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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 6705 | . . 3 ⊢ (𝑋 ∈ (LBasis‘𝑊) → 𝑊 ∈ V) | |
2 | islbs4.j | . . 3 ⊢ 𝐽 = (LBasis‘𝑊) | |
3 | 1, 2 | eleq2s 2933 | . 2 ⊢ (𝑋 ∈ 𝐽 → 𝑊 ∈ V) |
4 | elfvex 6705 | . . 3 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ V) | |
5 | 4 | adantr 483 | . 2 ⊢ ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵) → 𝑊 ∈ V) |
6 | islbs4.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
7 | eqid 2823 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
8 | eqid 2823 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
9 | eqid 2823 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
10 | islbs4.k | . . . 4 ⊢ 𝐾 = (LSpan‘𝑊) | |
11 | eqid 2823 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
12 | 6, 7, 8, 9, 2, 10, 11 | islbs 19850 | . . 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 | 6, 8, 10, 7, 9, 11 | islinds2 20959 | . . . . 5 ⊢ (𝑊 ∈ V → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))))) |
14 | 13 | 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))) |
15 | 3anan32 1093 | . . . 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)) | |
16 | 14, 15 | syl6rbbr 292 | . . 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 281 | . 2 ⊢ (𝑊 ∈ V → (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))) |
18 | 3, 5, 17 | pm5.21nii 382 | 1 ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 LSpanclspn 19745 LBasisclbs 19848 LIndSclinds 20951 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-lbs 19849 df-lindf 20952 df-linds 20953 |
This theorem is referenced by: lbslinds 20979 islinds3 20980 lmimlbs 20982 lindflbs 30942 rgmoddim 31010 dimkerim 31025 fedgmullem1 31027 fedgmul 31029 ccfldextdgrr 31059 lindsenlbs 34889 |
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