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Mirrors > Home > MPE Home > Th. List > lindff | Structured version Visualization version GIF version |
Description: Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
Ref | Expression |
---|---|
lindff.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
lindff | ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . 3 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹 LIndF 𝑊) | |
2 | rellindf 20946 | . . . . . 6 ⊢ Rel LIndF | |
3 | 2 | brrelex1i 5603 | . . . . 5 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V) |
4 | lindff.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
5 | eqid 2821 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | eqid 2821 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
7 | eqid 2821 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
8 | eqid 2821 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
9 | eqid 2821 | . . . . . 6 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
10 | 4, 5, 6, 7, 8, 9 | islindf 20950 | . . . . 5 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))) |
11 | 3, 10 | sylan2 594 | . . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 LIndF 𝑊) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))) |
12 | 11 | ancoms 461 | . . 3 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))) |
13 | 1, 12 | mpbid 234 | . 2 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))) |
14 | 13 | simpld 497 | 1 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 ∖ cdif 3933 {csn 4561 class class class wbr 5059 dom cdm 5550 “ cima 5553 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 LSpanclspn 19737 LIndF clindf 20942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-lindf 20944 |
This theorem is referenced by: lindfind2 20956 lindff1 20958 lindfrn 20959 f1lindf 20960 indlcim 20978 |
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