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Theorem xmettx 12952
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 12951 . 2 (𝜑𝐿 ⊆ (𝐽 ×t 𝐾))
8 eqid 2157 . . . . . . . . . . . 12 (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
98elrnmpog 5934 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠)))
109elv 2716 . . . . . . . . . 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 4601 . . . . . . . . . 10 (𝑟 = 𝑥 → (𝑟 × 𝑠) = (𝑥 × 𝑠))
1413eqeq2d 2169 . . . . . . . . 9 (𝑟 = 𝑥 → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑠)))
15 xpeq2 4602 . . . . . . . . . 10 (𝑠 = 𝑦 → (𝑥 × 𝑠) = (𝑥 × 𝑦))
1615eqeq2d 2169 . . . . . . . . 9 (𝑠 = 𝑦 → (𝑤 = (𝑥 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑦)))
1714, 16cbvrex2v 2692 . . . . . . . 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 523 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝜑)
21 simplrl 525 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑥𝐽)
22 simplrr 526 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑦𝐾)
234mopntopon 12885 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
242, 23syl 14 . . . . . . . . . . . . . . . . 17 (𝜑𝐽 ∈ (TopOn‘𝑋))
2524adantr 274 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
26 simprl 521 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑥𝐽)
27 toponss 12466 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
2825, 26, 27syl2anc 409 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑥𝑋)
295mopntopon 12885 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
303, 29syl 14 . . . . . . . . . . . . . . . . 17 (𝜑𝐾 ∈ (TopOn‘𝑌))
3130adantr 274 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
32 simprr 522 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑦𝐾)
33 toponss 12466 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦𝐾) → 𝑦𝑌)
3431, 32, 33syl2anc 409 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → 𝑦𝑌)
35 xpss12 4694 . . . . . . . . . . . . . . 15 ((𝑥𝑋𝑦𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
3628, 34, 35syl2anc 409 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
371, 2, 3xmetxp 12949 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
38 unirnbl 12865 . . . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 × 𝑦) ⊆ ran (ball‘𝑃))
422ad2antrr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
43 simplrl 525 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑥𝐽)
44 xp1st 6114 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (𝑥 × 𝑦) → (1st𝑗) ∈ 𝑥)
4544adantl 275 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st𝑗) ∈ 𝑥)
464mopni2 12925 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥𝐽 ∧ (1st𝑗) ∈ 𝑥) → ∃𝑚 ∈ ℝ+ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
4742, 43, 45, 46syl3anc 1220 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑚 ∈ ℝ+ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
483ad2antrr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑁 ∈ (∞Met‘𝑌))
49 simplrr 526 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑦𝐾)
50 xp2nd 6115 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (𝑥 × 𝑦) → (2nd𝑗) ∈ 𝑦)
5150adantl 275 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd𝑗) ∈ 𝑦)
525mopni2 12925 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑦𝐾 ∧ (2nd𝑗) ∈ 𝑦) → ∃𝑛 ∈ ℝ+ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
5348, 49, 51, 52syl3anc 1220 . . . . . . . . . . . . . . . . 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 12852 . . . . . . . . . . . . . . . . . . . . 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 5321 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ball‘𝑃) Fn ((𝑋 × 𝑌) × ℝ*))
5857ad4antr 486 . . . . . . . . . . . . . . . . . 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 3128 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑗 ∈ (𝑋 × 𝑌))
6059ad2antrr 480 . . . . . . . . . . . . . . . . . 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 9569 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℝ+𝑚 ∈ ℝ*)
6261ad2antrl 482 . . . . . . . . . . . . . . . . . . . 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 9569 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℝ+𝑛 ∈ ℝ*)
6564ad2antrl 482 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ*)
66 xrmincl 11167 . . . . . . . . . . . . . . . . . . 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 5969 . . . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . . . 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 2221 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑗𝑘𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < ))))
71 sseq1 3151 . . . . . . . . . . . . . . . . . . 19 (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑘 ⊆ (𝑥 × 𝑦) ↔ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦)))
7270, 71anbi12d 465 . . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑚 ∈ ℝ+)
76 simprl 521 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ+)
77 xrminrpcl 11175 . . . . . . . . . . . . . . . . . . . 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 12858 . . . . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
8248ad2antrr 480 . . . . . . . . . . . . . . . . . . . 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 12950 . . . . . . . . . . . . . . . . . . 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 3129 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st𝑗) ∈ 𝑋)
8685ad2antrr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (1st𝑗) ∈ 𝑋)
87 xrmin1inf 11168 . . . . . . . . . . . . . . . . . . . . . . 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 12868 . . . . . . . . . . . . . . . . . . . . . 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 1231 . . . . . . . . . . . . . . . . . . . . 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 526 . . . . . . . . . . . . . . . . . . . . 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 3138 . . . . . . . . . . . . . . . . . . . 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 3129 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd𝑗) ∈ 𝑌)
9594ad2antrr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (2nd𝑗) ∈ 𝑌)
96 xrmin2inf 11169 . . . . . . . . . . . . . . . . . . . . . . 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 12868 . . . . . . . . . . . . . . . . . . . . . 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 1231 . . . . . . . . . . . . . . . . . . . . 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 522 . . . . . . . . . . . . . . . . . . . . 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 3138 . . . . . . . . . . . . . . . . . . . 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 4694 . . . . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . . . 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 2579 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
10847, 107rexlimddv 2579 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗𝑘𝑘 ⊆ (𝑥 × 𝑦)))
109108ralrimiva 2530 . . . . . . . . . . . . 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 12829 . . . . . . . . . . . . . . 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 4852 . . . . . . . . . . . . 13 ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
115 eltg2 12495 . . . . . . . . . . . . 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 1218 . . . . . . . . . 10 ((((𝜑𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥𝐽𝑦𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → (𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)))
11919, 118eqeltrd 2234 . . . . . . . . 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 2582 . . . . . . 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 3134 . . . 4 (𝜑 → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃)))
1254mopntop 12886 . . . . . . . 8 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
1262, 125syl 14 . . . . . . 7 (𝜑𝐽 ∈ Top)
1275mopntop 12886 . . . . . . . 8 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
1283, 127syl 14 . . . . . . 7 (𝜑𝐾 ∈ Top)
129 mpoexga 6161 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
130126, 128, 129syl2anc 409 . . . . . 6 (𝜑 → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
131 rnexg 4852 . . . . . 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 12520 . . . . 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 2157 . . . . 5 ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
138137txval 12697 . . . 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 12884 . . . 4 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → 𝐿 = (topGen‘ran (ball‘𝑃)))
14137, 140syl 14 . . 3 (𝜑𝐿 = (topGen‘ran (ball‘𝑃)))
142136, 139, 1413sstr4d 3173 . 2 (𝜑 → (𝐽 ×t 𝐾) ⊆ 𝐿)
1437, 142eqssd 3145 1 (𝜑𝐿 = (𝐽 ×t 𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wcel 2128  wral 2435  wrex 2436  Vcvv 2712  wss 3102  𝒫 cpw 3543  {cpr 3561   cuni 3773   class class class wbr 3966   × cxp 4585  ran crn 4588   Fn wfn 5166  wf 5167  cfv 5171  (class class class)co 5825  cmpo 5827  1st c1st 6087  2nd c2nd 6088  supcsup 6927  infcinf 6928  *cxr 7912   < clt 7913  cle 7914  +crp 9561  topGenctg 12408  ∞Metcxmet 12422  ballcbl 12424  MetOpencmopn 12427  Topctop 12437  TopOnctopon 12450   ×t ctx 12694
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851  ax-arch 7852  ax-caucvg 7853
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-isom 5180  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-frec 6339  df-map 6596  df-sup 6929  df-inf 6930  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-n0 9092  df-z 9169  df-uz 9441  df-q 9530  df-rp 9562  df-xneg 9680  df-xadd 9681  df-seqfrec 10349  df-exp 10423  df-cj 10746  df-re 10747  df-im 10748  df-rsqrt 10902  df-abs 10903  df-topgen 12414  df-psmet 12429  df-xmet 12430  df-bl 12432  df-mopn 12433  df-top 12438  df-topon 12451  df-bases 12483  df-tx 12695
This theorem is referenced by:  txmetcnp  12960
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