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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 6684 | . . . . 5 ⊢ 𝑋 ∈ V |
5 | reex 10628 | . . . . 5 ⊢ ℝ ∈ V | |
6 | fex2 7638 | . . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V) | |
7 | 4, 5, 6 | mp3an23 1449 | . . . 4 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V) |
8 | tngngp.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
9 | tngngp.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
10 | 8, 9 | tngds 23257 | . . . 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 23184 | . . 3 ⊢ (𝜑 → (𝑁 ∘ − ) ∈ (Met‘𝑋)) |
16 | 11, 15 | eqeltrrd 2914 | . 2 ⊢ (𝜑 → (dist‘𝑇) ∈ (Met‘𝑋)) |
17 | eqid 2821 | . . . 4 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
18 | 8, 3, 17 | tngngp2 23261 | . . 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 711 | 1 ⊢ (𝜑 → 𝑇 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 + caddc 10540 ≤ cle 10676 Basecbs 16483 distcds 16574 0gc0g 16713 Grpcgrp 18103 -gcsg 18105 Metcmet 20531 NrmGrpcngp 23187 toNrmGrp ctng 23188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-tset 16584 df-ds 16587 df-rest 16696 df-topn 16697 df-0g 16715 df-topgen 16717 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-xms 22930 df-ms 22931 df-nm 23192 df-ngp 23193 df-tng 23194 |
This theorem is referenced by: tngngp 23263 tngngp3 23265 tcphcph 23840 |
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