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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 5059 | . 2 ⊢ (𝐺CVecOLD𝑆 ↔ 〈𝐺, 𝑆〉 ∈ CVecOLD) | |
2 | vcrel 28331 | . . 3 ⊢ Rel CVecOLD | |
3 | 2 | brrelex12i 5601 | . 2 ⊢ (𝐺CVecOLD𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
4 | 1, 3 | sylbir 237 | 1 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 Vcvv 3494 〈cop 4566 class class class wbr 5058 CVecOLDcvc 28329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-vc 28330 |
This theorem is referenced by: isvcOLD 28350 nvex 28382 isnv 28383 |
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