . 11 ⊢ ((((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ)
35	34	ex 411	. . . . . . . . . 10 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ∏𝑚 ∈ 𝑦 (𝐹‘𝑚) ∈ ℕ))
36	35	adantr 479	. . . . . . . . 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 479	. . . . . . 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 405	. . . . . 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 12623	. . . . 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	ffvelcdmd 7100	. . . . . . . 8 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → (𝐹‘𝑧) ∈ ℕ)
42	41	nnzd 12623	. . . . . . 7 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → (𝐹‘𝑧) ∈ ℤ)
43	42	adantr 479	. . . . . 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 480	. . . . 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 12617	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ)
46	45	adantr 479	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → 𝐾 ∈ ℤ)
47	46	adantl 480	. . . . . . 7 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → 𝐾 ∈ ℤ)
48	47	adantr 479	. . . . . 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 480	. . . . 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 1125	. . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑦 ∪ {𝑧}) ⊆ ℕ) → 𝐹:ℕ⟶ℕ)
52	9	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑦 ∪ {𝑧}) ⊆ ℕ) → 𝑧 ∈ ℕ)
53	51, 52	ffvelcdmd 7100	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑦 ∪ {𝑧}) ⊆ ℕ) → (𝐹‘𝑧) ∈ ℕ)
54	53	ex 411	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ℕ⟶ℕ → ((𝑦 ∪ {𝑧}) ⊆ ℕ → (𝐹‘𝑧) ∈ ℕ))
55	54	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ) → ((𝑦 ∪ {𝑧}) ⊆ ℕ → (𝐹‘𝑧) ∈ ℕ))
56	55	impcom 406	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ)) → (𝐹‘𝑧) ∈ ℕ)
57	56	adantl 480	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝐹‘𝑧) ∈ ℕ)
58	3, 18, 57	3jca 1125	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝑦 ∈ Fin ∧ 𝑦 ⊆ ℕ ∧ (𝐹‘𝑧) ∈ ℕ))
59	58	adantr 479	. . . . . . . . . 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 479	. . . . . . . . . 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		vsnid 4670	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ {𝑧}
62	61	olci 864	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧})
63		elun 4149	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧 ∈ 𝑦 ∨ 𝑧 ∈ {𝑧}))
64	62, 63	mpbir 230	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
65	64	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
66		snssi 4816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ 𝑦 → {𝑚} ⊆ 𝑦)
67	66	ssneld 3984	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ 𝑦 → (¬ 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑚}))
68	67	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑚 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑚}))
69	68	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝑚 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑚}))
70	69	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) → (𝑚 ∈ 𝑦 → ¬ 𝑧 ∈ {𝑚}))
71	70	imp 405	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → ¬ 𝑧 ∈ {𝑚})
72	65, 71	eldifd 3960	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ ((𝑦 ∪ {𝑧}) ⊆ ℕ ∧ (𝐾 ∈ ℕ ∧ 𝐹:ℕ⟶ℕ))) ∧ 𝑚 ∈ 𝑦) → 𝑧 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚}))
73		fveq2 6902	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑧 → (𝐹‘𝑛) = (𝐹‘𝑧))
74	73	oveq2d 7442	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑧 → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = ((𝐹‘𝑚) gcd (𝐹‘𝑧)))
75	74	eqeq1d 2730	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑧 → (((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ↔ ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
76	75	rspcv 3607	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚}) → (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
77	72, 76	syl 17	. . . . . . . . . . . 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 → ((𝐹‘𝑚) gcd (𝐹‘𝑧)) = 1))
78	77	ralimdva 3164	. . . . . . . . . . 11 ⊢ (((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))
79		ralunb 4193	. . . . . . . . . . . 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 ∧ ∀𝑚 ∈ {𝑧}∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
80	79	simplbi 496	. . . . . . . . . . 11 ⊢ (∀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)
81	78, 80	impel 504	. . . . . . . . . 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)
82		raldifb 4145	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})(𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
83		ralunb 4193	. . . . . . . . . . . . . . . 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)))
84		raldifb 4145	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
85	84	biimpi 215	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
86	85	adantr 479	. . . . . . . . . . . . . . . 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)
87	83, 86	sylbi 216	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})(𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
88	82, 87	sylbir 234	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
89	88	ralimi 3080	. . . . . . . . . . . . 13 ⊢ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
90	89	adantr 479	. . . . . . . . . . . 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)
91	79, 90	sylbi 216	. . . . . . . . . . 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)
92	91	adantl 480	. . . . . . . . . 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)
93		coprmprod 16639	. . . . . . . . . . 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))
94	93	imp 405	. . . . . . . . . 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)
95	59, 60, 81, 92, 94	syl31anc 1370	. . . . . . . . 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)
96	95	ex 411	. . . . . . . 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))
97	96	adantrd 490	. . . . . . 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))
98	97	expimpd 452	. . . . . 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))
99	98	adantr 479	. . . . 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))
100	99	imp 405	. . . 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)
101	83	simplbi 496	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ (𝑦 ∪ {𝑧})(𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) → ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
102	82, 101	sylbir 234	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 → ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1))
103	102	ralimi 3080	. . . . . . . 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))
104	103	adantr 479	. . . . . . 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))
105	79, 104	sylbi 216	. . . . . 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))
106		ralunb 4193	. . . . . . 7 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾 ↔ (∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾 ∧ ∀𝑚 ∈ {𝑧} (𝐹‘𝑚) ∥ 𝐾))
107	106	simplbi 496	. . . . . 6 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾 → ∀𝑚 ∈ 𝑦 (𝐹‘𝑚) ∥ 𝐾)
108	84	ralbii 3090	. . . . . . . 8 ⊢ (∀𝑚 ∈ 𝑦 ∀𝑛 ∈ 𝑦 (𝑛 ∉ {𝑚} → ((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1) ↔ ∀𝑚 ∈ 𝑦 ∀𝑛 ∈ (𝑦 ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1)
109	108	anbi1i 622	. . . . . . 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))
110	17	adantl 480	. . . . . . . . . . . . 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)
111		simprrl 779	. . . . . . . . . . . . 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)
112		simprrr 780	. . . . . . . . . . . . 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)
113	110, 111, 112	jca32 514	. . . . . . . . . . . 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)))
114		simplr 767	. . . . . . . . . . . 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))
115		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))
116	113, 114, 115	syl2anc 582	. . . . . . . . . . 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))
117	116	exp31 418	. . . . . . . . . 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))))
118	117	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))))
119	118	imp 405	. . . . . . . 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)))
120	119	imp 405	. . . . . . 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))
121	109, 120	biimtrid 241	. . . . . 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))
122	105, 107, 121	syl2ani 605	. . . . 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))
123	122	impr 453	. . . 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)
124	22	breq1d 5162	. . . . . . . . 9 ⊢ (𝑚 = 𝑧 → ((𝐹‘𝑚) ∥ 𝐾 ↔ (𝐹‘𝑧) ∥ 𝐾))
125	124	rspcv 3607	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾 → (𝐹‘𝑧) ∥ 𝐾))
126	64, 125	ax-mp 5	. . . . . . 7 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾 → (𝐹‘𝑧) ∥ 𝐾)
127	126	adantl 480	. . . . . 6 ⊢ ((∀𝑚 ∈ (𝑦 ∪ {𝑧})∀𝑛 ∈ ((𝑦 ∪ {𝑧}) ∖ {𝑚})((𝐹‘𝑚) gcd (𝐹‘𝑛)) = 1 ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝐹‘𝑚) ∥ 𝐾) → (𝐹‘𝑧) ∥ 𝐾)
128	127	adantl 480	. . . . 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)
129	128	adantl 480	. . . 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)
130		coprmdvds2 16632	. . . . 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))
131	130	imp 405	. . . 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)
132	50, 100, 123, 129, 131	syl22anc 837	. . 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)
133	26, 132	eqbrtrd 5174	. 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)
134	133	exp31 418	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 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∉ wnel 3043 ∀wral 3058 ∖ cdif 3946 ∪ cun 3947 ⊆ wss 3949 {csn 4632 class class class wbr 5152 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 1c1 11147 · cmul 11151 ℕcn 12250 ℤcz 12596 ∏cprod 15889 ∥ cdvds 16238 gcd cgcd 16476

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-prod 15890 df-dvds 16239 df-gcd 16477

This theorem is referenced by: coprmproddvds 16641

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