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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 6920 | . . . . 5 ⊢ 𝑋 ∈ V |
5 | reex 11243 | . . . . 5 ⊢ ℝ ∈ V | |
6 | fex2 7956 | . . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V) | |
7 | 4, 5, 6 | mp3an23 1452 | . . . 4 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V) |
8 | tngngp.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
9 | tngngp.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
10 | 8, 9 | tngds 24683 | . . . 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 24602 | . . 3 ⊢ (𝜑 → (𝑁 ∘ − ) ∈ (Met‘𝑋)) |
16 | 11, 15 | eqeltrrd 2839 | . 2 ⊢ (𝜑 → (dist‘𝑇) ∈ (Met‘𝑋)) |
17 | eqid 2734 | . . . 4 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
18 | 8, 3, 17 | tngngp2 24688 | . . 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 713 | 1 ⊢ (𝜑 → 𝑇 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 class class class wbr 5147 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 + caddc 11155 ≤ cle 11293 Basecbs 17244 distcds 17306 0gc0g 17485 Grpcgrp 18963 -gcsg 18965 Metcmet 21367 NrmGrpcngp 24605 toNrmGrp ctng 24606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-tset 17316 df-ds 17319 df-rest 17468 df-topn 17469 df-0g 17487 df-topgen 17489 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-xms 24345 df-ms 24346 df-nm 24610 df-ngp 24611 df-tng 24612 |
This theorem is referenced by: tngngp 24690 tngngp3 24692 tcphcph 25284 |
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