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Mirrors > Home > MPE Home > Th. List > hlvc | Structured version Visualization version GIF version |
Description: Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlvc.1 | ⊢ 𝑊 = (1st ‘𝑈) |
Ref | Expression |
---|---|
hlvc | ⊢ (𝑈 ∈ CHilOLD → 𝑊 ∈ CVecOLD) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlnv 28319 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
2 | hlvc.1 | . . 3 ⊢ 𝑊 = (1st ‘𝑈) | |
3 | 2 | nvvc 28042 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝑈 ∈ CHilOLD → 𝑊 ∈ CVecOLD) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 1st c1st 7443 CVecOLDcvc 27985 NrmCVeccnv 28011 CHilOLDchlo 28313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-1st 7445 df-2nd 7446 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-nmcv 28027 df-cbn 28291 df-hlo 28314 |
This theorem is referenced by: (None) |
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