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| Mirrors > Home > MPE Home > Th. List > islbs4 | Structured version Visualization version GIF version | ||
| Description: A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ (((LSpan‘𝑤) ‘𝑏) = (Base‘𝑤) ∧ 𝑏 ∈ (LIndS‘𝑤))}). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
| Ref | Expression |
|---|---|
| islbs4.b | ⊢ 𝐵 = (Base‘𝑊) |
| islbs4.j | ⊢ 𝐽 = (LBasis‘𝑊) |
| islbs4.k | ⊢ 𝐾 = (LSpan‘𝑊) |
| Ref | Expression |
|---|---|
| islbs4 | ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfvex 6944 | . . 3 ⊢ (𝑋 ∈ (LBasis‘𝑊) → 𝑊 ∈ V) | |
| 2 | islbs4.j | . . 3 ⊢ 𝐽 = (LBasis‘𝑊) | |
| 3 | 1, 2 | eleq2s 2859 | . 2 ⊢ (𝑋 ∈ 𝐽 → 𝑊 ∈ V) |
| 4 | elfvex 6944 | . . 3 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ V) | |
| 5 | 4 | adantr 480 | . 2 ⊢ ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵) → 𝑊 ∈ V) |
| 6 | islbs4.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
| 7 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 8 | eqid 2737 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 9 | eqid 2737 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 10 | islbs4.k | . . . 4 ⊢ 𝐾 = (LSpan‘𝑊) | |
| 11 | eqid 2737 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
| 12 | 6, 7, 8, 9, 2, 10, 11 | islbs 21075 | . . 3 ⊢ (𝑊 ∈ V → (𝑋 ∈ 𝐽 ↔ (𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))))) |
| 13 | 3anan32 1097 | . . . 4 ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ↔ ((𝑋 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ∧ (𝐾‘𝑋) = 𝐵)) | |
| 14 | 6, 8, 10, 7, 9, 11 | islinds2 21833 | . . . . 5 ⊢ (𝑊 ∈ V → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))))) |
| 15 | 14 | anbi1d 631 | . . . 4 ⊢ (𝑊 ∈ V → ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵) ↔ ((𝑋 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ∧ (𝐾‘𝑋) = 𝐵))) |
| 16 | 13, 15 | bitr4id 290 | . . 3 ⊢ (𝑊 ∈ V → ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))) |
| 17 | 12, 16 | bitrd 279 | . 2 ⊢ (𝑊 ∈ V → (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))) |
| 18 | 3, 5, 17 | pm5.21nii 378 | 1 ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LSpanclspn 20969 LBasisclbs 21073 LIndSclinds 21825 |
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-rab 3437 df-v 3482 df-sbc 3789 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-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-lbs 21074 df-lindf 21826 df-linds 21827 |
| This theorem is referenced by: lbslinds 21853 islinds3 21854 lmimlbs 21856 lindflbs 33407 rlmdim 33660 rgmoddimOLD 33661 dimkerim 33678 fedgmullem1 33680 fedgmul 33682 ccfldextdgrr 33722 lindsenlbs 37622 |
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