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Mirrors > Home > MPE Home > Th. List > nrmtngdist | Structured version Visualization version GIF version |
Description: The augmentation of a normed group by its own norm has the same distance function as the normed group (restricted to the base set). (Contributed by AV, 15-Oct-2021.) |
Ref | Expression |
---|---|
nrmtngdist.t | ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) |
nrmtngdist.x | ⊢ 𝑋 = (Base‘𝐺) |
Ref | Expression |
---|---|
nrmtngdist | ⊢ (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6683 | . . 3 ⊢ (norm‘𝐺) ∈ V | |
2 | nrmtngdist.t | . . . 4 ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) | |
3 | eqid 2821 | . . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
4 | 2, 3 | tngds 23257 | . . 3 ⊢ ((norm‘𝐺) ∈ V → ((norm‘𝐺) ∘ (-g‘𝐺)) = (dist‘𝑇)) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ ((norm‘𝐺) ∘ (-g‘𝐺)) = (dist‘𝑇) |
6 | eqid 2821 | . . . 4 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
7 | eqid 2821 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
8 | nrmtngdist.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
9 | eqid 2821 | . . . 4 ⊢ ((dist‘𝐺) ↾ (𝑋 × 𝑋)) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) | |
10 | 6, 3, 7, 8, 9 | isngp2 23206 | . . 3 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)))) |
11 | 10 | simp3bi 1143 | . 2 ⊢ (𝐺 ∈ NrmGrp → ((norm‘𝐺) ∘ (-g‘𝐺)) = ((dist‘𝐺) ↾ (𝑋 × 𝑋))) |
12 | 5, 11 | syl5eqr 2870 | 1 ⊢ (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 × cxp 5553 ↾ cres 5557 ∘ ccom 5559 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 distcds 16574 Grpcgrp 18103 -gcsg 18105 MetSpcms 22928 normcnm 23186 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-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-sets 16490 df-tset 16584 df-ds 16587 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: nrmtngnrm 23267 |
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