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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 28377 | . . 3 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → 〈𝐺, 𝑆〉 ∈ CVecOLD) | |
2 | vcex 28361 | . . 3 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
4 | nvss 28376 | . . . 4 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
5 | 4 | sseli 3911 | . . 3 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → 〈〈𝐺, 𝑆〉, 𝑁〉 ∈ (CVecOLD × V)) |
6 | opelxp2 5561 | . . 3 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ (CVecOLD × V) → 𝑁 ∈ V) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → 𝑁 ∈ V) |
8 | df-3an 1086 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V)) | |
9 | 3, 7, 8 | sylanbrc 586 | 1 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 Vcvv 3441 〈cop 4531 × cxp 5517 CVecOLDcvc 28341 NrmCVeccnv 28367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-oprab 7139 df-vc 28342 df-nv 28375 |
This theorem is referenced by: isnv 28395 h2hva 28757 h2hsm 28758 h2hnm 28759 |
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