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Theorem metnrm 22665

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 22245	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
3		eqid 2622	. . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ^, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ^, < ))
4		simp1 1061	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝐷 ∈ (∞Met‘𝑋))
5		simp2l 1087	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑥 ∈ (Clsd‘𝐽))
6		simp2r 1088	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑦 ∈ (Clsd‘𝐽))
7		simp3 1063	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑦) = ∅)
8		eqid 2622	. . . . 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 2622	. . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ^, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ^, < ))
10		eqid 2622	. . . . 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 22664	. . . 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 1267	. . 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 2971	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ (Clsd‘𝐽)∀𝑦 ∈ (Clsd‘𝐽)((𝑥 ∩ 𝑦) = ∅ → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
14		isnrm3 21163	. 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 698	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ifcif 4086 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 ‘cfv 5888 (class class class)co 6650 infcinf 8347 1c1 9937 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 / cdiv 10684 2c2 11070 ∞Metcxmt 19731 ballcbl 19733 MetOpencmopn 19736 Topctop 20698 Clsdccld 20820 Nrmcnrm 21114

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-ec 7744 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nrm 21121

This theorem is referenced by: metreg 22666

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