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Mirrors > Home > MPE Home > Th. List > tngngpd | Structured version Visualization version GIF version |
Description: Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
tngngp.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
tngngp.x | ⊢ 𝑋 = (Base‘𝐺) |
tngngp.m | ⊢ − = (-g‘𝐺) |
tngngp.z | ⊢ 0 = (0g‘𝐺) |
tngngpd.1 | ⊢ (𝜑 → 𝐺 ∈ Grp) |
tngngpd.2 | ⊢ (𝜑 → 𝑁:𝑋⟶ℝ) |
tngngpd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 )) |
tngngpd.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
Ref | Expression |
---|---|
tngngpd | ⊢ (𝜑 → 𝑇 ∈ NrmGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngngpd.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | tngngpd.2 | . . . 4 ⊢ (𝜑 → 𝑁:𝑋⟶ℝ) | |
3 | tngngp.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
4 | fvex 6201 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
5 | 3, 4 | eqeltri 2697 | . . . . 5 ⊢ 𝑋 ∈ V |
6 | reex 10027 | . . . . 5 ⊢ ℝ ∈ V | |
7 | fex2 7121 | . . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V) | |
8 | 5, 6, 7 | mp3an23 1416 | . . . 4 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V) |
9 | tngngp.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
10 | tngngp.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
11 | 9, 10 | tngds 22452 | . . . 4 ⊢ (𝑁 ∈ V → (𝑁 ∘ − ) = (dist‘𝑇)) |
12 | 2, 8, 11 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝑁 ∘ − ) = (dist‘𝑇)) |
13 | tngngp.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
14 | tngngpd.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 )) | |
15 | tngngpd.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) | |
16 | 3, 10, 13, 1, 2, 14, 15 | nrmmetd 22379 | . . 3 ⊢ (𝜑 → (𝑁 ∘ − ) ∈ (Met‘𝑋)) |
17 | 12, 16 | eqeltrrd 2702 | . 2 ⊢ (𝜑 → (dist‘𝑇) ∈ (Met‘𝑋)) |
18 | eqid 2622 | . . . 4 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
19 | 9, 3, 18 | tngngp2 22456 | . . 3 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋)))) |
20 | 2, 19 | syl 17 | . 2 ⊢ (𝜑 → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋)))) |
21 | 1, 17, 20 | mpbir2and 957 | 1 ⊢ (𝜑 → 𝑇 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 + caddc 9939 ≤ cle 10075 Basecbs 15857 distcds 15950 0gc0g 16100 Grpcgrp 17422 -gcsg 17424 Metcme 19732 NrmGrpcngp 22382 toNrmGrp ctng 22383 |
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-iun 4522 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-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-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-tset 15960 df-ds 15964 df-rest 16083 df-topn 16084 df-0g 16102 df-topgen 16104 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-xms 22125 df-ms 22126 df-nm 22387 df-ngp 22388 df-tng 22389 |
This theorem is referenced by: tngngp 22458 tngngp3 22460 tchcph 23036 |
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