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Mirrors > Home > MPE Home > Th. List > nvvcop | Structured version Visualization version GIF version |
Description: A normed complex vector space is a vector space. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvvcop | ⊢ (〈𝑊, 𝑁〉 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvss 30622 | . . 3 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
2 | 1 | sseli 3991 | . 2 ⊢ (〈𝑊, 𝑁〉 ∈ NrmCVec → 〈𝑊, 𝑁〉 ∈ (CVecOLD × V)) |
3 | opelxp1 5731 | . 2 ⊢ (〈𝑊, 𝑁〉 ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD) | |
4 | 2, 3 | syl 17 | 1 ⊢ (〈𝑊, 𝑁〉 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 〈cop 4637 × cxp 5687 CVecOLDcvc 30587 NrmCVeccnv 30613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-oprab 7435 df-nv 30621 |
This theorem is referenced by: nvex 30640 |
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