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| Mirrors > Home > MPE Home > Th. List > isvcOLD | Structured version Visualization version GIF version | ||
| Description: The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete version of iscvsp 25161. (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| isvcOLD.1 | ⊢ 𝑋 = ran 𝐺 |
| Ref | Expression |
|---|---|
| isvcOLD | ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vcex 30597 | . 2 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) | |
| 2 | elex 3501 | . . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V) | |
| 3 | 2 | adantr 480 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝐺 ∈ V) |
| 4 | cnex 11236 | . . . . . . 7 ⊢ ℂ ∈ V | |
| 5 | ablogrpo 30566 | . . . . . . . 8 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) | |
| 6 | isvcOLD.1 | . . . . . . . . 9 ⊢ 𝑋 = ran 𝐺 | |
| 7 | rnexg 7924 | . . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
| 8 | 6, 7 | eqeltrid 2845 | . . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
| 9 | 5, 8 | syl 17 | . . . . . . 7 ⊢ (𝐺 ∈ AbelOp → 𝑋 ∈ V) |
| 10 | xpexg 7770 | . . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑋 ∈ V) → (ℂ × 𝑋) ∈ V) | |
| 11 | 4, 9, 10 | sylancr 587 | . . . . . 6 ⊢ (𝐺 ∈ AbelOp → (ℂ × 𝑋) ∈ V) |
| 12 | fex 7246 | . . . . . 6 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ (ℂ × 𝑋) ∈ V) → 𝑆 ∈ V) | |
| 13 | 11, 12 | sylan2 593 | . . . . 5 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ 𝐺 ∈ AbelOp) → 𝑆 ∈ V) |
| 14 | 13 | ancoms 458 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝑆 ∈ V) |
| 15 | 3, 14 | jca 511 | . . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
| 16 | 15 | 3adant3 1133 | . 2 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))) → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
| 17 | 6 | isvclem 30596 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (〈𝐺, 𝑆〉 ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))) |
| 18 | 1, 16, 17 | pm5.21nii 378 | 1 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 〈cop 4632 × cxp 5683 ran crn 5686 ⟶wf 6557 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 GrpOpcgr 30508 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-ablo 30564 df-vc 30578 |
| This theorem is referenced by: isvciOLD 30599 |
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