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| Mirrors > Home > MPE Home > Th. List > vcrel | Structured version Visualization version GIF version | ||
| Description: The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| vcrel | ⊢ Rel CVecOLD |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-vc 30578 | . 2 ⊢ CVecOLD = {〈𝑔, 𝑠〉 ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))} | |
| 2 | 1 | relopabiv 5830 | 1 ⊢ Rel CVecOLD |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 × cxp 5683 ran crn 5686 Rel wrel 5690 ⟶wf 6557 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 AbelOpcablo 30563 CVecOLDcvc 30577 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 df-rel 5692 df-vc 30578 |
| This theorem is referenced by: vcex 30597 nvvop 30628 phop 30837 |
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