Theorem txmetcnp 12726

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 2140	. . . 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 982	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → 𝐶 ∈ (∞Met‘𝑋))
3	2	adantr 274	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐶 ∈ (∞Met‘𝑋))
4		simp2 983	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → 𝐷 ∈ (∞Met‘𝑌))
5	4	adantr 274	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐷 ∈ (∞Met‘𝑌))
6	1, 3, 5	xmetxp 12715	. . 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 987	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐸 ∈ (∞Met‘𝑍))
8		simprl 521	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
9		simprr 522	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
10	8, 9	opelxpd 4580	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
11		eqid 2140	. . . 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 12720	. . 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 1217	. 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 12718	. . . . 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 5797	. . . 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 5431	. . 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 2210	. 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 5790	. . . . . . . . 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 3947	. . . . . . . 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 5429	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑡) = (𝐹‘⟨𝑢, 𝑣⟩))
24	23	oveq2d 5798	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)))
25	24	breq1d 3947	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧 ↔ ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧))
26	22, 25	imbi12d 233	. . . . . . 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 4690	. . . . . 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 486	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 4580	. . . . . . . . . . . . 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 520	. . . . . . . . . . . . . 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 109	. . . . . . . . . . . . . 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 4580	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 12558	. . . . . . . . . . . . . . . 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 6056	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
38	28, 29, 37	syl2anc 409	. . . . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . . . 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 6056	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → (1st ‘⟨𝑢, 𝑣⟩) = 𝑢)
41	40	adantll 468	. . . . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . . . 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	fovrnd 5923	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . 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 12558	. . . . . . . . . . . . . . . 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 6057	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
48	28, 29, 47	syl2anc 409	. . . . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . . . 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 6057	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣)
51	50	adantll 468	. . . . . . . . . . . . . . . 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 2217	. . . . . . . . . . . . . . 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	fovrnd 5923	. . . . . . . . . . . . . 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 11053	. . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 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 5429	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑟) = (1st ‘⟨𝐴, 𝐵⟩))
57		fveq2 5429	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑠) = (1st ‘⟨𝑢, 𝑣⟩))
58	56, 57	oveqan12d 5801	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((1st ‘𝑟)𝐶(1st ‘𝑠)) = ((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)))
59		fveq2 5429	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑟) = (2nd ‘⟨𝐴, 𝐵⟩))
60		fveq2 5429	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑠) = (2nd ‘⟨𝑢, 𝑣⟩))
61	59, 60	oveqan12d 5801	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((2nd ‘𝑟)𝐷(2nd ‘𝑠)) = ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩)))
62	58, 61	preq12d 3616	. . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . 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 5908	. . . . . . . . . . . . 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 1217	. . . . . . . . . . . 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 5800	. . . . . . . . . . . . . 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 5800	. . . . . . . . . . . . . 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 3616	. . . . . . . . . . . . 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 6882	. . . . . . . . . . . 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 2173	. . . . . . . . . . 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 3947	. . . . . . . . . 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 12560	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝐴𝐶𝑢) ∈ ℝ*)
73	34, 28, 31, 72	syl3anc 1217	. . . . . . . . . . 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 12560	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝐵𝐷𝑣) ∈ ℝ*)
75	44, 29, 32, 74	syl3anc 1217	. . . . . . . . . . 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 9478	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ+ → 𝑤 ∈ ℝ*)
77	76	ad3antlr 485	. . . . . . . . . . 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 11059	. . . . . . . . . . 11 ⊢ (((𝐴𝐶𝑢) ∈ ℝ* ∧ (𝐵𝐷𝑣) ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
79	73, 75, 77, 78	syl3anc 1217	. . . . . . . . . 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 187	. . . . . . . . 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 5785	. . . . . . . . . . . . 13 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
82		df-ov 5785	. . . . . . . . . . . . 13 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
83	81, 82	oveq12i 5794	. . . . . . . . . . . 12 ⊢ ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩))
84	83	breq1i 3944	. . . . . . . . . . 11 ⊢ (((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧 ↔ ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧)
85	84	bicomi 131	. . . . . . . . . 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 233	. . . . . . . 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 2434	. . . . . . 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 2434	. . . . . 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	syl5bb 191	. . . . 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 2435	. . . 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 2434	. . 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 460	. 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 217	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 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  {cpr 3533  ⟨cop 3535   class class class wbr 3937   × cxp 4545  ⟶wf 5127  ‘cfv 5131  (class class class)co 5782   ∈ cmpo 5784  1^st c1st 6044  2^nd c2nd 6045  supcsup 6877  ℝ^*cxr 7823   < clt 7824  ℝ⁺crp 9470  ∞Metcxmet 12188  MetOpencmopn 12193   CnP ccnp 12394   ×_t ctx 12460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-map 6552  df-sup 6879  df-inf 6880  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-xneg 9589  df-xadd 9590  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-topgen 12180  df-psmet 12195  df-xmet 12196  df-bl 12198  df-mopn 12199  df-top 12204  df-topon 12217  df-bases 12249  df-cnp 12397  df-tx 12461

This theorem is referenced by:  txmetcn  12727  limccnp2cntop  12854