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Theorem mulcncflem 14129

Description: Lemma for mulcncf 14130. (Contributed by Jim Kingdon, 29-May-2023.)

**Hypotheses**
Ref	Expression
mulcncflem.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))
mulcncflem.b	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))
mulcncflem.v	⊢ (𝜑 → 𝑉 ∈ 𝑋)
mulcncflem.e	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
mulcncflem.f	⊢ (𝜑 → 𝐹 ∈ ℝ⁺)
mulcncflem.g	⊢ (𝜑 → 𝐺 ∈ ℝ⁺)
mulcncflem.s	⊢ (𝜑 → 𝑆 ∈ ℝ⁺)
mulcncflem.t	⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
mulcncflem.acn	⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹))
mulcncflem.bcn	⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺))
mulcncflem.cn	⊢ (𝜑 → ∀𝑢 ∈ 𝑋 (((abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸))

**Assertion**
Ref	Expression
mulcncflem	⊢ (𝜑 → ∃𝑑 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑑 → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))

Distinct variable groups: 𝐴,𝑑,𝑢 𝐵,𝑑,𝑢 𝑢,𝑉 𝐸,𝑑,𝑢 𝑢,𝐹 𝑢,𝐺 𝑆,𝑑,𝑢 𝑇,𝑑,𝑢 𝑉,𝑑,𝑥,𝑢 𝑋,𝑑,𝑢,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑢,𝑑) 𝐴(𝑥) 𝐵(𝑥) 𝑆(𝑥) 𝑇(𝑥) 𝐸(𝑥) 𝐹(𝑥,𝑑) 𝐺(𝑥,𝑑)

Proof of Theorem mulcncflem Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcncflem.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℝ⁺)
2	1	rpred 9698	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℝ)
3		mulcncflem.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
4	3	rpred 9698	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ)
5		mincl 11241	. . . 4 ⊢ ((𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ) → inf({𝑆, 𝑇}, ℝ, < ) ∈ ℝ)
6	2, 4, 5	syl2anc 411	. . 3 ⊢ (𝜑 → inf({𝑆, 𝑇}, ℝ, < ) ∈ ℝ)
7	1	rpgt0d 9701	. . . 4 ⊢ (𝜑 → 0 < 𝑆)
8	3	rpgt0d 9701	. . . 4 ⊢ (𝜑 → 0 < 𝑇)
9		0red 7960	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
10		ltmininf 11245	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (0 < inf({𝑆, 𝑇}, ℝ, < ) ↔ (0 < 𝑆 ∧ 0 < 𝑇)))
11	9, 2, 4, 10	syl3anc 1238	. . . 4 ⊢ (𝜑 → (0 < inf({𝑆, 𝑇}, ℝ, < ) ↔ (0 < 𝑆 ∧ 0 < 𝑇)))
12	7, 8, 11	mpbir2and 944	. . 3 ⊢ (𝜑 → 0 < inf({𝑆, 𝑇}, ℝ, < ))
13	6, 12	elrpd 9695	. 2 ⊢ (𝜑 → inf({𝑆, 𝑇}, ℝ, < ) ∈ ℝ⁺)
14		simplr 528	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → 𝑧 ∈ 𝑋)
15		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
16		mulcncflem.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))
17		cncff 14103	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
18	16, 17	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
19		eqid 2177	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
20	19	fmpt 5668	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
21	18, 20	sylibr 134	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ)
22		mulcncflem.b	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))
23		cncff 14103	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
24	22, 23	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
25		eqid 2177	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
26	25	fmpt 5668	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
27	24, 26	sylibr 134	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
28		r19.26 2603	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑋 (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ↔ (∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ))
29	21, 27, 28	sylanbrc 417	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ))
30		mulcl 7940	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
31	30	ralimi 2540	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝑋 (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∀𝑥 ∈ 𝑋 (𝐴 · 𝐵) ∈ ℂ)
32	29, 31	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝐴 · 𝐵) ∈ ℂ)
33	32	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 (𝐴 · 𝐵) ∈ ℂ)
34		rspcsbela 3118	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝐴 · 𝐵) ∈ ℂ) → ⦋𝑧 / 𝑥⦌(𝐴 · 𝐵) ∈ ℂ)
35	15, 33, 34	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ⦋𝑧 / 𝑥⦌(𝐴 · 𝐵) ∈ ℂ)
36	35	adantr 276	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ⦋𝑧 / 𝑥⦌(𝐴 · 𝐵) ∈ ℂ)
37		eqid 2177	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))
38	37	fvmpts 5596	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑋 ∧ ⦋𝑧 / 𝑥⦌(𝐴 · 𝐵) ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) = ⦋𝑧 / 𝑥⦌(𝐴 · 𝐵))
39	14, 36, 38	syl2anc 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) = ⦋𝑧 / 𝑥⦌(𝐴 · 𝐵))
40		csbov12g 5916	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑋 → ⦋𝑧 / 𝑥⦌(𝐴 · 𝐵) = (⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵))
41	14, 40	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ⦋𝑧 / 𝑥⦌(𝐴 · 𝐵) = (⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵))
42	39, 41	eqtrd 2210	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) = (⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵))
43		mulcncflem.v	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ 𝑋)
44	43	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → 𝑉 ∈ 𝑋)
45		rspcsbela 3118	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝐴 · 𝐵) ∈ ℂ) → ⦋𝑉 / 𝑥⦌(𝐴 · 𝐵) ∈ ℂ)
46	43, 32, 45	syl2anc 411	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋𝑉 / 𝑥⦌(𝐴 · 𝐵) ∈ ℂ)
47	46	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ⦋𝑉 / 𝑥⦌(𝐴 · 𝐵) ∈ ℂ)
48	37	fvmpts 5596	. . . . . . . . . 10 ⊢ ((𝑉 ∈ 𝑋 ∧ ⦋𝑉 / 𝑥⦌(𝐴 · 𝐵) ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉) = ⦋𝑉 / 𝑥⦌(𝐴 · 𝐵))
49	44, 47, 48	syl2anc 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉) = ⦋𝑉 / 𝑥⦌(𝐴 · 𝐵))
50		csbov12g 5916	. . . . . . . . . 10 ⊢ (𝑉 ∈ 𝑋 → ⦋𝑉 / 𝑥⦌(𝐴 · 𝐵) = (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))
51	44, 50	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ⦋𝑉 / 𝑥⦌(𝐴 · 𝐵) = (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))
52	49, 51	eqtrd 2210	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉) = (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))
53	42, 52	oveq12d 5895	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → (((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉)) = ((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵)))
54	53	fveq2d 5521	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) = (abs‘((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))))
55		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ))
56		cncfrss 14101	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ) → 𝑋 ⊆ ℂ)
57	16, 56	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
58	57	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → 𝑋 ⊆ ℂ)
59	58, 14	sseldd 3158	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → 𝑧 ∈ ℂ)
60	57, 43	sseldd 3158	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ ℂ)
61	60	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → 𝑉 ∈ ℂ)
62	59, 61	subcld 8270	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → (𝑧 − 𝑉) ∈ ℂ)
63	62	abscld 11192	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → (abs‘(𝑧 − 𝑉)) ∈ ℝ)
64	2	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → 𝑆 ∈ ℝ)
65	4	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → 𝑇 ∈ ℝ)
66		ltmininf 11245	. . . . . . . . . 10 ⊢ (((abs‘(𝑧 − 𝑉)) ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑇 ∈ ℝ) → ((abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) ↔ ((abs‘(𝑧 − 𝑉)) < 𝑆 ∧ (abs‘(𝑧 − 𝑉)) < 𝑇)))
67	63, 64, 65, 66	syl3anc 1238	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ((abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) ↔ ((abs‘(𝑧 − 𝑉)) < 𝑆 ∧ (abs‘(𝑧 − 𝑉)) < 𝑇)))
68	55, 67	mpbid 147	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ((abs‘(𝑧 − 𝑉)) < 𝑆 ∧ (abs‘(𝑧 − 𝑉)) < 𝑇))
69		mulcncflem.acn	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹))
70		fvoveq1 5900	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑧 → (abs‘(𝑢 − 𝑉)) = (abs‘(𝑧 − 𝑉)))
71	70	breq1d 4015	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → ((abs‘(𝑢 − 𝑉)) < 𝑆 ↔ (abs‘(𝑧 − 𝑉)) < 𝑆))
72	71	imbrov2fvoveq 5902	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → (((abs‘(𝑢 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹) ↔ ((abs‘(𝑧 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹)))
73	72	cbvralv 2705	. . . . . . . . . . . . 13 ⊢ (∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹) ↔ ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹))
74	69, 73	sylib 122	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹))
75	74	r19.21bi 2565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs‘(𝑧 − 𝑉)) < 𝑆 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹))
76	21	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ)
77		rspcsbela 3118	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ) → ⦋𝑧 / 𝑥⦌𝐴 ∈ ℂ)
78	15, 76, 77	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ⦋𝑧 / 𝑥⦌𝐴 ∈ ℂ)
79	19	fvmpts 5596	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑋 ∧ ⦋𝑧 / 𝑥⦌𝐴 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) = ⦋𝑧 / 𝑥⦌𝐴)
80	15, 78, 79	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) = ⦋𝑧 / 𝑥⦌𝐴)
81	43	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑉 ∈ 𝑋)
82		rspcsbela 3118	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 𝐴 ∈ ℂ) → ⦋𝑉 / 𝑥⦌𝐴 ∈ ℂ)
83	81, 76, 82	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ⦋𝑉 / 𝑥⦌𝐴 ∈ ℂ)
84	19	fvmpts 5596	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑋 ∧ ⦋𝑉 / 𝑥⦌𝐴 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉) = ⦋𝑉 / 𝑥⦌𝐴)
85	81, 83, 84	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉) = ⦋𝑉 / 𝑥⦌𝐴)
86	80, 85	oveq12d 5895	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉)) = (⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴))
87	86	fveq2d 5521	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) = (abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)))
88	87	breq1d 4015	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs‘(((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐴)‘𝑉))) < 𝐹 ↔ (abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹))
89	75, 88	sylibd 149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs‘(𝑧 − 𝑉)) < 𝑆 → (abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹))
90		mulcncflem.bcn	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺))
91	70	breq1d 4015	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → ((abs‘(𝑢 − 𝑉)) < 𝑇 ↔ (abs‘(𝑧 − 𝑉)) < 𝑇))
92	91	imbrov2fvoveq 5902	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → (((abs‘(𝑢 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺) ↔ ((abs‘(𝑧 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺)))
93	92	cbvralv 2705	. . . . . . . . . . . . 13 ⊢ (∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑢) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺) ↔ ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺))
94	90, 93	sylib 122	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺))
95	94	r19.21bi 2565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs‘(𝑧 − 𝑉)) < 𝑇 → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺))
96	27	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
97		rspcsbela 3118	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ) → ⦋𝑧 / 𝑥⦌𝐵 ∈ ℂ)
98	15, 96, 97	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ⦋𝑧 / 𝑥⦌𝐵 ∈ ℂ)
99	25	fvmpts 5596	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑋 ∧ ⦋𝑧 / 𝑥⦌𝐵 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑧) = ⦋𝑧 / 𝑥⦌𝐵)
100	15, 98, 99	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑧) = ⦋𝑧 / 𝑥⦌𝐵)
101		rspcsbela 3118	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ) → ⦋𝑉 / 𝑥⦌𝐵 ∈ ℂ)
102	81, 96, 101	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ⦋𝑉 / 𝑥⦌𝐵 ∈ ℂ)
103	25	fvmpts 5596	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑋 ∧ ⦋𝑉 / 𝑥⦌𝐵 ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉) = ⦋𝑉 / 𝑥⦌𝐵)
104	81, 102, 103	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉) = ⦋𝑉 / 𝑥⦌𝐵)
105	100, 104	oveq12d 5895	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉)) = (⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵))
106	105	fveq2d 5521	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) = (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)))
107	106	breq1d 4015	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs‘(((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑧) − ((𝑥 ∈ 𝑋 ↦ 𝐵)‘𝑉))) < 𝐺 ↔ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺))
108	95, 107	sylibd 149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs‘(𝑧 − 𝑉)) < 𝑇 → (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺))
109	89, 108	anim12d 335	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (((abs‘(𝑧 − 𝑉)) < 𝑆 ∧ (abs‘(𝑧 − 𝑉)) < 𝑇) → ((abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺)))
110	109	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → (((abs‘(𝑧 − 𝑉)) < 𝑆 ∧ (abs‘(𝑧 − 𝑉)) < 𝑇) → ((abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺)))
111	68, 110	mpd 13	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → ((abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺))
112		mulcncflem.cn	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 (((abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸))
113		csbeq1 3062	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → ⦋𝑢 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
114	113	fvoveq1d 5899	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → (abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) = (abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)))
115	114	breq1d 4015	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → ((abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ↔ (abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹))
116		csbeq1 3062	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → ⦋𝑢 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
117	116	fvoveq1d 5899	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → (abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) = (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)))
118	117	breq1d 4015	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → ((abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺 ↔ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺))
119	115, 118	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → (((abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) ↔ ((abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺)))
120	113, 116	oveq12d 5895	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → (⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) = (⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵))
121	120	fvoveq1d 5899	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (abs‘((⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) = (abs‘((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))))
122	121	breq1d 4015	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → ((abs‘((⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸 ↔ (abs‘((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸))
123	119, 122	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → ((((abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸) ↔ (((abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸)))
124	123	cbvralv 2705	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑋 (((abs‘(⦋𝑢 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑢 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑢 / 𝑥⦌𝐴 · ⦋𝑢 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸) ↔ ∀𝑧 ∈ 𝑋 (((abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸))
125	112, 124	sylib 122	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (((abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸))
126	125	r19.21bi 2565	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (((abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸))
127	126	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → (((abs‘(⦋𝑧 / 𝑥⦌𝐴 − ⦋𝑉 / 𝑥⦌𝐴)) < 𝐹 ∧ (abs‘(⦋𝑧 / 𝑥⦌𝐵 − ⦋𝑉 / 𝑥⦌𝐵)) < 𝐺) → (abs‘((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸))
128	111, 127	mpd 13	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → (abs‘((⦋𝑧 / 𝑥⦌𝐴 · ⦋𝑧 / 𝑥⦌𝐵) − (⦋𝑉 / 𝑥⦌𝐴 · ⦋𝑉 / 𝑥⦌𝐵))) < 𝐸)
129	54, 128	eqbrtrd 4027	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸)
130	129	ex 115	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))
131	130	ralrimiva 2550	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))
132		fvoveq1 5900	. . . . . 6 ⊢ (𝑧 = 𝑢 → (abs‘(𝑧 − 𝑉)) = (abs‘(𝑢 − 𝑉)))
133	132	breq1d 4015	. . . . 5 ⊢ (𝑧 = 𝑢 → ((abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) ↔ (abs‘(𝑢 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )))
134	133	imbrov2fvoveq 5902	. . . 4 ⊢ (𝑧 = 𝑢 → (((abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸) ↔ ((abs‘(𝑢 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸)))
135	134	cbvralv 2705	. . 3 ⊢ (∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑧) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸) ↔ ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))
136	131, 135	sylib 122	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))
137		breq2 4009	. . 3 ⊢ (𝑑 = inf({𝑆, 𝑇}, ℝ, < ) → ((abs‘(𝑢 − 𝑉)) < 𝑑 ↔ (abs‘(𝑢 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < )))
138	137	rspceaimv 2851	. 2 ⊢ ((inf({𝑆, 𝑇}, ℝ, < ) ∈ ℝ⁺ ∧ ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < inf({𝑆, 𝑇}, ℝ, < ) → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸)) → ∃𝑑 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑑 → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))
139	13, 136, 138	syl2anc 411	1 ⊢ (𝜑 → ∃𝑑 ∈ ℝ⁺ ∀𝑢 ∈ 𝑋 ((abs‘(𝑢 − 𝑉)) < 𝑑 → (abs‘(((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑢) − ((𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵))‘𝑉))) < 𝐸))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ⦋csb 3059 ⊆ wss 3131 {cpr 3595 class class class wbr 4005 ↦ cmpt 4066 ⟶wf 5214 ‘cfv 5218 (class class class)co 5877 infcinf 6984 ℂcc 7811 ℝcr 7812 0cc0 7813 · cmul 7818 < clt 7994 − cmin 8130 ℝ⁺crp 9655 abscabs 11008 –cn→ccncf 14096

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-map 6652 df-sup 6985 df-inf 6986 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-rp 9656 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-cncf 14097

This theorem is referenced by: mulcncf 14130

W3C validator