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Theorem xmettx 12668
 Description: The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)
Hypotheses
Ref Expression
xmetxp.p 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))
xmetxp.1 (𝜑𝑀 ∈ (∞Met‘𝑋))
xmetxp.2 (𝜑𝑁 ∈ (∞Met‘𝑌))
xmettx.j 𝐽 = (MetOpen‘𝑀)
xmettx.k 𝐾 = (MetOpen‘𝑁)
xmettx.l 𝐿 = (MetOpen‘𝑃)
Assertion
Ref Expression
xmettx (𝜑𝐿 = (𝐽 ×t 𝐾))
Distinct variable groups:   𝑢,𝑀,𝑣   𝑢,𝑁,𝑣   𝑢,𝑋,𝑣   𝑢,𝑌,𝑣
Allowed substitution hints:   𝜑(𝑣,𝑢)   𝑃(𝑣,𝑢)   𝐽(𝑣,𝑢)   𝐾(𝑣,𝑢)   𝐿(𝑣,𝑢)

Proof of Theorem xmettx
Dummy variables 𝑗 𝑘 𝑚 𝑛 𝑥 𝑦 𝑟 𝑠 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmetxp.p . . 3 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))
2 xmetxp.1 . . 3 (𝜑𝑀 ∈ (∞Met‘𝑋))
3 xmetxp.2 . . 3 (𝜑𝑁 ∈ (∞Met‘𝑌))
4 xmettx.j . . 3 𝐽 = (MetOpen‘𝑀)
5 xmettx.k . . 3 𝐾 = (MetOpen‘𝑁)
6 xmettx.l . . 3 𝐿 = (MetOpen‘𝑃)
71, 2, 3, 4, 5, 6xmettxlem 12667 . 2 (𝜑𝐿 ⊆ (𝐽 ×t 𝐾))
8 eqid 2137 . . . . . . . . . . . 12 (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
98elrnmpog 5876 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠)))
109elv 2685 . . . . . . . . . 10 (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠))
1110biimpi 119 . . . . . . . . 9 (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) → ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠))
1211adantl 275 . . . . . . . 8 ((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) → ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠))
13 xpeq1 4548 . . . . . . . . . 10 (𝑟 = 𝑥 → (𝑟 × 𝑠) = (𝑥 × 𝑠))
1413eqeq2d 2149 . . . . . . . . 9 (𝑟 = 𝑥 → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑠)))
15 xpeq2 4549 . . . . . . . . . 10 (𝑠 = 𝑦 → (𝑥 × 𝑠) = (𝑥 × 𝑦))
1615eqeq2d 2149 . . . . . . . . 9 (𝑠 = 𝑦 → (𝑤 = (𝑥 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑦)))
1714, 16cbvrex2v 2661 . . . . . . . 8 (∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠) ↔ ∃𝑥𝐽𝑦𝐾 𝑤 = (𝑥 × 𝑦))
1812, 17sylib 121 . . . . . . 7 ((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) → ∃𝑥𝐽𝑦𝐾 𝑤 = (𝑥 × 𝑦))
19 simpr 109 . . . . . . . . . 10 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑤 = (𝑥 × 𝑦))
20 simplll 522 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝜑)
21 simplrl 524 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑥𝐽)
22 simplrr 525 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑦𝐾)
234mopntopon 12601 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
242, 23syl 14 . . . . . . . . . . . . . . . . 17 (𝜑𝐽 ∈ (TopOn‘𝑋))
2524adantr 274 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
26 simprl 520 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑥𝐽)
27 toponss 12182 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
2825, 26, 27syl2anc 408 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑥𝑋)
295mopntopon 12601 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
303, 29syl 14 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ (TopOn‘𝑌))
3130adantr 274 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
32 simprr 521 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑦𝐾)
33 toponss 12182 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦𝐾) → 𝑦𝑌)
3431, 32, 33syl2anc 408 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑦𝑌)
35 xpss12 4641 . . . . . . . . . . . . . . 15 ((𝑥𝑋𝑦𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
3628, 34, 35syl2anc 408 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
371, 2, 3xmetxp 12665 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
38 unirnbl 12581 . . . . . . . . . . . . . . . 16 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → ran (ball‘𝑃) = (𝑋 × 𝑌))
3937, 38syl 14 . . . . . . . . . . . . . . 15 (𝜑 ran (ball‘𝑃) = (𝑋 × 𝑌))
4039adantr 274 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ran (ball‘𝑃) = (𝑋 × 𝑌))
4136, 40sseqtrrd 3131 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ⊆ ran (ball‘𝑃))
422ad2antrr 479 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
43 simplrl 524 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑥𝐽)
44 xp1st 6056 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑥 × 𝑦) → (1st𝑗) ∈ 𝑥)
4544adantl 275 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st𝑗) ∈ 𝑥)
464mopni2 12641 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥𝐽 ∧ (1st𝑗) ∈ 𝑥) → ∃𝑚 ∈ ℝ+ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
4742, 43, 45, 46syl3anc 1216 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑚 ∈ ℝ+ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
483ad2antrr 479 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑁 ∈ (∞Met‘𝑌))
49 simplrr 525 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑦𝐾)
50 xp2nd 6057 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (𝑥 × 𝑦) → (2nd𝑗) ∈ 𝑦)
5150adantl 275 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd𝑗) ∈ 𝑦)
525mopni2 12641 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑦𝐾 ∧ (2nd𝑗) ∈ 𝑦) → ∃𝑛 ∈ ℝ+ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
5348, 49, 51, 52syl3anc 1216 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑛 ∈ ℝ+ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
5453adantr 274 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → ∃𝑛 ∈ ℝ+ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
55 blf 12568 . . . . . . . . . . . . . . . . . . . . 21 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃):((𝑋 × 𝑌) × ℝ*)⟶𝒫 (𝑋 × 𝑌))
5637, 55syl 14 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (ball‘𝑃):((𝑋 × 𝑌) × ℝ*)⟶𝒫 (𝑋 × 𝑌))
5756ffnd 5268 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ball‘𝑃) Fn ((𝑋 × 𝑌) × ℝ*))
5857ad4antr 485 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (ball‘𝑃) Fn ((𝑋 × 𝑌) × ℝ*))
5936sselda 3092 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑗 ∈ (𝑋 × 𝑌))
6059ad2antrr 479 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑗 ∈ (𝑋 × 𝑌))
61 rpxr 9442 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℝ+𝑚 ∈ ℝ*)
6261ad2antrl 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → 𝑚 ∈ ℝ*)
6362adantr 274 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑚 ∈ ℝ*)
64 rpxr 9442 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ+𝑛 ∈ ℝ*)
6564ad2antrl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ*)
66 xrmincl 11028 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*)
6763, 65, 66syl2anc 408 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*)
68 fnovrn 5911 . . . . . . . . . . . . . . . . . 18 (((ball‘𝑃) Fn ((𝑋 × 𝑌) × ℝ*) ∧ 𝑗 ∈ (𝑋 × 𝑌) ∧ inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*) → (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∈ ran (ball‘𝑃))
6958, 60, 67, 68syl3anc 1216 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∈ ran (ball‘𝑃))
70 eleq2 2201 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑗𝑘𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < ))))
71 sseq1 3115 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑘 ⊆ (𝑥 × 𝑦) ↔ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦)))
7270, 71anbi12d 464 . . . . . . . . . . . . . . . . . 18 (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → ((𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)) ↔ (𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∧ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦))))
7372adantl 275 . . . . . . . . . . . . . . . . 17 ((((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) ∧ 𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < ))) → ((𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)) ↔ (𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∧ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦))))
7437ad4antr 485 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
75 simplrl 524 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑚 ∈ ℝ+)
76 simprl 520 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ+)
77 xrminrpcl 11036 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℝ+𝑛 ∈ ℝ+) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ+)
7875, 76, 77syl2anc 408 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ+)
79 blcntr 12574 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝑗 ∈ (𝑋 × 𝑌) ∧ inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ+) → 𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )))
8074, 60, 78, 79syl3anc 1216 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )))
8142ad2antrr 479 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
8248ad2antrr 479 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑁 ∈ (∞Met‘𝑌))
831, 81, 82, 67, 60xmetxpbl 12666 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) = (((1st𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) × ((2nd𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < ))))
8428adantr 274 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑥𝑋)
8584, 45sseldd 3093 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st𝑗) ∈ 𝑋)
8685ad2antrr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (1st𝑗) ∈ 𝑋)
87 xrmin1inf 11029 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑚)
8863, 65, 87syl2anc 408 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑚)
89 ssbl 12584 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝑗) ∈ 𝑋) ∧ (inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*𝑚 ∈ ℝ*) ∧ inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑚) → ((1st𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ ((1st𝑗)(ball‘𝑀)𝑚))
9081, 86, 67, 63, 88, 89syl221anc 1227 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((1st𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ ((1st𝑗)(ball‘𝑀)𝑚))
91 simplrr 525 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
9290, 91sstrd 3102 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((1st𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ 𝑥)
9334adantr 274 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑦𝑌)
9493, 51sseldd 3093 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd𝑗) ∈ 𝑌)
9594ad2antrr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (2nd𝑗) ∈ 𝑌)
96 xrmin2inf 11030 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑛)
9763, 65, 96syl2anc 408 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑛)
98 ssbl 12584 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝑗) ∈ 𝑌) ∧ (inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*𝑛 ∈ ℝ*) ∧ inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑛) → ((2nd𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ ((2nd𝑗)(ball‘𝑁)𝑛))
9982, 95, 67, 65, 97, 98syl221anc 1227 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((2nd𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ ((2nd𝑗)(ball‘𝑁)𝑛))
100 simprr 521 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
10199, 100sstrd 3102 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((2nd𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ 𝑦)
102 xpss12 4641 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ 𝑥 ∧ ((2nd𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ 𝑦) → (((1st𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) × ((2nd𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < ))) ⊆ (𝑥 × 𝑦))
10392, 101, 102syl2anc 408 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (((1st𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) × ((2nd𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < ))) ⊆ (𝑥 × 𝑦))
10483, 103eqsstrd 3128 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦))
10580, 104jca 304 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∧ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦)))
10669, 73, 105rspcedvd 2790 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
10754, 106rexlimddv 2552 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
10847, 107rexlimddv 2552 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
109108ralrimiva 2503 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
11041, 109jca 304 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ((𝑥 × 𝑦) ⊆ ran (ball‘𝑃) ∧ ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦))))
111 blex 12545 . . . . . . . . . . . . . . 15 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃) ∈ V)
11237, 111syl 14 . . . . . . . . . . . . . 14 (𝜑 → (ball‘𝑃) ∈ V)
113112adantr 274 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (ball‘𝑃) ∈ V)
114 rnexg 4799 . . . . . . . . . . . . 13 ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
115 eltg2 12211 . . . . . . . . . . . . 13 (ran (ball‘𝑃) ∈ V → ((𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)) ↔ ((𝑥 × 𝑦) ⊆ ran (ball‘𝑃) ∧ ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))))
116113, 114, 1153syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ((𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)) ↔ ((𝑥 × 𝑦) ⊆ ran (ball‘𝑃) ∧ ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))))
117110, 116mpbird 166 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)))
11820, 21, 22, 117syl12anc 1214 . . . . . . . . . 10 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → (𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)))
11919, 118eqeltrd 2214 . . . . . . . . 9 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑤 ∈ (topGen‘ran (ball‘𝑃)))
120119ex 114 . . . . . . . 8 (((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) → (𝑤 = (𝑥 × 𝑦) → 𝑤 ∈ (topGen‘ran (ball‘𝑃))))
121120rexlimdvva 2555 . . . . . . 7 ((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) → (∃𝑥𝐽𝑦𝐾 𝑤 = (𝑥 × 𝑦) → 𝑤 ∈ (topGen‘ran (ball‘𝑃))))
12218, 121mpd 13 . . . . . 6 ((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) → 𝑤 ∈ (topGen‘ran (ball‘𝑃)))
123122ex 114 . . . . 5 (𝜑 → (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) → 𝑤 ∈ (topGen‘ran (ball‘𝑃))))
124123ssrdv 3098 . . . 4 (𝜑 → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃)))
1254mopntop 12602 . . . . . . . 8 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
1262, 125syl 14 . . . . . . 7 (𝜑𝐽 ∈ Top)
1275mopntop 12602 . . . . . . . 8 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
1283, 127syl 14 . . . . . . 7 (𝜑𝐾 ∈ Top)
129 mpoexga 6103 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
130126, 128, 129syl2anc 408 . . . . . 6 (𝜑 → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
131 rnexg 4799 . . . . . 6 ((𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
132130, 131syl 14 . . . . 5 (𝜑 → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
13337, 111, 1143syl 17 . . . . 5 (𝜑 → ran (ball‘𝑃) ∈ V)
134 tgss3 12236 . . . . 5 ((ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V ∧ ran (ball‘𝑃) ∈ V) → ((topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ⊆ (topGen‘ran (ball‘𝑃)) ↔ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃))))
135132, 133, 134syl2anc 408 . . . 4 (𝜑 → ((topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ⊆ (topGen‘ran (ball‘𝑃)) ↔ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃))))
136124, 135mpbird 166 . . 3 (𝜑 → (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ⊆ (topGen‘ran (ball‘𝑃)))
137 eqid 2137 . . . . 5 ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
138137txval 12413 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
139126, 128, 138syl2anc 408 . . 3 (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
1406mopnval 12600 . . . 4 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → 𝐿 = (topGen‘ran (ball‘𝑃)))
14137, 140syl 14 . . 3 (𝜑𝐿 = (topGen‘ran (ball‘𝑃)))
142136, 139, 1413sstr4d 3137 . 2 (𝜑 → (𝐽 ×t 𝐾) ⊆ 𝐿)
1437, 142eqssd 3109 1 (𝜑𝐿 = (𝐽 ×t 𝐾))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2414  ∃wrex 2415  Vcvv 2681   ⊆ wss 3066  𝒫 cpw 3505  {cpr 3523  ∪ cuni 3731   class class class wbr 3924   × cxp 4532  ran crn 4535   Fn wfn 5113  ⟶wf 5114  ‘cfv 5118  (class class class)co 5767   ∈ cmpo 5769  1st c1st 6029  2nd c2nd 6030  supcsup 6862  infcinf 6863  ℝ*cxr 7792   < clt 7793   ≤ cle 7794  ℝ+crp 9434  topGenctg 12124  ∞Metcxmet 12138  ballcbl 12140  MetOpencmopn 12143  Topctop 12153  TopOnctopon 12166   ×t ctx 12410 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733 This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-map 6537  df-sup 6864  df-inf 6865  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-xneg 9552  df-xadd 9553  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-topgen 12130  df-psmet 12145  df-xmet 12146  df-bl 12148  df-mopn 12149  df-top 12154  df-topon 12167  df-bases 12199  df-tx 12411 This theorem is referenced by:  txmetcnp  12676
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