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Mirrors > Home > MPE Home > Th. List > sgrimval | Structured version Visualization version GIF version |
Description: The induced metric on a subgroup in terms of the induced metric on the parent normed group. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.) |
Ref | Expression |
---|---|
sgrim.x | ⊢ 𝑋 = (𝑇 ↾s 𝑈) |
sgrim.d | ⊢ 𝐷 = (dist‘𝑇) |
sgrim.e | ⊢ 𝐸 = (dist‘𝑋) |
sgrimval.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
sgrimval.n | ⊢ 𝑁 = (norm‘𝐺) |
sgrimval.s | ⊢ 𝑆 = (SubGrp‘𝑇) |
Ref | Expression |
---|---|
sgrimval | ⊢ (((𝐺 ∈ NrmGrp ∧ 𝑈 ∈ 𝑆) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈)) → (𝐴𝐸𝐵) = (𝐴𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgrim.x | . . . 4 ⊢ 𝑋 = (𝑇 ↾s 𝑈) | |
2 | sgrim.d | . . . 4 ⊢ 𝐷 = (dist‘𝑇) | |
3 | sgrim.e | . . . 4 ⊢ 𝐸 = (dist‘𝑋) | |
4 | 1, 2, 3 | sgrim 22656 | . . 3 ⊢ (𝑈 ∈ 𝑆 → 𝐸 = 𝐷) |
5 | 4 | oveqd 6831 | . 2 ⊢ (𝑈 ∈ 𝑆 → (𝐴𝐸𝐵) = (𝐴𝐷𝐵)) |
6 | 5 | ad2antlr 765 | 1 ⊢ (((𝐺 ∈ NrmGrp ∧ 𝑈 ∈ 𝑆) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈)) → (𝐴𝐸𝐵) = (𝐴𝐷𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ‘cfv 6049 (class class class)co 6814 ↾s cress 16080 distcds 16172 SubGrpcsubg 17809 normcnm 22602 NrmGrpcngp 22603 toNrmGrp ctng 22604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-ds 16186 |
This theorem is referenced by: (None) |
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