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Mirrors > Home > MPE Home > Th. List > nrmtngnrm | Structured version Visualization version GIF version |
Description: The augmentation of a normed group by its own norm is a normed group with the same norm. (Contributed by AV, 15-Oct-2021.) |
Ref | Expression |
---|---|
nrmtngdist.t | ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) |
Ref | Expression |
---|---|
nrmtngnrm | ⊢ (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpgrp 24463 | . . 3 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
2 | nrmtngdist.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) | |
3 | eqid 2726 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 2, 3 | nrmtngdist 24529 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) |
5 | eqid 2726 | . . . . 5 ⊢ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) | |
6 | 3, 5 | ngpmet 24467 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (Met‘(Base‘𝐺))) |
7 | 4, 6 | eqeltrd 2827 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (dist‘𝑇) ∈ (Met‘(Base‘𝐺))) |
8 | eqid 2726 | . . . . 5 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
9 | 3, 8 | nmf 24479 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (norm‘𝐺):(Base‘𝐺)⟶ℝ) |
10 | eqid 2726 | . . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
11 | 2, 3, 10 | tngngp2 24524 | . . . 4 ⊢ ((norm‘𝐺):(Base‘𝐺)⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘(Base‘𝐺))))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘(Base‘𝐺))))) |
13 | 1, 7, 12 | mpbir2and 710 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝑇 ∈ NrmGrp) |
14 | 1, 9 | jca 511 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (𝐺 ∈ Grp ∧ (norm‘𝐺):(Base‘𝐺)⟶ℝ)) |
15 | reex 11203 | . . . . 5 ⊢ ℝ ∈ V | |
16 | 2, 3, 15 | tngnm 24523 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (norm‘𝐺):(Base‘𝐺)⟶ℝ) → (norm‘𝐺) = (norm‘𝑇)) |
17 | 14, 16 | syl 17 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (norm‘𝐺) = (norm‘𝑇)) |
18 | 17 | eqcomd 2732 | . 2 ⊢ (𝐺 ∈ NrmGrp → (norm‘𝑇) = (norm‘𝐺)) |
19 | 13, 18 | jca 511 | 1 ⊢ (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 × cxp 5667 ↾ cres 5671 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 Basecbs 17153 distcds 17215 Grpcgrp 18863 Metcmet 21226 normcnm 24440 NrmGrpcngp 24441 toNrmGrp ctng 24442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-tset 17225 df-ds 17228 df-rest 17377 df-topn 17378 df-0g 17396 df-topgen 17398 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-xms 24181 df-ms 24182 df-nm 24446 df-ngp 24447 df-tng 24448 |
This theorem is referenced by: tngngpim 24531 |
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