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Theorem ldepslinc 47689
Description: For (left) vector spaces, isldepslvec2 47665 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 47688 indicates that there is not an analogous alternative definition for arbitrary (left) modules. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
Assertion
Ref Expression
ldepslinc (βˆ€π‘š ∈ LVec βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∧ Β¬ βˆ€π‘š ∈ LMod βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
Distinct variable group:   𝑓,π‘š,𝑠,𝑣

Proof of Theorem ldepslinc
StepHypRef Expression
1 eqid 2725 . . . . 5 (Baseβ€˜π‘š) = (Baseβ€˜π‘š)
2 eqid 2725 . . . . 5 (0gβ€˜π‘š) = (0gβ€˜π‘š)
3 eqid 2725 . . . . 5 (Scalarβ€˜π‘š) = (Scalarβ€˜π‘š)
4 eqid 2725 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘š)) = (Baseβ€˜(Scalarβ€˜π‘š))
5 eqid 2725 . . . . 5 (0gβ€˜(Scalarβ€˜π‘š)) = (0gβ€˜(Scalarβ€˜π‘š))
61, 2, 3, 4, 5isldepslvec2 47665 . . . 4 ((π‘š ∈ LVec ∧ 𝑠 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ↔ 𝑠 linDepS π‘š))
76bicomd 222 . . 3 ((π‘š ∈ LVec ∧ 𝑠 ∈ 𝒫 (Baseβ€˜π‘š)) β†’ (𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
87rgen2 3188 . 2 βˆ€π‘š ∈ LVec βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
9 ldepsnlinc 47688 . . . . . . 7 βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ βˆ€π‘£ ∈ 𝑠 βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣))
10 df-ne 2931 . . . . . . . . . . . . . . 15 ((𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣 ↔ Β¬ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)
1110imbi2i 335 . . . . . . . . . . . . . 14 ((𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣) ↔ (𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ Β¬ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
12 imnan 398 . . . . . . . . . . . . . 14 ((𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ Β¬ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ↔ Β¬ (𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
1311, 12bitri 274 . . . . . . . . . . . . 13 ((𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣) ↔ Β¬ (𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
1413ralbii 3083 . . . . . . . . . . . 12 (βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣) ↔ βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣})) Β¬ (𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
15 ralnex 3062 . . . . . . . . . . . 12 (βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣})) Β¬ (𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ↔ Β¬ βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
1614, 15bitri 274 . . . . . . . . . . 11 (βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣) ↔ Β¬ βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
1716ralbii 3083 . . . . . . . . . 10 (βˆ€π‘£ ∈ 𝑠 βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣) ↔ βˆ€π‘£ ∈ 𝑠 Β¬ βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
18 ralnex 3062 . . . . . . . . . 10 (βˆ€π‘£ ∈ 𝑠 Β¬ βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ↔ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
1917, 18bitri 274 . . . . . . . . 9 (βˆ€π‘£ ∈ 𝑠 βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣) ↔ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
2019anbi2i 621 . . . . . . . 8 ((𝑠 linDepS π‘š ∧ βˆ€π‘£ ∈ 𝑠 βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣)) ↔ (𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
21202rexbii 3119 . . . . . . 7 (βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ βˆ€π‘£ ∈ 𝑠 βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) β†’ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣)) ↔ βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
229, 21mpbi 229 . . . . . 6 βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
2322orci 863 . . . . 5 (βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š))
24 r19.43 3112 . . . . 5 (βˆƒπ‘š ∈ LMod (βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)) ↔ (βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)))
2523, 24mpbir 230 . . . 4 βˆƒπ‘š ∈ LMod (βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š))
26 r19.43 3112 . . . . 5 (βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)((𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)) ↔ (βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)))
2726rexbii 3084 . . . 4 (βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)((𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)) ↔ βˆƒπ‘š ∈ LMod (βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)(βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)))
2825, 27mpbir 230 . . 3 βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)((𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š))
29 xor 1012 . . . . . . . 8 (Β¬ (𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ↔ ((𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)))
3029bicomi 223 . . . . . . 7 (((𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)) ↔ Β¬ (𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
3130rexbii 3084 . . . . . 6 (βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)((𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)) ↔ βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š) Β¬ (𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
32 rexnal 3090 . . . . . 6 (βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š) Β¬ (𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ↔ Β¬ βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
3331, 32bitri 274 . . . . 5 (βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)((𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)) ↔ Β¬ βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
3433rexbii 3084 . . . 4 (βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)((𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)) ↔ βˆƒπ‘š ∈ LMod Β¬ βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
35 rexnal 3090 . . . 4 (βˆƒπ‘š ∈ LMod Β¬ βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ↔ Β¬ βˆ€π‘š ∈ LMod βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
3634, 35bitri 274 . . 3 (βˆƒπ‘š ∈ LMod βˆƒπ‘  ∈ 𝒫 (Baseβ€˜π‘š)((𝑠 linDepS π‘š ∧ Β¬ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∨ (βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣) ∧ Β¬ 𝑠 linDepS π‘š)) ↔ Β¬ βˆ€π‘š ∈ LMod βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
3728, 36mpbi 229 . 2 Β¬ βˆ€π‘š ∈ LMod βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣))
388, 37pm3.2i 469 1 (βˆ€π‘š ∈ LVec βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)) ∧ Β¬ βˆ€π‘š ∈ LMod βˆ€π‘  ∈ 𝒫 (Baseβ€˜π‘š)(𝑠 linDepS π‘š ↔ βˆƒπ‘£ ∈ 𝑠 βˆƒπ‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m (𝑠 βˆ– {𝑣}))(𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  βˆƒwrex 3060   βˆ– cdif 3936  π’« cpw 4598  {csn 4624   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416   ↑m cmap 8843   finSupp cfsupp 9385  Basecbs 17179  Scalarcsca 17235  0gc0g 17420  LModclmod 20747  LVecclvec 20991   linC clinc 47584   linDepS clindeps 47621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216  ax-addf 11217  ax-mulf 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-supp 8164  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8723  df-map 8845  df-ixp 8915  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-fsupp 9386  df-sup 9465  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-rp 13007  df-fz 13517  df-fzo 13660  df-seq 13999  df-exp 14059  df-hash 14322  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-dvds 16231  df-prm 16642  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-mulr 17246  df-starv 17247  df-sca 17248  df-vsca 17249  df-ip 17250  df-tset 17251  df-ple 17252  df-ds 17254  df-unif 17255  df-hom 17256  df-cco 17257  df-0g 17422  df-gsum 17423  df-prds 17428  df-pws 17430  df-mre 17565  df-mrc 17566  df-acs 17568  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18739  df-submnd 18740  df-grp 18897  df-minusg 18898  df-sbg 18899  df-mulg 19028  df-subg 19082  df-ghm 19172  df-cntz 19272  df-cmn 19741  df-abl 19742  df-mgp 20079  df-rng 20097  df-ur 20126  df-ring 20179  df-cring 20180  df-oppr 20277  df-dvdsr 20300  df-unit 20301  df-invr 20331  df-nzr 20456  df-subrng 20487  df-subrg 20512  df-drng 20630  df-lmod 20749  df-lss 20820  df-lvec 20992  df-sra 21062  df-rgmod 21063  df-cnfld 21284  df-zring 21377  df-dsmm 21670  df-frlm 21685  df-linc 47586  df-lininds 47622  df-lindeps 47624
This theorem is referenced by: (None)
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