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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 27414 | . 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 | relopabi 5245 | 1 ⊢ Rel CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 × cxp 5112 ran crn 5115 Rel wrel 5119 ⟶wf 5884 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 AbelOpcablo 27398 CVecOLDcvc 27413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 df-rel 5121 df-vc 27414 |
This theorem is referenced by: vcex 27433 nvvop 27464 phop 27673 |
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