Theorem txmetcnp 15542

Description: Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)

Hypotheses

Ref	Expression
metcn.2	⊢ 𝐽 = (MetOpen‘𝐶)
metcn.4	⊢ 𝐾 = (MetOpen‘𝐷)
txmetcnp.4	⊢ 𝐿 = (MetOpen‘𝐸)

Assertion

Ref	Expression
txmetcnp	⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))

Distinct variable groups:   𝑣,𝑢,𝑤,𝑧,𝐹   𝑢,𝐽,𝑣,𝑤,𝑧   𝑢,𝐾,𝑣,𝑤,𝑧   𝑢,𝑋,𝑣,𝑤,𝑧   𝑢,𝑌,𝑣,𝑤,𝑧   𝑢,𝑍,𝑣,𝑤,𝑧   𝑢,𝐴,𝑣,𝑤,𝑧   𝑢,𝐶,𝑣,𝑤,𝑧   𝑢,𝐷,𝑣,𝑤,𝑧   𝑢,𝐵,𝑣,𝑤,𝑧   𝑢,𝐸,𝑣,𝑤,𝑧   𝑤,𝐿,𝑧

Allowed substitution hints:   𝐿(𝑣,𝑢)

Proof of Theorem txmetcnp

Dummy variables 𝑡 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4 ⊢ (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < )) = (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))
2		simp1 1028	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → 𝐶 ∈ (∞Met‘𝑋))
3	2	adantr 276	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐶 ∈ (∞Met‘𝑋))
4		simp2 1029	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → 𝐷 ∈ (∞Met‘𝑌))
5	4	adantr 276	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐷 ∈ (∞Met‘𝑌))
6	1, 3, 5	xmetxp 15531	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < )) ∈ (∞Met‘(𝑋 × 𝑌)))
7		simpl3 1033	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐸 ∈ (∞Met‘𝑍))
8		simprl 535	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
9		simprr 537	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
10	8, 9	opelxpd 4802	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
11		eqid 2238	. . . 4 ⊢ (MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))) = (MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < )))
12		txmetcnp.4	. . . 4 ⊢ 𝐿 = (MetOpen‘𝐸)
13	11, 12	metcnp 15536	. . 3 ⊢ (((𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < )) ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝐸 ∈ (∞Met‘𝑍) ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → (𝐹 ∈ (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧))))
14	6, 7, 10, 13	syl3anc 1278	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐹 ∈ (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧))))
15		metcn.2	. . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐶)
16		metcn.4	. . . . . 6 ⊢ 𝐾 = (MetOpen‘𝐷)
17	1, 3, 5, 15, 16, 11	xmettx 15534	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))) = (𝐽 ×t 𝐾))
18	17	oveq1d 6090	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))) CnP 𝐿) = ((𝐽 ×t 𝐾) CnP 𝐿))
19	18	fveq1d 5692	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))) CnP 𝐿)‘⟨𝐴, 𝐵⟩) = (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))
20	19	eleq2d 2308	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐹 ∈ (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ 𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩)))
21		oveq2 6083	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) = (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩))
22	21	breq1d 4135	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) < 𝑤 ↔ (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤))
23		fveq2 5690	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑡) = (𝐹‘⟨𝑢, 𝑣⟩))
24	23	oveq2d 6091	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)))
25	24	breq1d 4135	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧 ↔ ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧))
26	22, 25	imbi12d 234	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧) ↔ ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧)))
27	26	ralxp 4918	. . . . . 6 ⊢ (∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧))
28	8	ad4antr 498	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → 𝐴 ∈ 𝑋)
29	9	ad4antr 498	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → 𝐵 ∈ 𝑌)
30	28, 29	opelxpd 4802	. . . . . . . . . . . . 13 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
31		simplr 533	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → 𝑢 ∈ 𝑋)
32		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → 𝑣 ∈ 𝑌)
33	31, 32	opelxpd 4802	. . . . . . . . . . . . 13 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ⟨𝑢, 𝑣⟩ ∈ (𝑋 × 𝑌))
34	2	ad5antr 500	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → 𝐶 ∈ (∞Met‘𝑋))
35		xmetf 15374	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐶:(𝑋 × 𝑋)⟶ℝ*)
36	34, 35	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → 𝐶:(𝑋 × 𝑋)⟶ℝ*)
37		op1stg 6374	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
38	28, 29, 37	syl2anc 415	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
39	38, 28	eqeltrd 2315	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (1st ‘⟨𝐴, 𝐵⟩) ∈ 𝑋)
40		op1stg 6374	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → (1st ‘⟨𝑢, 𝑣⟩) = 𝑢)
41	40	adantll 480	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (1st ‘⟨𝑢, 𝑣⟩) = 𝑢)
42	41, 31	eqeltrd 2315	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (1st ‘⟨𝑢, 𝑣⟩) ∈ 𝑋)
43	36, 39, 42	fovcdmd 6224	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)) ∈ ℝ*)
44	4	ad5antr 500	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → 𝐷 ∈ (∞Met‘𝑌))
45		xmetf 15374	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ*)
46	44, 45	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ*)
47		op2ndg 6375	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
48	28, 29, 47	syl2anc 415	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	48, 29	eqeltrd 2315	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (2nd ‘⟨𝐴, 𝐵⟩) ∈ 𝑌)
50		op2ndg 6375	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣)
51	50	adantll 480	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣)
52	51, 32	eqeltrd 2315	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (2nd ‘⟨𝑢, 𝑣⟩) ∈ 𝑌)
53	46, 49, 52	fovcdmd 6224	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩)) ∈ ℝ*)
54		xrmaxcl 11996	. . . . . . . . . . . . . 14 ⊢ ((((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)) ∈ ℝ* ∧ ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩)) ∈ ℝ*) → sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ) ∈ ℝ*)
55	43, 53, 54	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ) ∈ ℝ*)
56		fveq2 5690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑟) = (1st ‘⟨𝐴, 𝐵⟩))
57		fveq2 5690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑠) = (1st ‘⟨𝑢, 𝑣⟩))
58	56, 57	oveqan12d 6094	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((1st ‘𝑟)𝐶(1st ‘𝑠)) = ((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)))
59		fveq2 5690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑟) = (2nd ‘⟨𝐴, 𝐵⟩))
60		fveq2 5690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑠) = (2nd ‘⟨𝑢, 𝑣⟩))
61	59, 60	oveqan12d 6094	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((2nd ‘𝑟)𝐷(2nd ‘𝑠)) = ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩)))
62	58, 61	preq12d 3792	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → {((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))} = {((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))})
63	62	supeq1d 7317	. . . . . . . . . . . . . 14 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ) = sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ))
64	63, 1	ovmpoga 6208	. . . . . . . . . . . . 13 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝑋 × 𝑌) ∧ sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ) ∈ ℝ*) → (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) = sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ))
65	30, 33, 55, 64	syl3anc 1278	. . . . . . . . . . . 12 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) = sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ))
66	38, 41	oveq12d 6093	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)) = (𝐴𝐶𝑢))
67	48, 51	oveq12d 6093	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩)) = (𝐵𝐷𝑣))
68	66, 67	preq12d 3792	. . . . . . . . . . . . 13 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → {((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))} = {(𝐴𝐶𝑢), (𝐵𝐷𝑣)})
69	68	supeq1d 7317	. . . . . . . . . . . 12 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ) = sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ))
70	65, 69	eqtrd 2271	. . . . . . . . . . 11 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) = sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ))
71	70	breq1d 4135	. . . . . . . . . 10 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 ↔ sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ) < 𝑤))
72		xmetcl 15376	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝐴𝐶𝑢) ∈ ℝ*)
73	34, 28, 31, 72	syl3anc 1278	. . . . . . . . . . 11 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (𝐴𝐶𝑢) ∈ ℝ*)
74		xmetcl 15376	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝐵𝐷𝑣) ∈ ℝ*)
75	44, 29, 32, 74	syl3anc 1278	. . . . . . . . . . 11 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (𝐵𝐷𝑣) ∈ ℝ*)
76		rpxr 10041	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ+ → 𝑤 ∈ ℝ*)
77	76	ad3antlr 497	. . . . . . . . . . 11 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → 𝑤 ∈ ℝ*)
78		xrmaxltsup 12002	. . . . . . . . . . 11 ⊢ (((𝐴𝐶𝑢) ∈ ℝ* ∧ (𝐵𝐷𝑣) ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
79	73, 75, 77, 78	syl3anc 1278	. . . . . . . . . 10 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
80	71, 79	bitrd 188	. . . . . . . . 9 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
81		df-ov 6078	. . . . . . . . . . . . 13 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
82		df-ov 6078	. . . . . . . . . . . . 13 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
83	81, 82	oveq12i 6087	. . . . . . . . . . . 12 ⊢ ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩))
84	83	breq1i 4132	. . . . . . . . . . 11 ⊢ (((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧 ↔ ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧)
85	84	bicomi 132	. . . . . . . . . 10 ⊢ (((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧 ↔ ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)
86	85	a1i 9	. . . . . . . . 9 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧 ↔ ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))
87	80, 86	imbi12d 234	. . . . . . . 8 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧) ↔ (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
88	87	ralbidva 2546	. . . . . . 7 ⊢ ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢 ∈ 𝑋) → (∀𝑣 ∈ 𝑌 ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧) ↔ ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
89	88	ralbidva 2546	. . . . . 6 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) → (∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
90	27, 89	bitrid 192	. . . . 5 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) → (∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
91	90	rexbidva 2547	. . . 4 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ+) → (∃𝑤 ∈ ℝ+ ∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧) ↔ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
92	91	ralbidva 2546	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧) ↔ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
93	92	anbi2d 468	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑟)𝐶(1st ‘𝑠)), ((2nd ‘𝑟)𝐷(2nd ‘𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧)) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))
94	14, 20, 93	3bitr3d 218	1 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {cpr 3706  ⟨cop 3708   class class class wbr 4125   × cxp 4767  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  1^st c1st 6362  2^nd c2nd 6363  supcsup 7312  ℝ^*cxr 8349   < clt 8350  ℝ⁺crp 10033  ∞Metcxmet 14845  MetOpencmopn 14850   CnP ccnp 15210   ×_t ctx 15276

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-map 6914  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-xneg 10153  df-xadd 10154  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-topgen 13591  df-psmet 14852  df-xmet 14853  df-bl 14855  df-mopn 14856  df-top 15022  df-topon 15035  df-bases 15067  df-cnp 15213  df-tx 15277

This theorem is referenced by:  txmetcn  15543  limccnp2cntop  15701