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Theorem metnrmlem3 22786
Description: Lemma for metnrm 22787. (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 3913 . . . . 5 (𝑇𝑆) = (𝑆𝑇)
7 metnrmlem.4 . . . . 5 (𝜑 → (𝑆𝑇) = ∅)
86, 7syl5eq 2770 . . . 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 22785 . . 3 (𝜑 → (𝑉𝐽𝑆𝑉))
1110simpld 477 . 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 22785 . . 3 (𝜑 → (𝑈𝐽𝑇𝑈))
1514simpld 477 . 2 (𝜑𝑈𝐽)
1610simprd 482 . 2 (𝜑𝑆𝑉)
1714simprd 482 . 2 (𝜑𝑇𝑈)
189ineq1i 3918 . . . 4 (𝑉𝑈) = ( 𝑠𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈)
19 iunin1 4693 . . . 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 2749 . . 3 (𝑉𝑈) = 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈)
2113ineq2i 3919 . . . . . . . 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 4692 . . . . . . . 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 2749 . . . . . . 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 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 𝐷 ∈ (∞Met‘𝑋))
25 eqid 2724 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
2625cldss 20956 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
275, 26syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑆 𝐽)
282mopnuni 22368 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
293, 28syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 = 𝐽)
3027, 29sseqtr4d 3748 . . . . . . . . . . . . . 14 (𝜑𝑆𝑋)
3130sselda 3709 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → 𝑠𝑋)
3231adantrr 755 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 𝑠𝑋)
3325cldss 20956 . . . . . . . . . . . . . . . 16 (𝑇 ∈ (Clsd‘𝐽) → 𝑇 𝐽)
344, 33syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑇 𝐽)
3534, 29sseqtr4d 3748 . . . . . . . . . . . . . 14 (𝜑𝑇𝑋)
3635sselda 3709 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → 𝑡𝑋)
3736adantrl 754 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 𝑡𝑋)
381, 2, 3, 4, 5, 8metnrmlem1a 22783 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (0 < (𝐺𝑠) ∧ if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ+))
3938simprd 482 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ+)
4039adantrr 755 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ+)
4140rphalfcld 11998 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ+)
4241rpxrd 11987 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ*)
4312, 2, 3, 5, 4, 7metnrmlem1a 22783 . . . . . . . . . . . . . . . 16 ((𝜑𝑡𝑇) → (0 < (𝐹𝑡) ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ+))
4443adantrl 754 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (0 < (𝐹𝑡) ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ+))
4544simprd 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ+)
4645rphalfcld 11998 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ+)
4746rpxrd 11987 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ*)
4840rpred 11986 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ)
4948rehalfcld 11392 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ)
5045rpred 11986 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ)
5150rehalfcld 11392 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ)
52 rexadd 12177 . . . . . . . . . . . . . . 15 (((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)) = ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) + (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)))
5349, 51, 52syl2anc 696 . . . . . . . . . . . . . 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)))
5448recnd 10181 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℂ)
5550recnd 10181 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℂ)
56 2cnd 11206 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 2 ∈ ℂ)
57 2ne0 11226 . . . . . . . . . . . . . . . 16 2 ≠ 0
5857a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 2 ≠ 0)
5954, 55, 56, 58divdird 10952 . . . . . . . . . . . . . 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)))
6053, 59eqtr4d 2761 . . . . . . . . . . . . 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))
611, 2, 3, 4, 5, 8metnrmlem1 22784 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑠𝑆)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑡𝐷𝑠))
6261ancom2s 879 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑡𝐷𝑠))
63 xmetsym 22274 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡𝑋𝑠𝑋) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
6424, 37, 32, 63syl3anc 1439 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
6562, 64breqtrd 4786 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑠𝐷𝑡))
6612, 2, 3, 5, 4, 7metnrmlem1 22784 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ≤ (𝑠𝐷𝑡))
6740rpxrd 11987 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ*)
6845rpxrd 11987 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ*)
69 xmetcl 22258 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠𝑋𝑡𝑋) → (𝑠𝐷𝑡) ∈ ℝ*)
7024, 32, 37, 69syl3anc 1439 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (𝑠𝐷𝑡) ∈ ℝ*)
71 xle2add 12203 . . . . . . . . . . . . . . . . 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))))
7267, 68, 70, 70, 71syl22anc 1440 . . . . . . . . . . . . . . . 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))))
7365, 66, 72mp2and 717 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) +𝑒 if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ≤ ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡)))
7448, 50readdcld 10182 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ∈ ℝ)
7574recnd 10181 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ∈ ℂ)
7675, 56, 58divcan2d 10916 . . . . . . . . . . . . . . . 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))))
77 2re 11203 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
7874rehalfcld 11392 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ∈ ℝ)
79 rexmul 12215 . . . . . . . . . . . . . . . . 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)))
8077, 78, 79sylancr 698 . . . . . . . . . . . . . . . 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)))
81 rexadd 12177 . . . . . . . . . . . . . . . . 17 ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) +𝑒 if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) = (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))))
8248, 50, 81syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) +𝑒 if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) = (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))))
8376, 80, 823eqtr4d 2768 . . . . . . . . . . . . . . 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))))
84 x2times 12243 . . . . . . . . . . . . . . . 16 ((𝑠𝐷𝑡) ∈ ℝ* → (2 ·e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡)))
8570, 84syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (2 ·e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡)))
8673, 83, 853brtr4d 4792 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (2 ·e ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)) ≤ (2 ·e (𝑠𝐷𝑡)))
8778rexrd 10202 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ∈ ℝ*)
88 2rp 11951 . . . . . . . . . . . . . . . 16 2 ∈ ℝ+
8988a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 2 ∈ ℝ+)
90 xlemul2 12235 . . . . . . . . . . . . . . 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))))
9187, 70, 89, 90syl3anc 1439 . . . . . . . . . . . . . 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))))
9286, 91mpbird 247 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ≤ (𝑠𝐷𝑡))
9360, 92eqbrtrd 4782 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) +𝑒 (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)) ≤ (𝑠𝐷𝑡))
94 bldisj 22325 . . . . . . . . . . . 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))) = ∅)
9524, 32, 37, 42, 47, 93, 94syl33anc 1454 . . . . . . . . . . 11 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) = ∅)
96 eqimss 3763 . . . . . . . . . . 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))) ⊆ ∅)
9795, 96syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
9897anassrs 683 . . . . . . . . 9 (((𝜑𝑠𝑆) ∧ 𝑡𝑇) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
9998ralrimiva 3068 . . . . . . . 8 ((𝜑𝑠𝑆) → ∀𝑡𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
100 iunss 4669 . . . . . . . 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))) ⊆ ∅)
10199, 100sylibr 224 . . . . . . 7 ((𝜑𝑠𝑆) → 𝑡𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
10223, 101syl5eqss 3755 . . . . . 6 ((𝜑𝑠𝑆) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
103102ralrimiva 3068 . . . . 5 (𝜑 → ∀𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
104 iunss 4669 . . . . 5 ( 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅ ↔ ∀𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
105103, 104sylibr 224 . . . 4 (𝜑 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
106 ss0 4082 . . . 4 ( 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅ → 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) = ∅)
107105, 106syl 17 . . 3 (𝜑 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) = ∅)
10820, 107syl5eq 2770 . 2 (𝜑 → (𝑉𝑈) = ∅)
109 sseq2 3733 . . . 4 (𝑧 = 𝑉 → (𝑆𝑧𝑆𝑉))
110 ineq1 3915 . . . . 5 (𝑧 = 𝑉 → (𝑧𝑤) = (𝑉𝑤))
111110eqeq1d 2726 . . . 4 (𝑧 = 𝑉 → ((𝑧𝑤) = ∅ ↔ (𝑉𝑤) = ∅))
112109, 1113anbi13d 1514 . . 3 (𝑧 = 𝑉 → ((𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅) ↔ (𝑆𝑉𝑇𝑤 ∧ (𝑉𝑤) = ∅)))
113 sseq2 3733 . . . 4 (𝑤 = 𝑈 → (𝑇𝑤𝑇𝑈))
114 ineq2 3916 . . . . 5 (𝑤 = 𝑈 → (𝑉𝑤) = (𝑉𝑈))
115114eqeq1d 2726 . . . 4 (𝑤 = 𝑈 → ((𝑉𝑤) = ∅ ↔ (𝑉𝑈) = ∅))
116113, 1153anbi23d 1515 . . 3 (𝑤 = 𝑈 → ((𝑆𝑉𝑇𝑤 ∧ (𝑉𝑤) = ∅) ↔ (𝑆𝑉𝑇𝑈 ∧ (𝑉𝑈) = ∅)))
117112, 116rspc2ev 3428 . 2 ((𝑉𝐽𝑈𝐽 ∧ (𝑆𝑉𝑇𝑈 ∧ (𝑉𝑈) = ∅)) → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
11811, 15, 16, 17, 108, 117syl113anc 1451 1 (𝜑 → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wne 2896  wral 3014  wrex 3015  cin 3679  wss 3680  c0 4023  ifcif 4194   cuni 4544   ciun 4628   class class class wbr 4760  cmpt 4837  ran crn 5219  cfv 6001  (class class class)co 6765  infcinf 8463  cr 10048  0cc0 10049  1c1 10050   + caddc 10052   · cmul 10054  *cxr 10186   < clt 10187  cle 10188   / cdiv 10797  2c2 11183  +crp 11946   +𝑒 cxad 12058   ·e cxmu 12059  ∞Metcxmt 19854  ballcbl 19856  MetOpencmopn 19859  Clsdccld 20943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-ec 7864  df-map 7976  df-en 8073  df-dom 8074  df-sdom 8075  df-sup 8464  df-inf 8465  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-n0 11406  df-z 11491  df-uz 11801  df-q 11903  df-rp 11947  df-xneg 12060  df-xadd 12061  df-xmul 12062  df-icc 12296  df-topgen 16227  df-psmet 19861  df-xmet 19862  df-bl 19864  df-mopn 19865  df-top 20822  df-topon 20839  df-bases 20873  df-cld 20946  df-ntr 20947  df-cls 20948
This theorem is referenced by:  metnrm  22787
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