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Mirrors > Home > MPE Home > Th. List > vcex | Structured version Visualization version GIF version |
Description: The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vcex | ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5041 | . 2 ⊢ (𝐺CVecOLD𝑆 ↔ 〈𝐺, 𝑆〉 ∈ CVecOLD) | |
2 | vcrel 28507 | . . 3 ⊢ Rel CVecOLD | |
3 | 2 | brrelex12i 5588 | . 2 ⊢ (𝐺CVecOLD𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
4 | 1, 3 | sylbir 238 | 1 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 Vcvv 3400 〈cop 4532 class class class wbr 5040 CVecOLDcvc 28505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-xp 5541 df-rel 5542 df-vc 28506 |
This theorem is referenced by: isvcOLD 28526 nvex 28558 isnv 28559 |
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