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Theorem ldepslinc 47465
Description: For (left) vector spaces, isldepslvec2 47441 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 47464 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 2726 . . . . 5 (Baseβ€˜π‘š) = (Baseβ€˜π‘š)
2 eqid 2726 . . . . 5 (0gβ€˜π‘š) = (0gβ€˜π‘š)
3 eqid 2726 . . . . 5 (Scalarβ€˜π‘š) = (Scalarβ€˜π‘š)
4 eqid 2726 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘š)) = (Baseβ€˜(Scalarβ€˜π‘š))
5 eqid 2726 . . . . 5 (0gβ€˜(Scalarβ€˜π‘š)) = (0gβ€˜(Scalarβ€˜π‘š))
61, 2, 3, 4, 5isldepslvec2 47441 . . . 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 3191 . 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 47464 . . . . . . 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 2935 . . . . . . . . . . . . . . 15 ((𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) β‰  𝑣 ↔ Β¬ (𝑓( linC β€˜π‘š)(𝑠 βˆ– {𝑣})) = 𝑣)
1110imbi2i 336 . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . 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 3087 . . . . . . . . . . . 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 3066 . . . . . . . . . . . 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 275 . . . . . . . . . . 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 3087 . . . . . . . . . 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 3066 . . . . . . . . . 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 275 . . . . . . . . 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 622 . . . . . . . 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 3123 . . . . . . 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 862 . . . . 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 3116 . . . . 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 3116 . . . . 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 3088 . . . 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 1011 . . . . . . . 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 3088 . . . . . 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 3094 . . . . . 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 275 . . . . 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 3088 . . . 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 3094 . . . 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 275 . . 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 470 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 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064   βˆ– cdif 3940  π’« cpw 4597  {csn 4623   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405   ↑m cmap 8822   finSupp cfsupp 9363  Basecbs 17153  Scalarcsca 17209  0gc0g 17394  LModclmod 20706  LVecclvec 20950   linC clinc 47360   linDepS clindeps 47397
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-rp 12981  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-dvds 16205  df-prm 16616  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-starv 17221  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-hom 17230  df-cco 17231  df-0g 17396  df-gsum 17397  df-prds 17402  df-pws 17404  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18996  df-subg 19050  df-ghm 19139  df-cntz 19233  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-ring 20140  df-cring 20141  df-oppr 20236  df-dvdsr 20259  df-unit 20260  df-invr 20290  df-nzr 20415  df-subrng 20446  df-subrg 20471  df-drng 20589  df-lmod 20708  df-lss 20779  df-lvec 20951  df-sra 21021  df-rgmod 21022  df-cnfld 21241  df-zring 21334  df-dsmm 21627  df-frlm 21642  df-linc 47362  df-lininds 47398  df-lindeps 47400
This theorem is referenced by: (None)
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