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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 24491. (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 29520 | . 2 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) | |
2 | elex 3463 | . . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V) | |
3 | 2 | adantr 481 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝐺 ∈ V) |
4 | cnex 11132 | . . . . . . 7 ⊢ ℂ ∈ V | |
5 | ablogrpo 29489 | . . . . . . . 8 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) | |
6 | isvcOLD.1 | . . . . . . . . 9 ⊢ 𝑋 = ran 𝐺 | |
7 | rnexg 7841 | . . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
8 | 6, 7 | eqeltrid 2842 | . . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
9 | 5, 8 | syl 17 | . . . . . . 7 ⊢ (𝐺 ∈ AbelOp → 𝑋 ∈ V) |
10 | xpexg 7684 | . . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑋 ∈ V) → (ℂ × 𝑋) ∈ V) | |
11 | 4, 9, 10 | sylancr 587 | . . . . . 6 ⊢ (𝐺 ∈ AbelOp → (ℂ × 𝑋) ∈ V) |
12 | fex 7176 | . . . . . 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 459 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝑆 ∈ V) |
15 | 3, 14 | jca 512 | . . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
16 | 15 | 3adant3 1132 | . 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 29519 | . 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 379 | 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 Vcvv 3445 〈cop 4592 × cxp 5631 ran crn 5634 ⟶wf 6492 (class class class)co 7357 ℂcc 11049 1c1 11052 + caddc 11054 · cmul 11056 GrpOpcgr 29431 AbelOpcablo 29486 CVecOLDcvc 29500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-ablo 29487 df-vc 29501 |
This theorem is referenced by: isvciOLD 29522 |
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