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Theorem xmettx 13903
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 13902 . 2 (𝜑𝐿 ⊆ (𝐽 ×t 𝐾))
8 eqid 2177 . . . . . . . . . . . 12 (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
98elrnmpog 5986 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠)))
109elv 2741 . . . . . . . . . 10 (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠))
1110biimpi 120 . . . . . . . . 9 (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) → ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠))
1211adantl 277 . . . . . . . 8 ((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) → ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠))
13 xpeq1 4640 . . . . . . . . . 10 (𝑟 = 𝑥 → (𝑟 × 𝑠) = (𝑥 × 𝑠))
1413eqeq2d 2189 . . . . . . . . 9 (𝑟 = 𝑥 → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑠)))
15 xpeq2 4641 . . . . . . . . . 10 (𝑠 = 𝑦 → (𝑥 × 𝑠) = (𝑥 × 𝑦))
1615eqeq2d 2189 . . . . . . . . 9 (𝑠 = 𝑦 → (𝑤 = (𝑥 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑦)))
1714, 16cbvrex2v 2717 . . . . . . . 8 (∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠) ↔ ∃𝑥𝐽𝑦𝐾 𝑤 = (𝑥 × 𝑦))
1812, 17sylib 122 . . . . . . 7 ((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) → ∃𝑥𝐽𝑦𝐾 𝑤 = (𝑥 × 𝑦))
19 simpr 110 . . . . . . . . . 10 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑤 = (𝑥 × 𝑦))
20 simplll 533 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝜑)
21 simplrl 535 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑥𝐽)
22 simplrr 536 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑦𝐾)
234mopntopon 13836 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
242, 23syl 14 . . . . . . . . . . . . . . . . 17 (𝜑𝐽 ∈ (TopOn‘𝑋))
2524adantr 276 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
26 simprl 529 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑥𝐽)
27 toponss 13417 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
2825, 26, 27syl2anc 411 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑥𝑋)
295mopntopon 13836 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
303, 29syl 14 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ (TopOn‘𝑌))
3130adantr 276 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
32 simprr 531 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑦𝐾)
33 toponss 13417 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦𝐾) → 𝑦𝑌)
3431, 32, 33syl2anc 411 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑦𝑌)
35 xpss12 4733 . . . . . . . . . . . . . . 15 ((𝑥𝑋𝑦𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
3628, 34, 35syl2anc 411 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
371, 2, 3xmetxp 13900 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
38 unirnbl 13816 . . . . . . . . . . . . . . . 16 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → ran (ball‘𝑃) = (𝑋 × 𝑌))
3937, 38syl 14 . . . . . . . . . . . . . . 15 (𝜑 ran (ball‘𝑃) = (𝑋 × 𝑌))
4039adantr 276 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ran (ball‘𝑃) = (𝑋 × 𝑌))
4136, 40sseqtrrd 3194 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ⊆ ran (ball‘𝑃))
422ad2antrr 488 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
43 simplrl 535 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑥𝐽)
44 xp1st 6165 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑥 × 𝑦) → (1st𝑗) ∈ 𝑥)
4544adantl 277 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st𝑗) ∈ 𝑥)
464mopni2 13876 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥𝐽 ∧ (1st𝑗) ∈ 𝑥) → ∃𝑚 ∈ ℝ+ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
4742, 43, 45, 46syl3anc 1238 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑚 ∈ ℝ+ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
483ad2antrr 488 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑁 ∈ (∞Met‘𝑌))
49 simplrr 536 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑦𝐾)
50 xp2nd 6166 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (𝑥 × 𝑦) → (2nd𝑗) ∈ 𝑦)
5150adantl 277 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd𝑗) ∈ 𝑦)
525mopni2 13876 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑦𝐾 ∧ (2nd𝑗) ∈ 𝑦) → ∃𝑛 ∈ ℝ+ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
5348, 49, 51, 52syl3anc 1238 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑛 ∈ ℝ+ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
5453adantr 276 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → ∃𝑛 ∈ ℝ+ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
55 blf 13803 . . . . . . . . . . . . . . . . . . . . 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 5366 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ball‘𝑃) Fn ((𝑋 × 𝑌) × ℝ*))
5857ad4antr 494 . . . . . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑗 ∈ (𝑋 × 𝑌))
6059ad2antrr 488 . . . . . . . . . . . . . . . . . 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 9659 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℝ+𝑚 ∈ ℝ*)
6261ad2antrl 490 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → 𝑚 ∈ ℝ*)
6362adantr 276 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑚 ∈ ℝ*)
64 rpxr 9659 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ+𝑛 ∈ ℝ*)
6564ad2antrl 490 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ*)
66 xrmincl 11269 . . . . . . . . . . . . . . . . . . 19 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*)
6763, 65, 66syl2anc 411 . . . . . . . . . . . . . . . . . 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 6021 . . . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . . . 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 2241 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑗𝑘𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < ))))
71 sseq1 3178 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑘 ⊆ (𝑥 × 𝑦) ↔ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦)))
7270, 71anbi12d 473 . . . . . . . . . . . . . . . . . 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 277 . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑚 ∈ ℝ+)
76 simprl 529 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ+)
77 xrminrpcl 11277 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℝ+𝑛 ∈ ℝ+) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ+)
7875, 76, 77syl2anc 411 . . . . . . . . . . . . . . . . . . 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 13809 . . . . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
8248ad2antrr 488 . . . . . . . . . . . . . . . . . . . 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 13901 . . . . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑥𝑋)
8584, 45sseldd 3156 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st𝑗) ∈ 𝑋)
8685ad2antrr 488 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (1st𝑗) ∈ 𝑋)
87 xrmin1inf 11270 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑚)
8863, 65, 87syl2anc 411 . . . . . . . . . . . . . . . . . . . . . 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 13819 . . . . . . . . . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑦𝑌)
9493, 51sseldd 3156 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd𝑗) ∈ 𝑌)
9594ad2antrr 488 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (2nd𝑗) ∈ 𝑌)
96 xrmin2inf 11271 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℝ*𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑛)
9763, 65, 96syl2anc 411 . . . . . . . . . . . . . . . . . . . . . 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 13819 . . . . . . . . . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . . . . . . . . . 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 531 . . . . . . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . . . . . . . . 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 4733 . . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . . . . 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 3191 . . . . . . . . . . . . . . . . . 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 306 . . . . . . . . . . . . . . . . 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 2847 . . . . . . . . . . . . . . . 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 2599 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
10847, 107rexlimddv 2599 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
109108ralrimiva 2550 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
11041, 109jca 306 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ((𝑥 × 𝑦) ⊆ ran (ball‘𝑃) ∧ ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦))))
111 blex 13780 . . . . . . . . . . . . . . 15 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃) ∈ V)
11237, 111syl 14 . . . . . . . . . . . . . 14 (𝜑 → (ball‘𝑃) ∈ V)
113112adantr 276 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (ball‘𝑃) ∈ V)
114 rnexg 4892 . . . . . . . . . . . . 13 ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
115 eltg2 13446 . . . . . . . . . . . . 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 167 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)))
11820, 21, 22, 117syl12anc 1236 . . . . . . . . . 10 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → (𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)))
11919, 118eqeltrd 2254 . . . . . . . . 9 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑤 ∈ (topGen‘ran (ball‘𝑃)))
120119ex 115 . . . . . . . 8 (((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) → (𝑤 = (𝑥 × 𝑦) → 𝑤 ∈ (topGen‘ran (ball‘𝑃))))
121120rexlimdvva 2602 . . . . . . 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 115 . . . . 5 (𝜑 → (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) → 𝑤 ∈ (topGen‘ran (ball‘𝑃))))
124123ssrdv 3161 . . . 4 (𝜑 → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃)))
1254mopntop 13837 . . . . . . . 8 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
1262, 125syl 14 . . . . . . 7 (𝜑𝐽 ∈ Top)
1275mopntop 13837 . . . . . . . 8 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
1283, 127syl 14 . . . . . . 7 (𝜑𝐾 ∈ Top)
129 mpoexga 6212 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
130126, 128, 129syl2anc 411 . . . . . 6 (𝜑 → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
131 rnexg 4892 . . . . . 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 13471 . . . . 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 411 . . . 4 (𝜑 → ((topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ⊆ (topGen‘ran (ball‘𝑃)) ↔ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃))))
136124, 135mpbird 167 . . 3 (𝜑 → (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ⊆ (topGen‘ran (ball‘𝑃)))
137 eqid 2177 . . . . 5 ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
138137txval 13648 . . . 4 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
139126, 128, 138syl2anc 411 . . 3 (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
1406mopnval 13835 . . . 4 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → 𝐿 = (topGen‘ran (ball‘𝑃)))
14137, 140syl 14 . . 3 (𝜑𝐿 = (topGen‘ran (ball‘𝑃)))
142136, 139, 1413sstr4d 3200 . 2 (𝜑 → (𝐽 ×t 𝐾) ⊆ 𝐿)
1437, 142eqssd 3172 1 (𝜑𝐿 = (𝐽 ×t 𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  wrex 2456  Vcvv 2737  wss 3129  𝒫 cpw 3575  {cpr 3593   cuni 3809   class class class wbr 4003   × cxp 4624  ran crn 4627   Fn wfn 5211  wf 5212  cfv 5216  (class class class)co 5874  cmpo 5876  1st c1st 6138  2nd c2nd 6139  supcsup 6980  infcinf 6981  *cxr 7989   < clt 7990  cle 7991  +crp 9651  topGenctg 12693  ∞Metcxmet 13331  ballcbl 13333  MetOpencmopn 13336  Topctop 13388  TopOnctopon 13401   ×t ctx 13645
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-map 6649  df-sup 6982  df-inf 6983  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-xneg 9770  df-xadd 9771  df-seqfrec 10443  df-exp 10517  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-topgen 12699  df-psmet 13338  df-xmet 13339  df-bl 13341  df-mopn 13342  df-top 13389  df-topon 13402  df-bases 13434  df-tx 13646
This theorem is referenced by:  txmetcnp  13911
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