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Theorem ldepslinc 41560
Description: For (left) vector spaces, isldepslvec2 41536 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 41559 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 (∀𝑚 ∈ LVec ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∧ ¬ ∀𝑚 ∈ LMod ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
Distinct variable group:   𝑓,𝑚,𝑠,𝑣

Proof of Theorem ldepslinc
StepHypRef Expression
1 eqid 2626 . . . . 5 (Base‘𝑚) = (Base‘𝑚)
2 eqid 2626 . . . . 5 (0g𝑚) = (0g𝑚)
3 eqid 2626 . . . . 5 (Scalar‘𝑚) = (Scalar‘𝑚)
4 eqid 2626 . . . . 5 (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑚))
5 eqid 2626 . . . . 5 (0g‘(Scalar‘𝑚)) = (0g‘(Scalar‘𝑚))
61, 2, 3, 4, 5isldepslvec2 41536 . . . 4 ((𝑚 ∈ LVec ∧ 𝑠 ∈ 𝒫 (Base‘𝑚)) → (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ↔ 𝑠 linDepS 𝑚))
76bicomd 213 . . 3 ((𝑚 ∈ LVec ∧ 𝑠 ∈ 𝒫 (Base‘𝑚)) → (𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
87rgen2 2974 . 2 𝑚 ∈ LVec ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
9 ldepsnlinc 41559 . . . . . . 7 𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ∀𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣))
10 df-ne 2797 . . . . . . . . . . . . . . 15 ((𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣 ↔ ¬ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)
1110imbi2i 326 . . . . . . . . . . . . . 14 ((𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ (𝑓 finSupp (0g‘(Scalar‘𝑚)) → ¬ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
12 imnan 438 . . . . . . . . . . . . . 14 ((𝑓 finSupp (0g‘(Scalar‘𝑚)) → ¬ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ↔ ¬ (𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
1311, 12bitri 264 . . . . . . . . . . . . 13 ((𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ ¬ (𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
1413ralbii 2979 . . . . . . . . . . . 12 (∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣})) ¬ (𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
15 ralnex 2991 . . . . . . . . . . . 12 (∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣})) ¬ (𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ↔ ¬ ∃𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
1614, 15bitri 264 . . . . . . . . . . 11 (∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ ¬ ∃𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
1716ralbii 2979 . . . . . . . . . 10 (∀𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ ∀𝑣𝑠 ¬ ∃𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
18 ralnex 2991 . . . . . . . . . 10 (∀𝑣𝑠 ¬ ∃𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ↔ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
1917, 18bitri 264 . . . . . . . . 9 (∀𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
2019anbi2i 729 . . . . . . . 8 ((𝑠 linDepS 𝑚 ∧ ∀𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣)) ↔ (𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
21202rexbii 3040 . . . . . . 7 (∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ∀𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣)) ↔ ∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
229, 21mpbi 220 . . . . . 6 𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
2322orci 405 . . . . 5 (∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ ∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚))
24 r19.43 3090 . . . . 5 (∃𝑚 ∈ LMod (∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ ∃𝑠 ∈ 𝒫 (Base‘𝑚)(∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)) ↔ (∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ ∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)))
2523, 24mpbir 221 . . . 4 𝑚 ∈ LMod (∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ ∃𝑠 ∈ 𝒫 (Base‘𝑚)(∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚))
26 r19.43 3090 . . . . 5 (∃𝑠 ∈ 𝒫 (Base‘𝑚)((𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)) ↔ (∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ ∃𝑠 ∈ 𝒫 (Base‘𝑚)(∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)))
2726rexbii 3039 . . . 4 (∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)((𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)) ↔ ∃𝑚 ∈ LMod (∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ ∃𝑠 ∈ 𝒫 (Base‘𝑚)(∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)))
2825, 27mpbir 221 . . 3 𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)((𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚))
29 xor 934 . . . . . . . 8 (¬ (𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ↔ ((𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)))
3029bicomi 214 . . . . . . 7 (((𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)) ↔ ¬ (𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
3130rexbii 3039 . . . . . 6 (∃𝑠 ∈ 𝒫 (Base‘𝑚)((𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)) ↔ ∃𝑠 ∈ 𝒫 (Base‘𝑚) ¬ (𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
32 rexnal 2994 . . . . . 6 (∃𝑠 ∈ 𝒫 (Base‘𝑚) ¬ (𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ↔ ¬ ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
3331, 32bitri 264 . . . . 5 (∃𝑠 ∈ 𝒫 (Base‘𝑚)((𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)) ↔ ¬ ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
3433rexbii 3039 . . . 4 (∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)((𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)) ↔ ∃𝑚 ∈ LMod ¬ ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
35 rexnal 2994 . . . 4 (∃𝑚 ∈ LMod ¬ ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ↔ ¬ ∀𝑚 ∈ LMod ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
3634, 35bitri 264 . . 3 (∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)((𝑠 linDepS 𝑚 ∧ ¬ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∨ (∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣) ∧ ¬ 𝑠 linDepS 𝑚)) ↔ ¬ ∀𝑚 ∈ LMod ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
3728, 36mpbi 220 . 2 ¬ ∀𝑚 ∈ LMod ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣))
388, 37pm3.2i 471 1 (∀𝑚 ∈ LVec ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∧ ¬ ∀𝑚 ∈ LMod ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  cdif 3557  𝒫 cpw 4135  {csn 4153   class class class wbr 4618  cfv 5850  (class class class)co 6605  𝑚 cmap 7803   finSupp cfsupp 8220  Basecbs 15776  Scalarcsca 15860  0gc0g 16016  LModclmod 18779  LVecclvec 19016   linC clinc 41455   linDepS clindeps 41492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959  ax-addf 9960  ax-mulf 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-tpos 7298  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fsupp 8221  df-sup 8293  df-inf 8294  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-rp 11777  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-dvds 14903  df-prm 15305  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-starv 15872  df-sca 15873  df-vsca 15874  df-ip 15875  df-tset 15876  df-ple 15877  df-ds 15880  df-unif 15881  df-hom 15882  df-cco 15883  df-0g 16018  df-gsum 16019  df-prds 16024  df-pws 16026  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-sbg 17343  df-mulg 17457  df-subg 17507  df-ghm 17574  df-cntz 17666  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-ring 18465  df-cring 18466  df-oppr 18539  df-dvdsr 18557  df-unit 18558  df-invr 18588  df-drng 18665  df-subrg 18694  df-lmod 18781  df-lss 18847  df-lvec 19017  df-sra 19086  df-rgmod 19087  df-nzr 19172  df-cnfld 19661  df-zring 19733  df-dsmm 19990  df-frlm 20005  df-linc 41457  df-lininds 41493  df-lindeps 41495
This theorem is referenced by: (None)
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