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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 6750 | . . 3 ⊢ (𝑋 ∈ (LBasis‘𝑊) → 𝑊 ∈ V) | |
2 | islbs4.j | . . 3 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | 1, 2 | eleq2s 2856 | . 2 ⊢ (𝑋 ∈ 𝐽 → 𝑊 ∈ V) |
4 | elfvex 6750 | . . 3 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ V) | |
5 | 4 | adantr 484 | . 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 20113 | . . 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 1099 | . . . 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 20775 | . . . . 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 633 | . . . 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 293 | . . 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 282 | . 2 ⊢ (𝑊 ∈ V → (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))) |
18 | 3, 5, 17 | pm5.21nii 383 | 1 ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ∖ cdif 3863 ⊆ wss 3866 {csn 4541 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 Scalarcsca 16805 ·𝑠 cvsca 16806 0gc0g 16944 LSpanclspn 20008 LBasisclbs 20111 LIndSclinds 20767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-lbs 20112 df-lindf 20768 df-linds 20769 |
This theorem is referenced by: lbslinds 20795 islinds3 20796 lmimlbs 20798 lindflbs 31288 rgmoddim 31407 dimkerim 31422 fedgmullem1 31424 fedgmul 31426 ccfldextdgrr 31456 lindsenlbs 35509 |
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