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Theorem xmettxlem 13676
Description: Lemma for xmettx 13677. (Contributed by Jim Kingdon, 15-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
xmettxlem (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 xmettxlem
Dummy variables 𝑝 𝑟 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xmetxp.p . . . . . . . . 9 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))
2 xmetxp.1 . . . . . . . . 9 (𝜑𝑀 ∈ (∞Met‘𝑋))
3 xmetxp.2 . . . . . . . . 9 (𝜑𝑁 ∈ (∞Met‘𝑌))
41, 2, 3xmetxp 13674 . . . . . . . 8 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5 blrn 13579 . . . . . . . 8 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (𝑤 ∈ ran (ball‘𝑃) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)∃𝑝 ∈ ℝ* 𝑤 = (𝑧(ball‘𝑃)𝑝)))
64, 5syl 14 . . . . . . 7 (𝜑 → (𝑤 ∈ ran (ball‘𝑃) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)∃𝑝 ∈ ℝ* 𝑤 = (𝑧(ball‘𝑃)𝑝)))
76biimpa 296 . . . . . 6 ((𝜑𝑤 ∈ ran (ball‘𝑃)) → ∃𝑧 ∈ (𝑋 × 𝑌)∃𝑝 ∈ ℝ* 𝑤 = (𝑧(ball‘𝑃)𝑝))
8 xmettx.j . . . . . . . . . . . . . . 15 𝐽 = (MetOpen‘𝑀)
98mopntop 13611 . . . . . . . . . . . . . 14 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
102, 9syl 14 . . . . . . . . . . . . 13 (𝜑𝐽 ∈ Top)
11 xmettx.k . . . . . . . . . . . . . . 15 𝐾 = (MetOpen‘𝑁)
1211mopntop 13611 . . . . . . . . . . . . . 14 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
133, 12syl 14 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ Top)
14 mpoexga 6207 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
1510, 13, 14syl2anc 411 . . . . . . . . . . . 12 (𝜑 → (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
16 rnexg 4888 . . . . . . . . . . . 12 ((𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
1715, 16syl 14 . . . . . . . . . . 11 (𝜑 → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
1817ad3antrrr 492 . . . . . . . . . 10 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
19 bastg 13228 . . . . . . . . . 10 (ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
2018, 19syl 14 . . . . . . . . 9 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
212ad3antrrr 492 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑀 ∈ (∞Met‘𝑋))
22 simplrl 535 . . . . . . . . . . . . 13 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑧 ∈ (𝑋 × 𝑌))
23 xp1st 6160 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
2422, 23syl 14 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → (1st𝑧) ∈ 𝑋)
25 simplrr 536 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑝 ∈ ℝ*)
268blopn 13657 . . . . . . . . . . . 12 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝑧) ∈ 𝑋𝑝 ∈ ℝ*) → ((1st𝑧)(ball‘𝑀)𝑝) ∈ 𝐽)
2721, 24, 25, 26syl3anc 1238 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ((1st𝑧)(ball‘𝑀)𝑝) ∈ 𝐽)
283ad3antrrr 492 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑁 ∈ (∞Met‘𝑌))
29 xp2nd 6161 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
3022, 29syl 14 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → (2nd𝑧) ∈ 𝑌)
3111blopn 13657 . . . . . . . . . . . 12 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝑧) ∈ 𝑌𝑝 ∈ ℝ*) → ((2nd𝑧)(ball‘𝑁)𝑝) ∈ 𝐾)
3228, 30, 25, 31syl3anc 1238 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ((2nd𝑧)(ball‘𝑁)𝑝) ∈ 𝐾)
33 simpr 110 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑤 = (𝑧(ball‘𝑃)𝑝))
341, 21, 28, 25, 22xmetxpbl 13675 . . . . . . . . . . . 12 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → (𝑧(ball‘𝑃)𝑝) = (((1st𝑧)(ball‘𝑀)𝑝) × ((2nd𝑧)(ball‘𝑁)𝑝)))
3533, 34eqtrd 2210 . . . . . . . . . . 11 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑤 = (((1st𝑧)(ball‘𝑀)𝑝) × ((2nd𝑧)(ball‘𝑁)𝑝)))
36 xpeq1 4637 . . . . . . . . . . . . 13 (𝑟 = ((1st𝑧)(ball‘𝑀)𝑝) → (𝑟 × 𝑠) = (((1st𝑧)(ball‘𝑀)𝑝) × 𝑠))
3736eqeq2d 2189 . . . . . . . . . . . 12 (𝑟 = ((1st𝑧)(ball‘𝑀)𝑝) → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (((1st𝑧)(ball‘𝑀)𝑝) × 𝑠)))
38 xpeq2 4638 . . . . . . . . . . . . 13 (𝑠 = ((2nd𝑧)(ball‘𝑁)𝑝) → (((1st𝑧)(ball‘𝑀)𝑝) × 𝑠) = (((1st𝑧)(ball‘𝑀)𝑝) × ((2nd𝑧)(ball‘𝑁)𝑝)))
3938eqeq2d 2189 . . . . . . . . . . . 12 (𝑠 = ((2nd𝑧)(ball‘𝑁)𝑝) → (𝑤 = (((1st𝑧)(ball‘𝑀)𝑝) × 𝑠) ↔ 𝑤 = (((1st𝑧)(ball‘𝑀)𝑝) × ((2nd𝑧)(ball‘𝑁)𝑝))))
4037, 39rspc2ev 2856 . . . . . . . . . . 11 ((((1st𝑧)(ball‘𝑀)𝑝) ∈ 𝐽 ∧ ((2nd𝑧)(ball‘𝑁)𝑝) ∈ 𝐾𝑤 = (((1st𝑧)(ball‘𝑀)𝑝) × ((2nd𝑧)(ball‘𝑁)𝑝))) → ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠))
4127, 32, 35, 40syl3anc 1238 . . . . . . . . . 10 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠))
42 eqid 2177 . . . . . . . . . . . 12 (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
4342elrnmpog 5981 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠)))
4443elv 2741 . . . . . . . . . 10 (𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟𝐽𝑠𝐾 𝑤 = (𝑟 × 𝑠))
4541, 44sylibr 134 . . . . . . . . 9 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑤 ∈ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)))
4620, 45sseldd 3156 . . . . . . . 8 ((((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) ∧ 𝑤 = (𝑧(ball‘𝑃)𝑝)) → 𝑤 ∈ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
4746ex 115 . . . . . . 7 (((𝜑𝑤 ∈ ran (ball‘𝑃)) ∧ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑝 ∈ ℝ*)) → (𝑤 = (𝑧(ball‘𝑃)𝑝) → 𝑤 ∈ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)))))
4847rexlimdvva 2602 . . . . . 6 ((𝜑𝑤 ∈ ran (ball‘𝑃)) → (∃𝑧 ∈ (𝑋 × 𝑌)∃𝑝 ∈ ℝ* 𝑤 = (𝑧(ball‘𝑃)𝑝) → 𝑤 ∈ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)))))
497, 48mpd 13 . . . . 5 ((𝜑𝑤 ∈ ran (ball‘𝑃)) → 𝑤 ∈ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
5049ex 115 . . . 4 (𝜑 → (𝑤 ∈ ran (ball‘𝑃) → 𝑤 ∈ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)))))
5150ssrdv 3161 . . 3 (𝜑 → ran (ball‘𝑃) ⊆ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
52 blex 13554 . . . . 5 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃) ∈ V)
53 rnexg 4888 . . . . 5 ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
544, 52, 533syl 17 . . . 4 (𝜑 → ran (ball‘𝑃) ∈ V)
55 tgss3 13245 . . . 4 ((ran (ball‘𝑃) ∈ V ∧ ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) ∈ V) → ((topGen‘ran (ball‘𝑃)) ⊆ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ↔ ran (ball‘𝑃) ⊆ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)))))
5654, 17, 55syl2anc 411 . . 3 (𝜑 → ((topGen‘ran (ball‘𝑃)) ⊆ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))) ↔ ran (ball‘𝑃) ⊆ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)))))
5751, 56mpbird 167 . 2 (𝜑 → (topGen‘ran (ball‘𝑃)) ⊆ (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
58 xmettx.l . . . 4 𝐿 = (MetOpen‘𝑃)
5958mopnval 13609 . . 3 (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → 𝐿 = (topGen‘ran (ball‘𝑃)))
604, 59syl 14 . 2 (𝜑𝐿 = (topGen‘ran (ball‘𝑃)))
61 eqid 2177 . . . 4 ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))
6261txval 13422 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
6310, 13, 62syl2anc 411 . 2 (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟𝐽, 𝑠𝐾 ↦ (𝑟 × 𝑠))))
6457, 60, 633sstr4d 3200 1 (𝜑𝐿 ⊆ (𝐽 ×t 𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wrex 2456  Vcvv 2737  wss 3129  {cpr 3592   × cxp 4621  ran crn 4624  cfv 5212  (class class class)co 5869  cmpo 5871  1st c1st 6133  2nd c2nd 6134  supcsup 6975  *cxr 7981   < clt 7982  topGenctg 12651  ∞Metcxmet 13147  ballcbl 13149  MetOpencmopn 13152  Topctop 13162   ×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-tx 13420
This theorem is referenced by:  xmettx  13677
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