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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 23776 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | |
2 | nlmngp 23286 | . 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 2114 NrmGrpcngp 23187 NrmModcnlm 23190 ℂPreHilccph 23770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fv 6363 df-ov 7159 df-nlm 23196 df-cph 23772 |
This theorem is referenced by: cphnmf 23799 reipcl 23801 ipge0 23802 cphipval2 23844 4cphipval2 23845 cphipval 23846 ipcn 23849 cnmpt1ip 23850 cnmpt2ip 23851 clsocv 23853 minveclem1 24027 minveclem2 24029 minveclem3b 24031 minveclem3 24032 minveclem4a 24033 minveclem4 24035 minveclem6 24037 minveclem7 24038 pjthlem1 24040 rrxngp 42590 |
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