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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 23018 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | nlmngp 22528 | . 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 2030 NrmGrpcngp 22429 NrmModcnlm 22432 ℂPreHilccph 23012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-nul 4822 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-xp 5149 df-cnv 5151 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fv 5934 df-ov 6693 df-nlm 22438 df-cph 23014 |
This theorem is referenced by: cphnmf 23041 reipcl 23043 ipge0 23044 cphipval2 23086 4cphipval2 23087 cphipval 23088 ipcn 23091 cnmpt1ip 23092 cnmpt2ip 23093 clsocv 23095 minveclem1 23241 minveclem2 23243 minveclem3b 23245 minveclem3 23246 minveclem4a 23247 minveclem4 23249 minveclem6 23251 minveclem7 23252 pjthlem1 23254 rrxngp 40820 |
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