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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 23703 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | nlmngp 23213 | . 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 2105 NrmGrpcngp 23114 NrmModcnlm 23117 ℂPreHilccph 23697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 df-ov 7148 df-nlm 23123 df-cph 23699 |
This theorem is referenced by: cphnmf 23726 reipcl 23728 ipge0 23729 cphipval2 23771 4cphipval2 23772 cphipval 23773 ipcn 23776 cnmpt1ip 23777 cnmpt2ip 23778 clsocv 23780 minveclem1 23954 minveclem2 23956 minveclem3b 23958 minveclem3 23959 minveclem4a 23960 minveclem4 23962 minveclem6 23964 minveclem7 23965 pjthlem1 23967 rrxngp 42447 |
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