Theorem txmetcnp 15329

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 2231	. . . 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 1024	. . . . 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 1025	. . . . 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 15318	. . 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 1029	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐸 ∈ (∞Met‘𝑍))
8		simprl 531	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
9		simprr 533	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
10	8, 9	opelxpd 4764	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
11		eqid 2231	. . . 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 15323	. . 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 1274	. 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 15321	. . . . 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 6043	. . . 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 5650	. . 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 2301	. 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 6036	. . . . . . . . 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 4103	. . . . . . . 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 5648	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑡) = (𝐹‘⟨𝑢, 𝑣⟩))
24	23	oveq2d 6044	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)))
25	24	breq1d 4103	. . . . . . . 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 4879	. . . . . 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 494	. . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . 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 4764	. . . . . . . . . . . . 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 529	. . . . . . . . . . . . . 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 4764	. . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . 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 15161	. . . . . . . . . . . . . . . 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 6322	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
38	28, 29, 37	syl2anc 411	. . . . . . . . . . . . . . . 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 2308	. . . . . . . . . . . . . . 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 6322	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → (1st ‘⟨𝑢, 𝑣⟩) = 𝑢)
41	40	adantll 476	. . . . . . . . . . . . . . . 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 2308	. . . . . . . . . . . . . . 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 6177	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . 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 15161	. . . . . . . . . . . . . . . 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 6323	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
48	28, 29, 47	syl2anc 411	. . . . . . . . . . . . . . . 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 2308	. . . . . . . . . . . . . . 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 6323	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣)
51	50	adantll 476	. . . . . . . . . . . . . . . 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 2308	. . . . . . . . . . . . . . 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 6177	. . . . . . . . . . . . . 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 11892	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 5648	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑟) = (1st ‘⟨𝐴, 𝐵⟩))
57		fveq2 5648	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑠) = (1st ‘⟨𝑢, 𝑣⟩))
58	56, 57	oveqan12d 6047	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((1st ‘𝑟)𝐶(1st ‘𝑠)) = ((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)))
59		fveq2 5648	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑟) = (2nd ‘⟨𝐴, 𝐵⟩))
60		fveq2 5648	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑠) = (2nd ‘⟨𝑢, 𝑣⟩))
61	59, 60	oveqan12d 6047	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((2nd ‘𝑟)𝐷(2nd ‘𝑠)) = ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩)))
62	58, 61	preq12d 3760	. . . . . . . . . . . . . . 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 7246	. . . . . . . . . . . . . 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 6161	. . . . . . . . . . . . 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 1274	. . . . . . . . . . . 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 6046	. . . . . . . . . . . . . 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 6046	. . . . . . . . . . . . . 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 3760	. . . . . . . . . . . . 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 7246	. . . . . . . . . . . 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 2264	. . . . . . . . . . 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 4103	. . . . . . . . . 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 15163	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝐴𝐶𝑢) ∈ ℝ*)
73	34, 28, 31, 72	syl3anc 1274	. . . . . . . . . . 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 15163	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝐵𝐷𝑣) ∈ ℝ*)
75	44, 29, 32, 74	syl3anc 1274	. . . . . . . . . . 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 9957	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℝ+ → 𝑤 ∈ ℝ*)
77	76	ad3antlr 493	. . . . . . . . . . 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 11898	. . . . . . . . . . 11 ⊢ (((𝐴𝐶𝑢) ∈ ℝ* ∧ (𝐵𝐷𝑣) ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
79	73, 75, 77, 78	syl3anc 1274	. . . . . . . . . 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 6031	. . . . . . . . . . . . 13 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
82		df-ov 6031	. . . . . . . . . . . . 13 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
83	81, 82	oveq12i 6040	. . . . . . . . . . . 12 ⊢ ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩))
84	83	breq1i 4100	. . . . . . . . . . 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 2529	. . . . . . 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 2529	. . . . . 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 2530	. . . 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 2529	. . 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 464	. 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 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  {cpr 3674  ⟨cop 3676   class class class wbr 4093   × cxp 4729  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028   ∈ cmpo 6030  1^st c1st 6310  2^nd c2nd 6311  supcsup 7241  ℝ^*cxr 8272   < clt 8273  ℝ⁺crp 9949  ∞Metcxmet 14632  MetOpencmopn 14637   CnP ccnp 14997   ×_t ctx 15063

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-map 6862  df-sup 7243  df-inf 7244  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-xneg 10068  df-xadd 10069  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-topgen 13423  df-psmet 14639  df-xmet 14640  df-bl 14642  df-mopn 14643  df-top 14809  df-topon 14822  df-bases 14854  df-cnp 15000  df-tx 15064

This theorem is referenced by:  txmetcn  15330  limccnp2cntop  15488