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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 28373 | . . 3 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → 〈𝐺, 𝑆〉 ∈ CVecOLD) | |
2 | vcex 28357 | . . 3 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
4 | nvss 28372 | . . . 4 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
5 | 4 | sseli 3965 | . . 3 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → 〈〈𝐺, 𝑆〉, 𝑁〉 ∈ (CVecOLD × V)) |
6 | opelxp2 5599 | . . 3 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ (CVecOLD × V) → 𝑁 ∈ V) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → 𝑁 ∈ V) |
8 | df-3an 1085 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑁 ∈ V)) | |
9 | 3, 7, 8 | sylanbrc 585 | 1 ⊢ (〈〈𝐺, 𝑆〉, 𝑁〉 ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 Vcvv 3496 〈cop 4575 × cxp 5555 CVecOLDcvc 28337 NrmCVeccnv 28363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-oprab 7162 df-vc 28338 df-nv 28371 |
This theorem is referenced by: isnv 28391 h2hva 28753 h2hsm 28754 h2hnm 28755 |
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