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Theorem txmetcnp 13685
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)) ∧ (𝐴𝑋𝐵𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((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.
StepHypRef Expression
1 eqid 2177 . . . 4 (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < )) = (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))
2 simp1 997 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → 𝐶 ∈ (∞Met‘𝑋))
32adantr 276 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐶 ∈ (∞Met‘𝑋))
4 simp2 998 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → 𝐷 ∈ (∞Met‘𝑌))
54adantr 276 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐷 ∈ (∞Met‘𝑌))
61, 3, 5xmetxp 13674 . . 3 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < )) ∈ (∞Met‘(𝑋 × 𝑌)))
7 simpl3 1002 . . 3 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐸 ∈ (∞Met‘𝑍))
8 simprl 529 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
9 simprr 531 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
108, 9opelxpd 4656 . . 3 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
11 eqid 2177 . . . 4 (MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))) = (MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < )))
12 txmetcnp.4 . . . 4 𝐿 = (MetOpen‘𝐸)
1311, 12metcnp 13679 . . 3 (((𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < )) ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝐸 ∈ (∞Met‘𝑍) ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → (𝐹 ∈ (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, 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({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) < 𝑧))))
146, 7, 10, 13syl3anc 1238 . 2 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (𝐹 ∈ (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, 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({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) < 𝑧))))
15 metcn.2 . . . . . 6 𝐽 = (MetOpen‘𝐶)
16 metcn.4 . . . . . 6 𝐾 = (MetOpen‘𝐷)
171, 3, 5, 15, 16, 11xmettx 13677 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))) = (𝐽 ×t 𝐾))
1817oveq1d 5884 . . . 4 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → ((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))) CnP 𝐿) = ((𝐽 ×t 𝐾) CnP 𝐿))
1918fveq1d 5513 . . 3 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))) CnP 𝐿)‘⟨𝐴, 𝐵⟩) = (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))
2019eleq2d 2247 . 2 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (𝐹 ∈ (((MetOpen‘(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ 𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩)))
21 oveq2 5877 . . . . . . . . 9 (𝑡 = ⟨𝑢, 𝑣⟩ → (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) = (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩))
2221breq1d 4010 . . . . . . . 8 (𝑡 = ⟨𝑢, 𝑣⟩ → ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) < 𝑤 ↔ (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤))
23 fveq2 5511 . . . . . . . . . 10 (𝑡 = ⟨𝑢, 𝑣⟩ → (𝐹𝑡) = (𝐹‘⟨𝑢, 𝑣⟩))
2423oveq2d 5885 . . . . . . . . 9 (𝑡 = ⟨𝑢, 𝑣⟩ → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)))
2524breq1d 4010 . . . . . . . 8 (𝑡 = ⟨𝑢, 𝑣⟩ → (((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) < 𝑧 ↔ ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧))
2622, 25imbi12d 234 . . . . . . 7 (𝑡 = ⟨𝑢, 𝑣⟩ → (((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) < 𝑧) ↔ ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧)))
2726ralxp 4766 . . . . . 6 (∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) < 𝑧) ↔ ∀𝑢𝑋𝑣𝑌 ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧))
288ad4antr 494 . . . . . . . . . . . . . 14 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → 𝐴𝑋)
299ad4antr 494 . . . . . . . . . . . . . 14 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → 𝐵𝑌)
3028, 29opelxpd 4656 . . . . . . . . . . . . 13 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
31 simplr 528 . . . . . . . . . . . . . 14 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → 𝑢𝑋)
32 simpr 110 . . . . . . . . . . . . . 14 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → 𝑣𝑌)
3331, 32opelxpd 4656 . . . . . . . . . . . . 13 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → ⟨𝑢, 𝑣⟩ ∈ (𝑋 × 𝑌))
342ad5antr 496 . . . . . . . . . . . . . . . 16 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → 𝐶 ∈ (∞Met‘𝑋))
35 xmetf 13517 . . . . . . . . . . . . . . . 16 (𝐶 ∈ (∞Met‘𝑋) → 𝐶:(𝑋 × 𝑋)⟶ℝ*)
3634, 35syl 14 . . . . . . . . . . . . . . 15 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → 𝐶:(𝑋 × 𝑋)⟶ℝ*)
37 op1stg 6145 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝐵𝑌) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
3828, 29, 37syl2anc 411 . . . . . . . . . . . . . . . 16 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
3938, 28eqeltrd 2254 . . . . . . . . . . . . . . 15 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (1st ‘⟨𝐴, 𝐵⟩) ∈ 𝑋)
40 op1stg 6145 . . . . . . . . . . . . . . . . 17 ((𝑢𝑋𝑣𝑌) → (1st ‘⟨𝑢, 𝑣⟩) = 𝑢)
4140adantll 476 . . . . . . . . . . . . . . . 16 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (1st ‘⟨𝑢, 𝑣⟩) = 𝑢)
4241, 31eqeltrd 2254 . . . . . . . . . . . . . . 15 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (1st ‘⟨𝑢, 𝑣⟩) ∈ 𝑋)
4336, 39, 42fovcdmd 6013 . . . . . . . . . . . . . 14 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → ((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)) ∈ ℝ*)
444ad5antr 496 . . . . . . . . . . . . . . . 16 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → 𝐷 ∈ (∞Met‘𝑌))
45 xmetf 13517 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (∞Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ*)
4644, 45syl 14 . . . . . . . . . . . . . . 15 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ*)
47 op2ndg 6146 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝐵𝑌) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
4828, 29, 47syl2anc 411 . . . . . . . . . . . . . . . 16 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
4948, 29eqeltrd 2254 . . . . . . . . . . . . . . 15 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (2nd ‘⟨𝐴, 𝐵⟩) ∈ 𝑌)
50 op2ndg 6146 . . . . . . . . . . . . . . . . 17 ((𝑢𝑋𝑣𝑌) → (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣)
5150adantll 476 . . . . . . . . . . . . . . . 16 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣)
5251, 32eqeltrd 2254 . . . . . . . . . . . . . . 15 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (2nd ‘⟨𝑢, 𝑣⟩) ∈ 𝑌)
5346, 49, 52fovcdmd 6013 . . . . . . . . . . . . . 14 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩)) ∈ ℝ*)
54 xrmaxcl 11244 . . . . . . . . . . . . . 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*)
5543, 53, 54syl2anc 411 . . . . . . . . . . . . 13 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ) ∈ ℝ*)
56 fveq2 5511 . . . . . . . . . . . . . . . . 17 (𝑟 = ⟨𝐴, 𝐵⟩ → (1st𝑟) = (1st ‘⟨𝐴, 𝐵⟩))
57 fveq2 5511 . . . . . . . . . . . . . . . . 17 (𝑠 = ⟨𝑢, 𝑣⟩ → (1st𝑠) = (1st ‘⟨𝑢, 𝑣⟩))
5856, 57oveqan12d 5888 . . . . . . . . . . . . . . . 16 ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((1st𝑟)𝐶(1st𝑠)) = ((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)))
59 fveq2 5511 . . . . . . . . . . . . . . . . 17 (𝑟 = ⟨𝐴, 𝐵⟩ → (2nd𝑟) = (2nd ‘⟨𝐴, 𝐵⟩))
60 fveq2 5511 . . . . . . . . . . . . . . . . 17 (𝑠 = ⟨𝑢, 𝑣⟩ → (2nd𝑠) = (2nd ‘⟨𝑢, 𝑣⟩))
6159, 60oveqan12d 5888 . . . . . . . . . . . . . . . 16 ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → ((2nd𝑟)𝐷(2nd𝑠)) = ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩)))
6258, 61preq12d 3676 . . . . . . . . . . . . . . 15 ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → {((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))} = {((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))})
6362supeq1d 6980 . . . . . . . . . . . . . 14 ((𝑟 = ⟨𝐴, 𝐵⟩ ∧ 𝑠 = ⟨𝑢, 𝑣⟩) → sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ) = sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ))
6463, 1ovmpoga 5998 . . . . . . . . . . . . 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({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) = sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ))
6530, 33, 55, 64syl3anc 1238 . . . . . . . . . . . 12 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) = sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ))
6638, 41oveq12d 5887 . . . . . . . . . . . . . 14 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → ((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)) = (𝐴𝐶𝑢))
6748, 51oveq12d 5887 . . . . . . . . . . . . . 14 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩)) = (𝐵𝐷𝑣))
6866, 67preq12d 3676 . . . . . . . . . . . . 13 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → {((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))} = {(𝐴𝐶𝑢), (𝐵𝐷𝑣)})
6968supeq1d 6980 . . . . . . . . . . . 12 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → sup({((1st ‘⟨𝐴, 𝐵⟩)𝐶(1st ‘⟨𝑢, 𝑣⟩)), ((2nd ‘⟨𝐴, 𝐵⟩)𝐷(2nd ‘⟨𝑢, 𝑣⟩))}, ℝ*, < ) = sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ))
7065, 69eqtrd 2210 . . . . . . . . . . 11 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) = sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ))
7170breq1d 4010 . . . . . . . . . 10 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 ↔ sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ) < 𝑤))
72 xmetcl 13519 . . . . . . . . . . . 12 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐴𝑋𝑢𝑋) → (𝐴𝐶𝑢) ∈ ℝ*)
7334, 28, 31, 72syl3anc 1238 . . . . . . . . . . 11 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (𝐴𝐶𝑢) ∈ ℝ*)
74 xmetcl 13519 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑌) ∧ 𝐵𝑌𝑣𝑌) → (𝐵𝐷𝑣) ∈ ℝ*)
7544, 29, 32, 74syl3anc 1238 . . . . . . . . . . 11 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (𝐵𝐷𝑣) ∈ ℝ*)
76 rpxr 9648 . . . . . . . . . . . 12 (𝑤 ∈ ℝ+𝑤 ∈ ℝ*)
7776ad3antlr 493 . . . . . . . . . . 11 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → 𝑤 ∈ ℝ*)
78 xrmaxltsup 11250 . . . . . . . . . . 11 (((𝐴𝐶𝑢) ∈ ℝ* ∧ (𝐵𝐷𝑣) ∈ ℝ*𝑤 ∈ ℝ*) → (sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
7973, 75, 77, 78syl3anc 1238 . . . . . . . . . 10 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (sup({(𝐴𝐶𝑢), (𝐵𝐷𝑣)}, ℝ*, < ) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
8071, 79bitrd 188 . . . . . . . . 9 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 ↔ ((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤)))
81 df-ov 5872 . . . . . . . . . . . . 13 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
82 df-ov 5872 . . . . . . . . . . . . 13 (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
8381, 82oveq12i 5881 . . . . . . . . . . . 12 ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) = ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩))
8483breq1i 4007 . . . . . . . . . . 11 (((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧 ↔ ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧)
8584bicomi 132 . . . . . . . . . 10 (((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧 ↔ ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)
8685a1i 9 . . . . . . . . 9 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧 ↔ ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))
8780, 86imbi12d 234 . . . . . . . 8 (((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) ∧ 𝑣𝑌) → (((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧) ↔ (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
8887ralbidva 2473 . . . . . . 7 ((((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) ∧ 𝑢𝑋) → (∀𝑣𝑌 ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧) ↔ ∀𝑣𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
8988ralbidva 2473 . . . . . 6 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) → (∀𝑢𝑋𝑣𝑌 ((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))⟨𝑢, 𝑣⟩) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹‘⟨𝑢, 𝑣⟩)) < 𝑧) ↔ ∀𝑢𝑋𝑣𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
9027, 89bitrid 192 . . . . 5 (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) ∧ 𝑤 ∈ ℝ+) → (∀𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) < 𝑧) ↔ ∀𝑢𝑋𝑣𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
9190rexbidva 2474 . . . 4 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑧 ∈ ℝ+) → (∃𝑤 ∈ ℝ+𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) < 𝑧) ↔ ∃𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
9291ralbidva 2473 . . 3 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (∀𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) < 𝑧) ↔ ∀𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧)))
9392anbi2d 464 . 2 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → ((𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑡 ∈ (𝑋 × 𝑌)((⟨𝐴, 𝐵⟩(𝑟 ∈ (𝑋 × 𝑌), 𝑠 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑟)𝐶(1st𝑠)), ((2nd𝑟)𝐷(2nd𝑠))}, ℝ*, < ))𝑡) < 𝑤 → ((𝐹‘⟨𝐴, 𝐵⟩)𝐸(𝐹𝑡)) < 𝑧)) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))
9414, 20, 933bitr3d 218 1 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((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 3592  cop 3594   class class class wbr 4000   × cxp 4621  wf 5208  cfv 5212  (class class class)co 5869  cmpo 5871  1st c1st 6133  2nd c2nd 6134  supcsup 6975  *cxr 7981   < clt 7982  +crp 9640  ∞Metcxmet 13147  MetOpencmopn 13152   CnP ccnp 13353   ×t ctx 13419
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-map 6644  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-xneg 9759  df-xadd 9760  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-topgen 12657  df-psmet 13154  df-xmet 13155  df-bl 13157  df-mopn 13158  df-top 13163  df-topon 13176  df-bases 13208  df-cnp 13356  df-tx 13420
This theorem is referenced by:  txmetcn  13686  limccnp2cntop  13813
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