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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 30351 | . . 3 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
2 | 1 | sseli 3973 | . 2 ⊢ (⟨𝑊, 𝑁⟩ ∈ NrmCVec → ⟨𝑊, 𝑁⟩ ∈ (CVecOLD × V)) |
3 | opelxp1 5711 | . 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 2098 Vcvv 3468 ⟨cop 4629 × cxp 5667 CVecOLDcvc 30316 NrmCVeccnv 30342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5204 df-xp 5675 df-oprab 7408 df-nv 30350 |
This theorem is referenced by: nvex 30369 |
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