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Theorem metnrm 24751

Description: A metric space is normal. (Contributed by Jeff Hankins, 31-Aug-2013.) (Revised by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)

**Hypothesis**
Ref	Expression
metnrm.j	⊢ 𝐽 = (MetOpen‘𝐷)

**Assertion**
Ref	Expression
metnrm	⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm)

Proof of Theorem metnrm Dummy variables 𝑡 𝑠 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metnrm.j	. . 3 ⊢ 𝐽 = (MetOpen‘𝐷)
2	1	mopntop 24328	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
3		eqid 2729	. . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ^, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ^, < ))
4		simp1 1136	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝐷 ∈ (∞Met‘𝑋))
5		simp2l 1200	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑥 ∈ (Clsd‘𝐽))
6		simp2r 1201	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑦 ∈ (Clsd‘𝐽))
7		simp3 1138	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑦) = ∅)
8		eqid 2729	. . . . 5 ⊢ ∪ 𝑠 ∈ 𝑦 (𝑠(ball‘𝐷)(if(1 ≤ ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ^, < ))‘𝑠), 1, ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ^, < ))‘𝑠)) / 2)) = ∪ 𝑠 ∈ 𝑦 (𝑠(ball‘𝐷)(if(1 ≤ ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ^, < ))‘𝑠), 1, ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ^, < ))‘𝑠)) / 2))
9		eqid 2729	. . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ^, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ^, < ))
10		eqid 2729	. . . . 5 ⊢ ∪ 𝑡 ∈ 𝑥 (𝑡(ball‘𝐷)(if(1 ≤ ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ^, < ))‘𝑡), 1, ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ^, < ))‘𝑡)) / 2)) = ∪ 𝑡 ∈ 𝑥 (𝑡(ball‘𝐷)(if(1 ≤ ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ^, < ))‘𝑡), 1, ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ^, < ))‘𝑡)) / 2))
11	3, 1, 4, 5, 6, 7, 8, 9, 10	metnrmlem3 24750	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
12	11	3expia 1121	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽))) → ((𝑥 ∩ 𝑦) = ∅ → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
13	12	ralrimivva 3180	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ (Clsd‘𝐽)∀𝑦 ∈ (Clsd‘𝐽)((𝑥 ∩ 𝑦) = ∅ → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
14		isnrm3 23246	. 2 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ (Clsd‘𝐽)∀𝑦 ∈ (Clsd‘𝐽)((𝑥 ∩ 𝑦) = ∅ → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))))
15	2, 13, 14	sylanbrc 583	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 ifcif 4488 ∪ ciun 4955 class class class wbr 5107 ↦ cmpt 5188 ran crn 5639 ‘cfv 6511 (class class class)co 7387 infcinf 9392 1c1 11069 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 / cdiv 11835 2c2 12241 ∞Metcxmet 21249 ballcbl 21251 MetOpencmopn 21254 Topctop 22780 Clsdccld 22903 Nrmcnrm 23197

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-ec 8673 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-icc 13313 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-bl 21259 df-mopn 21260 df-top 22781 df-topon 22798 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nrm 23204

This theorem is referenced by: metreg 24752

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