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Theorem xmettx 13304
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 13303 . 2 (𝜑𝐿 ⊆ (𝐽 ×t 𝐾))
8 eqid 2170 . . . . . . . . . . . 12 (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
98elrnmpog 5965 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠)))
109elv 2734 . . . . . . . . . 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 4625 . . . . . . . . . 10 (𝑟 = 𝑥 → (𝑟 × 𝑠) = (𝑥 × 𝑠))
1413eqeq2d 2182 . . . . . . . . 9 (𝑟 = 𝑥 → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑠)))
15 xpeq2 4626 . . . . . . . . . 10 (𝑠 = 𝑦 → (𝑥 × 𝑠) = (𝑥 × 𝑦))
1615eqeq2d 2182 . . . . . . . . 9 (𝑠 = 𝑦 → (𝑤 = (𝑥 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑦)))
1714, 16cbvrex2v 2710 . . . . . . . 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 528 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝜑)
21 simplrl 530 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑥𝐽)
22 simplrr 531 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑦𝐾)
234mopntopon 13237 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
242, 23syl 14 . . . . . . . . . . . . . . . . 17 (𝜑𝐽 ∈ (TopOn‘𝑋))
2524adantr 274 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
26 simprl 526 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑥𝐽)
27 toponss 12818 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
2825, 26, 27syl2anc 409 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑥𝑋)
295mopntopon 13237 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
303, 29syl 14 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ (TopOn‘𝑌))
3130adantr 274 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
32 simprr 527 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑦𝐾)
33 toponss 12818 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦𝐾) → 𝑦𝑌)
3431, 32, 33syl2anc 409 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑦𝑌)
35 xpss12 4718 . . . . . . . . . . . . . . 15 ((𝑥𝑋𝑦𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
3628, 34, 35syl2anc 409 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
371, 2, 3xmetxp 13301 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
38 unirnbl 13217 . . . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ⊆ ran (ball‘𝑃))
422ad2antrr 485 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
43 simplrl 530 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑥𝐽)
44 xp1st 6144 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑥 × 𝑦) → (1st𝑗) ∈ 𝑥)
4544adantl 275 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st𝑗) ∈ 𝑥)
464mopni2 13277 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥𝐽 ∧ (1st𝑗) ∈ 𝑥) → ∃𝑚 ∈ ℝ+ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
4742, 43, 45, 46syl3anc 1233 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑚 ∈ ℝ+ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
483ad2antrr 485 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑁 ∈ (∞Met‘𝑌))
49 simplrr 531 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑦𝐾)
50 xp2nd 6145 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (𝑥 × 𝑦) → (2nd𝑗) ∈ 𝑦)
5150adantl 275 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd𝑗) ∈ 𝑦)
525mopni2 13277 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑦𝐾 ∧ (2nd𝑗) ∈ 𝑦) → ∃𝑛 ∈ ℝ+ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
5348, 49, 51, 52syl3anc 1233 . . . . . . . . . . . . . . . . 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 13204 . . . . . . . . . . . . . . . . . . . . 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 5348 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ball‘𝑃) Fn ((𝑋 × 𝑌) × ℝ*))
5857ad4antr 491 . . . . . . . . . . . . . . . . . 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 3147 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑗 ∈ (𝑋 × 𝑌))
6059ad2antrr 485 . . . . . . . . . . . . . . . . . 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 9618 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℝ+𝑚 ∈ ℝ*)
6261ad2antrl 487 . . . . . . . . . . . . . . . . . . . 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 9618 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ+𝑛 ∈ ℝ*)
6564ad2antrl 487 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ*)
66 xrmincl 11229 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*)
6763, 65, 66syl2anc 409 . . . . . . . . . . . . . . . . . 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 6000 . . . . . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . . . . . 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 2234 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑗𝑘𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < ))))
71 sseq1 3170 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑘 ⊆ (𝑥 × 𝑦) ↔ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦)))
7270, 71anbi12d 470 . . . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . . . . . 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 530 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑚 ∈ ℝ+)
76 simprl 526 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ+)
77 xrminrpcl 11237 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℝ+𝑛 ∈ ℝ+) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ+)
7875, 76, 77syl2anc 409 . . . . . . . . . . . . . . . . . . 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 13210 . . . . . . . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
8248ad2antrr 485 . . . . . . . . . . . . . . . . . . . 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 13302 . . . . . . . . . . . . . . . . . . 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 3148 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st𝑗) ∈ 𝑋)
8685ad2antrr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (1st𝑗) ∈ 𝑋)
87 xrmin1inf 11230 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑚)
8863, 65, 87syl2anc 409 . . . . . . . . . . . . . . . . . . . . . 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 13220 . . . . . . . . . . . . . . . . . . . . . 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 1244 . . . . . . . . . . . . . . . . . . . . 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 531 . . . . . . . . . . . . . . . . . . . . 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 3157 . . . . . . . . . . . . . . . . . . . 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 3148 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd𝑗) ∈ 𝑌)
9594ad2antrr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (2nd𝑗) ∈ 𝑌)
96 xrmin2inf 11231 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑛)
9763, 65, 96syl2anc 409 . . . . . . . . . . . . . . . . . . . . . 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 13220 . . . . . . . . . . . . . . . . . . . . . 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 1244 . . . . . . . . . . . . . . . . . . . . 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 527 . . . . . . . . . . . . . . . . . . . . 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 3157 . . . . . . . . . . . . . . . . . . . 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 4718 . . . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . . . . 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 3183 . . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . . 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 2592 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
10847, 107rexlimddv 2592 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
109108ralrimiva 2543 . . . . . . . . . . . . 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 13181 . . . . . . . . . . . . . . 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 4876 . . . . . . . . . . . . 13 ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
115 eltg2 12847 . . . . . . . . . . . . 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 1231 . . . . . . . . . 10 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → (𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)))
11919, 118eqeltrd 2247 . . . . . . . . 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 2595 . . . . . . 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 3153 . . . 4 (𝜑 → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃)))
1254mopntop 13238 . . . . . . . 8 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
1262, 125syl 14 . . . . . . 7 (𝜑𝐽 ∈ Top)
1275mopntop 13238 . . . . . . . 8 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
1283, 127syl 14 . . . . . . 7 (𝜑𝐾 ∈ Top)
129 mpoexga 6191 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
130126, 128, 129syl2anc 409 . . . . . 6 (𝜑 → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
131 rnexg 4876 . . . . . 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 12872 . . . . 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 409 . . . 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 2170 . . . . 5 ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
138137txval 13049 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
139126, 128, 138syl2anc 409 . . 3 (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
1406mopnval 13236 . . . 4 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → 𝐿 = (topGen‘ran (ball‘𝑃)))
14137, 140syl 14 . . 3 (𝜑𝐿 = (topGen‘ran (ball‘𝑃)))
142136, 139, 1413sstr4d 3192 . 2 (𝜑 → (𝐽 ×t 𝐾) ⊆ 𝐿)
1437, 142eqssd 3164 1 (𝜑𝐿 = (𝐽 ×t 𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  wral 2448  wrex 2449  Vcvv 2730  wss 3121  𝒫 cpw 3566  {cpr 3584   cuni 3796   class class class wbr 3989   × cxp 4609  ran crn 4612   Fn wfn 5193  wf 5194  cfv 5198  (class class class)co 5853  cmpo 5855  1st c1st 6117  2nd c2nd 6118  supcsup 6959  infcinf 6960  *cxr 7953   < clt 7954  cle 7955  +crp 9610  topGenctg 12594  ∞Metcxmet 12774  ballcbl 12776  MetOpencmopn 12779  Topctop 12789  TopOnctopon 12802   ×t ctx 13046
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835  df-tx 13047
This theorem is referenced by:  txmetcnp  13312
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