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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 23715. (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 28339 | . 2 ⊢ (〈𝐺, 𝑆〉 ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V)) | |
2 | elex 3504 | . . . . 5 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ V) | |
3 | 2 | adantr 483 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝐺 ∈ V) |
4 | cnex 10604 | . . . . . . 7 ⊢ ℂ ∈ V | |
5 | ablogrpo 28308 | . . . . . . . 8 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) | |
6 | isvcOLD.1 | . . . . . . . . 9 ⊢ 𝑋 = ran 𝐺 | |
7 | rnexg 7600 | . . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 ∈ V) | |
8 | 6, 7 | eqeltrid 2917 | . . . . . . . 8 ⊢ (𝐺 ∈ GrpOp → 𝑋 ∈ V) |
9 | 5, 8 | syl 17 | . . . . . . 7 ⊢ (𝐺 ∈ AbelOp → 𝑋 ∈ V) |
10 | xpexg 7459 | . . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑋 ∈ V) → (ℂ × 𝑋) ∈ V) | |
11 | 4, 9, 10 | sylancr 589 | . . . . . 6 ⊢ (𝐺 ∈ AbelOp → (ℂ × 𝑋) ∈ V) |
12 | fex 6975 | . . . . . 6 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ (ℂ × 𝑋) ∈ V) → 𝑆 ∈ V) | |
13 | 11, 12 | sylan2 594 | . . . . 5 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ 𝐺 ∈ AbelOp) → 𝑆 ∈ V) |
14 | 13 | ancoms 461 | . . . 4 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → 𝑆 ∈ V) |
15 | 3, 14 | jca 514 | . . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋) → (𝐺 ∈ V ∧ 𝑆 ∈ V)) |
16 | 15 | 3adant3 1128 | . 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 28338 | . 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 382 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3486 〈cop 4559 × cxp 5539 ran crn 5542 ⟶wf 6337 (class class class)co 7142 ℂcc 10521 1c1 10524 + caddc 10526 · cmul 10528 GrpOpcgr 28250 AbelOpcablo 28305 CVecOLDcvc 28319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-ablo 28306 df-vc 28320 |
This theorem is referenced by: isvciOLD 28341 |
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