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Theorem metnrmlem3 22567
Description: Lemma for metnrm 22568. (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 3788 . . . . 5 (𝑇𝑆) = (𝑆𝑇)
7 metnrmlem.4 . . . . 5 (𝜑 → (𝑆𝑇) = ∅)
86, 7syl5eq 2672 . . . 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 22566 . . 3 (𝜑 → (𝑉𝐽𝑆𝑉))
1110simpld 475 . 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 22566 . . 3 (𝜑 → (𝑈𝐽𝑇𝑈))
1514simpld 475 . 2 (𝜑𝑈𝐽)
1610simprd 479 . 2 (𝜑𝑆𝑉)
1714simprd 479 . 2 (𝜑𝑇𝑈)
189ineq1i 3793 . . . 4 (𝑉𝑈) = ( 𝑠𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈)
19 iunin1 4556 . . . 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 2651 . . 3 (𝑉𝑈) = 𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈)
2113ineq2i 3794 . . . . . . . 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 4555 . . . . . . . 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 2651 . . . . . . 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 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 𝐷 ∈ (∞Met‘𝑋))
25 eqid 2626 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
2625cldss 20738 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
275, 26syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑆 𝐽)
282mopnuni 22151 . . . . . . . . . . . . . . . 16 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
293, 28syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 = 𝐽)
3027, 29sseqtr4d 3626 . . . . . . . . . . . . . 14 (𝜑𝑆𝑋)
3130sselda 3588 . . . . . . . . . . . . 13 ((𝜑𝑠𝑆) → 𝑠𝑋)
3231adantrr 752 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 𝑠𝑋)
3325cldss 20738 . . . . . . . . . . . . . . . 16 (𝑇 ∈ (Clsd‘𝐽) → 𝑇 𝐽)
344, 33syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑇 𝐽)
3534, 29sseqtr4d 3626 . . . . . . . . . . . . . 14 (𝜑𝑇𝑋)
3635sselda 3588 . . . . . . . . . . . . 13 ((𝜑𝑡𝑇) → 𝑡𝑋)
3736adantrl 751 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 𝑡𝑋)
381, 2, 3, 4, 5, 8metnrmlem1a 22564 . . . . . . . . . . . . . . . 16 ((𝜑𝑠𝑆) → (0 < (𝐺𝑠) ∧ if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ+))
3938simprd 479 . . . . . . . . . . . . . . 15 ((𝜑𝑠𝑆) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ+)
4039adantrr 752 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ+)
4140rphalfcld 11828 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ+)
4241rpxrd 11817 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ*)
4312, 2, 3, 5, 4, 7metnrmlem1a 22564 . . . . . . . . . . . . . . . 16 ((𝜑𝑡𝑇) → (0 < (𝐹𝑡) ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ+))
4443adantrl 751 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (0 < (𝐹𝑡) ∧ if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ+))
4544simprd 479 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ+)
4645rphalfcld 11828 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ+)
4746rpxrd 11817 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ*)
4840rpred 11816 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ)
4948rehalfcld 11224 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) ∈ ℝ)
5045rpred 11816 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ)
5150rehalfcld 11224 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2) ∈ ℝ)
52 rexadd 12005 . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . 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 10013 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℂ)
5550recnd 10013 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℂ)
56 2cnd 11038 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 2 ∈ ℂ)
57 2ne0 11058 . . . . . . . . . . . . . . . 16 2 ≠ 0
5857a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 2 ≠ 0)
5954, 55, 56, 58divdird 10784 . . . . . . . . . . . . . 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 2663 . . . . . . . . . . . . 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 22565 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑡𝑇𝑠𝑆)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑡𝐷𝑠))
6261ancom2s 843 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑡𝐷𝑠))
63 xmetsym 22057 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡𝑋𝑠𝑋) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
6424, 37, 32, 63syl3anc 1323 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
6562, 64breqtrd 4644 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ≤ (𝑠𝐷𝑡))
6612, 2, 3, 5, 4, 7metnrmlem1 22565 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ≤ (𝑠𝐷𝑡))
6740rpxrd 11817 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) ∈ ℝ*)
6845rpxrd 11817 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) ∈ ℝ*)
69 xmetcl 22041 . . . . . . . . . . . . . . . . . 18 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠𝑋𝑡𝑋) → (𝑠𝐷𝑡) ∈ ℝ*)
7024, 32, 37, 69syl3anc 1323 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (𝑠𝐷𝑡) ∈ ℝ*)
71 xle2add 12029 . . . . . . . . . . . . . . . . 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 1324 . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) +𝑒 if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ≤ ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡)))
7448, 50readdcld 10014 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ∈ ℝ)
7574recnd 10013 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) ∈ ℂ)
7675, 56, 58divcan2d 10748 . . . . . . . . . . . . . . . 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 11035 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
7874rehalfcld 11224 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ∈ ℝ)
79 rexmul 12041 . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . 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 12005 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 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 2670 . . . . . . . . . . . . . . 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 12069 . . . . . . . . . . . . . . . 16 ((𝑠𝐷𝑡) ∈ ℝ* → (2 ·e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡)))
8570, 84syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (2 ·e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +𝑒 (𝑠𝐷𝑡)))
8673, 83, 853brtr4d 4650 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → (2 ·e ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2)) ≤ (2 ·e (𝑠𝐷𝑡)))
8778rexrd 10034 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) + if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡))) / 2) ∈ ℝ*)
88 2rp 11781 . . . . . . . . . . . . . . . 16 2 ∈ ℝ+
8988a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → 2 ∈ ℝ+)
90 xlemul2 12061 . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . 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 4640 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2) +𝑒 (if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2)) ≤ (𝑠𝐷𝑡))
94 bldisj 22108 . . . . . . . . . . . 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 1338 . . . . . . . . . . 11 ((𝜑 ∧ (𝑠𝑆𝑡𝑇)) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) = ∅)
96 eqimss 3641 . . . . . . . . . . 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 679 . . . . . . . . 9 (((𝜑𝑠𝑆) ∧ 𝑡𝑇) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
9998ralrimiva 2965 . . . . . . . 8 ((𝜑𝑠𝑆) → ∀𝑡𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))) ⊆ ∅)
100 iunss 4532 . . . . . . . 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 3633 . . . . . 6 ((𝜑𝑠𝑆) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
103102ralrimiva 2965 . . . . 5 (𝜑 → ∀𝑠𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
104 iunss 4532 . . . . 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 3951 . . . 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 2672 . 2 (𝜑 → (𝑉𝑈) = ∅)
109 sseq2 3611 . . . 4 (𝑧 = 𝑉 → (𝑆𝑧𝑆𝑉))
110 ineq1 3790 . . . . 5 (𝑧 = 𝑉 → (𝑧𝑤) = (𝑉𝑤))
111110eqeq1d 2628 . . . 4 (𝑧 = 𝑉 → ((𝑧𝑤) = ∅ ↔ (𝑉𝑤) = ∅))
112109, 1113anbi13d 1398 . . 3 (𝑧 = 𝑉 → ((𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅) ↔ (𝑆𝑉𝑇𝑤 ∧ (𝑉𝑤) = ∅)))
113 sseq2 3611 . . . 4 (𝑤 = 𝑈 → (𝑇𝑤𝑇𝑈))
114 ineq2 3791 . . . . 5 (𝑤 = 𝑈 → (𝑉𝑤) = (𝑉𝑈))
115114eqeq1d 2628 . . . 4 (𝑤 = 𝑈 → ((𝑉𝑤) = ∅ ↔ (𝑉𝑈) = ∅))
116113, 1153anbi23d 1399 . . 3 (𝑤 = 𝑈 → ((𝑆𝑉𝑇𝑤 ∧ (𝑉𝑤) = ∅) ↔ (𝑆𝑉𝑇𝑈 ∧ (𝑉𝑈) = ∅)))
117112, 116rspc2ev 3313 . 2 ((𝑉𝐽𝑈𝐽 ∧ (𝑆𝑉𝑇𝑈 ∧ (𝑉𝑈) = ∅)) → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
11811, 15, 16, 17, 108, 117syl113anc 1335 1 (𝜑 → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  cin 3559  wss 3560  c0 3896  ifcif 4063   cuni 4407   ciun 4490   class class class wbr 4618  cmpt 4678  ran crn 5080  cfv 5850  (class class class)co 6605  infcinf 8292  cr 9880  0cc0 9881  1c1 9882   + caddc 9884   · cmul 9886  *cxr 10018   < clt 10019  cle 10020   / cdiv 10629  2c2 11015  +crp 11776   +𝑒 cxad 11888   ·e cxmu 11889  ∞Metcxmt 19645  ballcbl 19647  MetOpencmopn 19650  Clsdccld 20725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-ec 7690  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-sup 8293  df-inf 8294  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-icc 12121  df-topgen 16020  df-psmet 19652  df-xmet 19653  df-bl 19655  df-mopn 19656  df-top 20616  df-bases 20617  df-topon 20618  df-cld 20728  df-ntr 20729  df-cls 20730
This theorem is referenced by:  metnrm  22568
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