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Mirrors > Home > MPE Home > Th. List > nvex | Structured version Visualization version GIF version |
Description: The components of a normed complex vector space are sets. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvex | ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvvcop 26833 | . . 3 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → 〈𝐺, 𝑆〉 ∈ CVecOLD) | |
2 | vcex 26817 | . . 3 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
4 | nvss 26832 | . . . 4 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
5 | 4 | sseli 3564 | . . 3 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → 〈〈𝐺, 𝑆〉, 𝑁〉 ∈ (CVecOLD × V)) |
6 | opelxp2 5075 | . . 3 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ (CVecOLD × V) → 𝑁 ∈ V) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → 𝑁 ∈ V) |
8 | df-3an 1033 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V)) | |
9 | 3, 7, 8 | sylanbrc 695 | 1 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 Vcvv 3173 〈cop 4131 × cxp 5036 CVecOLDcvc 26797 NrmCVeccnv 26823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-oprab 6553 df-vc 26798 df-nv 26831 |
This theorem is referenced by: isnv 26851 h2hva 27215 h2hsm 27216 h2hnm 27217 |
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