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Theorem txmetcnp 14057

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 2177	. . . 4 ⊢ (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < )) = (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))
2		simp1 997	. . . . 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 998	. . . . 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 14046	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^*, < )) ∈ (∞Met‘(𝑋 × 𝑌)))
7		simpl3 1002	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐸 ∈ (∞Met‘𝑍))
8		simprl 529	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
9		simprr 531	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
10	8, 9	opelxpd 4661	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
11		eqid 2177	. . . 4 ⊢ (MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))) = (MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < )))
12		txmetcnp.4	. . . 4 ⊢ 𝐿 = (MetOpen‘𝐸)
13	11, 12	metcnp 14051	. . 3 ⊢ (((𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < )) ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝐸 ∈ (∞Met‘𝑍) ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → (𝐹 ∈ (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧))))
14	6, 7, 10, 13	syl3anc 1238	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐹 ∈ (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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 14049	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^*, < ))) = (𝐽 ×_t 𝐾))
18	17	oveq1d 5892	. . . 4 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^*, < ))) CnP 𝐿) = ((𝐽 ×_t 𝐾) CnP 𝐿))
19	18	fveq1d 5519	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^*, < ))) CnP 𝐿)‘⟨𝐴, 𝐵⟩) = (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))
20	19	eleq2d 2247	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐹 ∈ (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^*, < ))) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ 𝐹 ∈ (((𝐽 ×_t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩)))
21		oveq2 5885	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))𝑡) = (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))⟨𝑢, 𝑣⟩))
22	21	breq1d 4015	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))𝑡) < 𝑤 ↔ (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))⟨𝑢, 𝑣⟩) < 𝑤))
23		fveq2 5517	. . . . . . . . . 10 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑡) = (𝐹‘⟨𝑢, 𝑣⟩))
24	23	oveq2d 5893	. . . . . . . . 9 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)))
25	24	breq1d 4015	. . . . . . . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧) ↔ ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧)))
27	26	ralxp 4772	. . . . . 6 ⊢ (∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘𝑡)) < 𝑧) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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 4661	. . . . . . . . . . . . 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 528	. . . . . . . . . . . . . 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 4661	. . . . . . . . . . . . 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 13889	. . . . . . . . . . . . . . . 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 6153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (1^st ‘⟨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) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
39	38, 28	eqeltrd 2254	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (1^st ‘⟨𝐴, 𝐵⟩) ∈ 𝑋)
40		op1stg 6153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → (1^st ‘⟨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) → (1^st ‘⟨𝑢, 𝑣⟩) = 𝑢)
42	41, 31	eqeltrd 2254	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (1^st ‘⟨𝑢, 𝑣⟩) ∈ 𝑋)
43	36, 39, 42	fovcdmd 6021	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨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 13889	. . . . . . . . . . . . . . . 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 6154	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (2^nd ‘⟨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) → (2^nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
49	48, 29	eqeltrd 2254	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (2^nd ‘⟨𝐴, 𝐵⟩) ∈ 𝑌)
50		op2ndg 6154	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝑋 ∧ 𝑣 ∈ 𝑌) → (2^nd ‘⟨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) → (2^nd ‘⟨𝑢, 𝑣⟩) = 𝑣)
52	51, 32	eqeltrd 2254	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → (2^nd ‘⟨𝑢, 𝑣⟩) ∈ 𝑌)
53	46, 49, 52	fovcdmd 6021	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩)) ∈ ℝ^*)
54		xrmaxcl 11262	. . . . . . . . . . . . . 14 ⊢ ((((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)) ∈ ℝ^* ∧ ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩)) ∈ ℝ^) → sup({((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)), ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨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({((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)), ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩))}, ℝ^, < ) ∈ ℝ^)
56		fveq2 5517	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ⟨𝐴, 𝐵⟩ → (1^st ‘𝑟) = (1^st ‘⟨𝐴, 𝐵⟩))
57		fveq2 5517	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨𝑢, 𝑣⟩ → (1^st ‘𝑠) = (1^st ‘⟨𝑢, 𝑣⟩))
58	56, 57	oveqan12d 5896	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((1^st ‘𝑟)𝐶(1^st ‘𝑠)) = ((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)))
59		fveq2 5517	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = ⟨𝐴, 𝐵⟩ → (2^nd ‘𝑟) = (2^nd ‘⟨𝐴, 𝐵⟩))
60		fveq2 5517	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨𝑢, 𝑣⟩ → (2^nd ‘𝑠) = (2^nd ‘⟨𝑢, 𝑣⟩))
61	59, 60	oveqan12d 5896	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠)) = ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩)))
62	58, 61	preq12d 3679	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → {((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))} = {((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)), ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩))})
63	62	supeq1d 6988	. . . . . . . . . . . . . 14 ⊢ ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ) = sup({((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)), ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩))}, ℝ^, < ))
64	63, 1	ovmpoga 6006	. . . . . . . . . . . . 13 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝑋 × 𝑌) ∧ sup({((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)), ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩))}, ℝ^, < ) ∈ ℝ^) → (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))⟨𝑢, 𝑣⟩) = sup({((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)), ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩))}, ℝ^, < ))
65	30, 33, 55, 64	syl3anc 1238	. . . . . . . . . . . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))⟨𝑢, 𝑣⟩) = sup({((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)), ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩))}, ℝ^, < ))
66	38, 41	oveq12d 5895	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)) = (𝐴𝐶𝑢))
67	48, 51	oveq12d 5895	. . . . . . . . . . . . . 14 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩)) = (𝐵𝐷𝑣))
68	66, 67	preq12d 3679	. . . . . . . . . . . . 13 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → {((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)), ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩))} = {(𝐴𝐶𝑢), (𝐵𝐷𝑣)})
69	68	supeq1d 6988	. . . . . . . . . . . 12 ⊢ (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑋) ∧ 𝑣 ∈ 𝑌) → sup({((1^st ‘⟨𝐴, 𝐵⟩)𝐶(1^st ‘⟨𝑢, 𝑣⟩)), ((2^nd ‘⟨𝐴, 𝐵⟩)𝐷(2^nd ‘⟨𝑢, 𝑣⟩))}, ℝ^, < ) = sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ^, < ))
70	65, 69	eqtrd 2210	. . . . . . . . . . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))⟨𝑢, 𝑣⟩) = sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ^, < ))
71	70	breq1d 4015	. . . . . . . . . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^, < ))⟨𝑢, 𝑣⟩) < 𝑤 ↔ sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ^, < ) < 𝑤))
72		xmetcl 13891	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑢 ∈ 𝑋) → (𝐴𝐶𝑢) ∈ ℝ^*)
73	34, 28, 31, 72	syl3anc 1238	. . . . . . . . . . 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 13891	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝐵𝐷𝑣) ∈ ℝ^*)
75	44, 29, 32, 74	syl3anc 1238	. . . . . . . . . . 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 9663	. . . . . . . . . . . 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 11268	. . . . . . . . . . 11 ⊢ (((𝐴𝐶𝑢) ∈ ℝ^* ∧ (𝐵𝐷𝑣) ∈ ℝ^* ∧ 𝑤 ∈ ℝ^) → (sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ^, < ) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
79	73, 75, 77, 78	syl3anc 1238	. . . . . . . . . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^*, < ))⟨𝑢, 𝑣⟩) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
81		df-ov 5880	. . . . . . . . . . . . 13 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
82		df-ov 5880	. . . . . . . . . . . . 13 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
83	81, 82	oveq12i 5889	. . . . . . . . . . . 12 ⊢ ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩))
84	83	breq1i 4012	. . . . . . . . . . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘𝑠))}, ℝ^*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧) ↔ (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
88	87	ralbidva 2473	. . . . . . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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 2473	. . . . . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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 2474	. . . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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 2473	. . 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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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({((1^st ‘𝑟)𝐶(1^st ‘𝑠)), ((2^nd ‘𝑟)𝐷(2^nd ‘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))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 {cpr 3595 ⟨cop 3597 class class class wbr 4005 × cxp 4626 ⟶wf 5214 ‘cfv 5218 (class class class)co 5877 ∈ cmpo 5879 1^st c1st 6141 2^nd c2nd 6142 supcsup 6983 ℝ^*cxr 7993 < clt 7994 ℝ⁺crp 9655 ∞Metcxmet 13479 MetOpencmopn 13484 CnP ccnp 13725 ×_t ctx 13791

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-stab 831 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-q 9622 df-rp 9656 df-xneg 9774 df-xadd 9775 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-topgen 12714 df-psmet 13486 df-xmet 13487 df-bl 13489 df-mopn 13490 df-top 13537 df-topon 13550 df-bases 13582 df-cnp 13728 df-tx 13792

This theorem is referenced by: txmetcn 14058 limccnp2cntop 14185

W3C validator