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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 | 3 | fvexi 6895 | . . . . 5 ⊢ 𝑋 ∈ V |
5 | reex 11188 | . . . . 5 ⊢ ℝ ∈ V | |
6 | fex2 7911 | . . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V) | |
7 | 4, 5, 6 | mp3an23 1454 | . . . 4 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V) |
8 | tngngp.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
9 | tngngp.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
10 | 8, 9 | tngds 24133 | . . . 4 ⊢ (𝑁 ∈ V → (𝑁 ∘ − ) = (dist‘𝑇)) |
11 | 2, 7, 10 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝑁 ∘ − ) = (dist‘𝑇)) |
12 | tngngp.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
13 | tngngpd.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 )) | |
14 | tngngpd.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) | |
15 | 3, 9, 12, 1, 2, 13, 14 | nrmmetd 24052 | . . 3 ⊢ (𝜑 → (𝑁 ∘ − ) ∈ (Met‘𝑋)) |
16 | 11, 15 | eqeltrrd 2835 | . 2 ⊢ (𝜑 → (dist‘𝑇) ∈ (Met‘𝑋)) |
17 | eqid 2733 | . . . 4 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
18 | 8, 3, 17 | tngngp2 24138 | . . 3 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋)))) |
19 | 2, 18 | syl 17 | . 2 ⊢ (𝜑 → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋)))) |
20 | 1, 16, 19 | mpbir2and 712 | 1 ⊢ (𝜑 → 𝑇 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 class class class wbr 5144 ∘ ccom 5676 ⟶wf 6531 ‘cfv 6535 (class class class)co 7396 ℝcr 11096 0cc0 11097 + caddc 11100 ≤ cle 11236 Basecbs 17131 distcds 17193 0gc0g 17372 Grpcgrp 18806 -gcsg 18808 Metcmet 20904 NrmGrpcngp 24055 toNrmGrp ctng 24056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-sup 9424 df-inf 9425 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-q 12920 df-rp 12962 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-plusg 17197 df-tset 17203 df-ds 17206 df-rest 17355 df-topn 17356 df-0g 17374 df-topgen 17376 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-grp 18809 df-minusg 18810 df-sbg 18811 df-psmet 20910 df-xmet 20911 df-met 20912 df-bl 20913 df-mopn 20914 df-top 22365 df-topon 22382 df-topsp 22404 df-bases 22418 df-xms 23795 df-ms 23796 df-nm 24060 df-ngp 24061 df-tng 24062 |
This theorem is referenced by: tngngp 24140 tngngp3 24142 tcphcph 24723 |
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