| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 11246 | . . . . 5 ⊢ ℝ ∈ V | |
| 6 | fex2 7958 | . . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V) | |
| 7 | 4, 5, 6 | mp3an23 1455 | . . . 4 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V) |
| 8 | tngngp.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
| 9 | tngngp.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
| 10 | 8, 9 | tngds 24668 | . . . 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 24587 | . . 3 ⊢ (𝜑 → (𝑁 ∘ − ) ∈ (Met‘𝑋)) |
| 16 | 11, 15 | eqeltrrd 2842 | . 2 ⊢ (𝜑 → (dist‘𝑇) ∈ (Met‘𝑋)) |
| 17 | eqid 2737 | . . . 4 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
| 18 | 8, 3, 17 | tngngp2 24673 | . . 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 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 + caddc 11158 ≤ cle 11296 Basecbs 17247 distcds 17306 0gc0g 17484 Grpcgrp 18951 -gcsg 18953 Metcmet 21350 NrmGrpcngp 24590 toNrmGrp ctng 24591 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-tset 17316 df-ds 17319 df-rest 17467 df-topn 17468 df-0g 17486 df-topgen 17488 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330 df-ms 24331 df-nm 24595 df-ngp 24596 df-tng 24597 |
| This theorem is referenced by: tngngp 24675 tngngp3 24677 tcphcph 25271 |
| Copyright terms: Public domain | W3C validator |