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Mirrors > Home > MPE Home > Th. List > cphngp | Structured version Visualization version GIF version |
Description: A subcomplex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cphngp | ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphnlm 24023 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | nlmngp 23529 | . 2 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 NrmGrpcngp 23429 NrmModcnlm 23432 ℂPreHilccph 24017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-xp 5542 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fv 6366 df-ov 7194 df-nlm 23438 df-cph 24019 |
This theorem is referenced by: cphnmf 24046 reipcl 24048 ipge0 24049 cphpyth 24067 cphipval2 24092 4cphipval2 24093 cphipval 24094 ipcn 24097 cnmpt1ip 24098 cnmpt2ip 24099 clsocv 24101 minveclem1 24275 minveclem2 24277 minveclem3b 24279 minveclem3 24280 minveclem4a 24281 minveclem4 24283 minveclem6 24285 minveclem7 24286 pjthlem1 24288 rrxngp 43444 |
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