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Theorem metnrmlem3 24883
Description: Lemma for metnrm 24884. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)
Hypotheses
Ref Expression
metdscn.f 𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))
metdscn.j 𝐽 = (MetOpen‘𝐷)
metnrmlem.1 (𝜑𝐷 ∈ (∞Met‘𝑋))
metnrmlem.2 (𝜑𝑆 ∈ (Clsd‘𝐽))
metnrmlem.3 (𝜑𝑇 ∈ (Clsd‘𝐽))
metnrmlem.4 (𝜑 → (𝑆𝑇) = ∅)
metnrmlem.u 𝑈 = 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))
metnrmlem.g 𝐺 = (𝑥𝑋 ↦ inf(ran (𝑦𝑇 ↦ (𝑥𝐷𝑦)), ℝ*, < ))
metnrmlem.v 𝑉 = 𝑠𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2))
Assertion
Ref Expression
metnrmlem3 (𝜑 → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑡,𝑠,𝑤,𝑥,𝑦,𝑧,𝐷   𝐽,𝑠,𝑡,𝑤,𝑦,𝑧   𝜑,𝑠,𝑡   𝐺,𝑠,𝑡   𝑇,𝑠,𝑡,𝑤,𝑥,𝑦,𝑧   𝑆,𝑠,𝑡,𝑤,𝑥,𝑦,𝑧   𝑈,𝑠,𝑤   𝑋,𝑠,𝑡,𝑤,𝑥,𝑦,𝑧   𝐹,𝑠,𝑡,𝑤,𝑧   𝑤,𝑉,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝑈(𝑥,𝑦,𝑧,𝑡)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦,𝑧,𝑤)   𝐽(𝑥)   𝑉(𝑥,𝑦,𝑡,𝑠)

Proof of Theorem metnrmlem3
StepHypRef Expression
1 metnrmlem.g . . . 4 𝐺 = (𝑥𝑋 ↦ inf(ran (𝑦𝑇 ↦ (𝑥𝐷𝑦)), ℝ*, < ))
2 metdscn.j . . . 4 𝐽 = (MetOpen‘𝐷)
3 metnrmlem.1 . . . 4 (𝜑𝐷 ∈ (∞Met‘𝑋))
4 metnrmlem.3 . . . 4 (𝜑𝑇 ∈ (Clsd‘𝐽))
5 metnrmlem.2 . . . 4 (𝜑𝑆 ∈ (Clsd‘𝐽))
6 incom 4209 . . . . 5 (𝑇𝑆) = (𝑆𝑇)
7 metnrmlem.4 . . . . 5 (𝜑 → (𝑆𝑇) = ∅)
86, 7eqtrid 2789 . . . 4 (𝜑 → (𝑇𝑆) = ∅)
9 metnrmlem.v . . . 4 𝑉 = 𝑠𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2))
101, 2, 3, 4, 5, 8, 9metnrmlem2 24882 . . 3 (𝜑 → (𝑉𝐽𝑆𝑉))
1110simpld 494 . 2 (𝜑𝑉𝐽)
12 metdscn.f . . . 4 𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))
13 metnrmlem.u . . . 4 𝑈 = 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))
1412, 2, 3, 5, 4, 7, 13metnrmlem2 24882 . . 3 (𝜑 → (𝑈𝐽𝑇𝑈))
1514simpld 494 . 2 (𝜑𝑈𝐽)
1610simprd 495 . 2 (𝜑𝑆𝑉)
1714simprd 495 . 2 (𝜑𝑇𝑈)
189ineq1i 4216 . . . 4 (𝑉𝑈) = ( 𝑠𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈)
19 iunin1 5072 . . . 4 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) = ( 𝑠𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈)
2018, 19eqtr4i 2768 . . 3 (𝑉𝑈) = 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈)
2113ineq2i 4217 . . . . . . . 8 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) = ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)))
22 iunin2 5071 . . . . . . . 8 𝑡𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) = ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)))
2321, 22eqtr4i 2768 . . . . . . 7 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) = 𝑡𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)))
243adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 𝐷 ∈ (∞Met‘𝑋))
25 eqid 2737 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
2625cldss 23037 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
275, 26syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑆 𝐽)
282mopnuni 24451 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
293, 28syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 = 𝐽)
3027, 29sseqtrrd 4021 . . . . . . . . . . . . . 14 (𝜑𝑆𝑋)
3130sselda 3983 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → 𝑠𝑋)
3231adantrr 717 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 𝑠𝑋)
3325cldss 23037 . . . . . . . . . . . . . . . 16 (𝑇 ∈ (Clsd‘𝐽) → 𝑇 𝐽)
344, 33syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑇 𝐽)
3534, 29sseqtrrd 4021 . . . . . . . . . . . . . 14 (𝜑𝑇𝑋)
3635sselda 3983 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → 𝑡𝑋)
3736adantrl 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 𝑡𝑋)
381, 2, 3, 4, 5, 8metnrmlem1a 24880 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (0 < (𝐺𝑠) ∧ if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ+))
3938simprd 495 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ+)
4039adantrr 717 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ+)
4140rphalfcld 13089 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ+)
4241rpxrd 13078 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ*)
4312, 2, 3, 5, 4, 7metnrmlem1a 24880 . . . . . . . . . . . . . . . 16 ((𝜑𝑡𝑇) → (0 < (𝐹𝑡) ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ+))
4443adantrl 716 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (0 < (𝐹𝑡) ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ+))
4544simprd 495 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ+)
4645rphalfcld 13089 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ+)
4746rpxrd 13078 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ*)
4840rpred 13077 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ)
4948rehalfcld 12513 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ)
5045rpred 13077 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ)
5150rehalfcld 12513 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ)
5249, 51rexaddd 13276 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) +𝑒 (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)) = ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) + (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)))
5348recnd 11289 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℂ)
5450recnd 11289 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℂ)
55 2cnd 12344 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 2 ∈ ℂ)
56 2ne0 12370 . . . . . . . . . . . . . . . 16 2 ≠ 0
5756a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 2 ≠ 0)
5853, 54, 55, 57divdird 12081 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) = ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) + (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)))
5952, 58eqtr4d 2780 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) +𝑒 (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)) = ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2))
601, 2, 3, 4, 5, 8metnrmlem1 24881 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑠𝑆)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑡𝐷𝑠))
6160ancom2s 650 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑡𝐷𝑠))
62 xmetsym 24357 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡𝑋𝑠𝑋) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
6324, 37, 32, 62syl3anc 1373 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
6461, 63breqtrd 5169 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑠𝐷𝑡))
6512, 2, 3, 5, 4, 7metnrmlem1 24881 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ≤ (𝑠𝐷𝑡))
6640rpxrd 13078 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ*)
6745rpxrd 13078 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ*)
68 xmetcl 24341 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠𝑋𝑡𝑋) → (𝑠𝐷𝑡) ∈ ℝ*)
6924, 32, 37, 68syl3anc 1373 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (𝑠𝐷𝑡) ∈ ℝ*)
70 xle2add 13301 . . . . . . . . . . . . . . . . 17 (((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ* ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ*) ∧ ((𝑠𝐷𝑡) ∈ ℝ* ∧ (𝑠𝐷𝑡) ∈ ℝ*)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑠𝐷𝑡) ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ≤ (𝑠𝐷𝑡)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) +𝑒 if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ≤ ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡))))
7166, 67, 69, 69, 70syl22anc 839 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑠𝐷𝑡) ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ≤ (𝑠𝐷𝑡)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) +𝑒 if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ≤ ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡))))
7264, 65, 71mp2and 699 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) +𝑒 if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ≤ ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡)))
7348, 50readdcld 11290 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ∈ ℝ)
7473recnd 11289 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ∈ ℂ)
7574, 55, 57divcan2d 12045 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (2 · ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)) = (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))))
76 2re 12340 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
7773rehalfcld 12513 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ∈ ℝ)
78 rexmul 13313 . . . . . . . . . . . . . . . . 17 ((2 ∈ ℝ ∧ ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ∈ ℝ) → (2 ·e ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)) = (2 · ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)))
7976, 77, 78sylancr 587 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (2 ·e ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)) = (2 · ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)))
8048, 50rexaddd 13276 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) +𝑒 if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) = (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))))
8175, 79, 803eqtr4d 2787 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (2 ·e ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)) = (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) +𝑒 if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))))
82 x2times 13341 . . . . . . . . . . . . . . . 16 ((𝑠𝐷𝑡) ∈ ℝ* → (2 ·e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡)))
8369, 82syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (2 ·e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡)))
8472, 81, 833brtr4d 5175 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (2 ·e ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)) ≤ (2 ·e (𝑠𝐷𝑡)))
8577rexrd 11311 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ∈ ℝ*)
86 2rp 13039 . . . . . . . . . . . . . . . 16 2 ∈ ℝ+
8786a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 2 ∈ ℝ+)
88 xlemul2 13333 . . . . . . . . . . . . . . 15 ((((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ∈ ℝ* ∧ (𝑠𝐷𝑡) ∈ ℝ* ∧ 2 ∈ ℝ+) → (((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ≤ (𝑠𝐷𝑡) ↔ (2 ·e ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)) ≤ (2 ·e (𝑠𝐷𝑡))))
8985, 69, 87, 88syl3anc 1373 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ≤ (𝑠𝐷𝑡) ↔ (2 ·e ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)) ≤ (2 ·e (𝑠𝐷𝑡))))
9084, 89mpbird 257 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ≤ (𝑠𝐷𝑡))
9159, 90eqbrtrd 5165 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) +𝑒 (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)) ≤ (𝑠𝐷𝑡))
92 bldisj 24408 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠𝑋𝑡𝑋) ∧ ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ* ∧ (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ* ∧ ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) +𝑒 (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)) ≤ (𝑠𝐷𝑡))) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) = ∅)
9324, 32, 37, 42, 47, 91, 92syl33anc 1387 . . . . . . . . . . 11 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) = ∅)
94 eqimss 4042 . . . . . . . . . . 11 (((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) = ∅ → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
9593, 94syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
9695anassrs 467 . . . . . . . . 9 (((𝜑𝑠𝑆) ∧ 𝑡𝑇) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
9796ralrimiva 3146 . . . . . . . 8 ((𝜑𝑠𝑆) → ∀𝑡𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
98 iunss 5045 . . . . . . . 8 ( 𝑡𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅ ↔ ∀𝑡𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
9997, 98sylibr 234 . . . . . . 7 ((𝜑𝑠𝑆) → 𝑡𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
10023, 99eqsstrid 4022 . . . . . 6 ((𝜑𝑠𝑆) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
101100ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
102 iunss 5045 . . . . 5 ( 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅ ↔ ∀𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
103101, 102sylibr 234 . . . 4 (𝜑 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
104 ss0 4402 . . . 4 ( 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅ → 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) = ∅)
105103, 104syl 17 . . 3 (𝜑 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) = ∅)
10620, 105eqtrid 2789 . 2 (𝜑 → (𝑉𝑈) = ∅)
107 sseq2 4010 . . . 4 (𝑧 = 𝑉 → (𝑆𝑧𝑆𝑉))
108 ineq1 4213 . . . . 5 (𝑧 = 𝑉 → (𝑧𝑤) = (𝑉𝑤))
109108eqeq1d 2739 . . . 4 (𝑧 = 𝑉 → ((𝑧𝑤) = ∅ ↔ (𝑉𝑤) = ∅))
110107, 1093anbi13d 1440 . . 3 (𝑧 = 𝑉 → ((𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅) ↔ (𝑆𝑉𝑇𝑤 ∧ (𝑉𝑤) = ∅)))
111 sseq2 4010 . . . 4 (𝑤 = 𝑈 → (𝑇𝑤𝑇𝑈))
112 ineq2 4214 . . . . 5 (𝑤 = 𝑈 → (𝑉𝑤) = (𝑉𝑈))
113112eqeq1d 2739 . . . 4 (𝑤 = 𝑈 → ((𝑉𝑤) = ∅ ↔ (𝑉𝑈) = ∅))
114111, 1133anbi23d 1441 . . 3 (𝑤 = 𝑈 → ((𝑆𝑉𝑇𝑤 ∧ (𝑉𝑤) = ∅) ↔ (𝑆𝑉𝑇𝑈 ∧ (𝑉𝑈) = ∅)))
115110, 114rspc2ev 3635 . 2 ((𝑉𝐽𝑈𝐽 ∧ (𝑆𝑉𝑇𝑈 ∧ (𝑉𝑈) = ∅)) → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
11611, 15, 16, 17, 106, 115syl113anc 1384 1 (𝜑 → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cin 3950  wss 3951  c0 4333  ifcif 4525   cuni 4907   ciun 4991   class class class wbr 5143  cmpt 5225  ran crn 5686  cfv 6561  (class class class)co 7431  infcinf 9481  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  *cxr 11294   < clt 11295  cle 11296   / cdiv 11920  2c2 12321  +crp 13034   +𝑒 cxad 13152   ·e cxmu 13153  ∞Metcxmet 21349  ballcbl 21351  MetOpencmopn 21354  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-ec 8747  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-icc 13394  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029
This theorem is referenced by:  metnrm  24884
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