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Theorem islbs2 21047
Description: An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.)
Hypotheses
Ref Expression
islbs2.v 𝑉 = (Baseβ€˜π‘Š)
islbs2.j 𝐽 = (LBasisβ€˜π‘Š)
islbs2.n 𝑁 = (LSpanβ€˜π‘Š)
Assertion
Ref Expression
islbs2 (π‘Š ∈ LVec β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝑁   π‘₯,𝑉   π‘₯,π‘Š   π‘₯,𝐽
Proof of Theorem islbs2
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 islbs2.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
2 islbs2.j . . . . 5 𝐽 = (LBasisβ€˜π‘Š)
31, 2lbsss 20967 . . . 4 (𝐡 ∈ 𝐽 β†’ 𝐡 βŠ† 𝑉)
43adantl 480 . . 3 ((π‘Š ∈ LVec ∧ 𝐡 ∈ 𝐽) β†’ 𝐡 βŠ† 𝑉)
5 islbs2.n . . . . 5 𝑁 = (LSpanβ€˜π‘Š)
61, 2, 5lbssp 20969 . . . 4 (𝐡 ∈ 𝐽 β†’ (π‘β€˜π΅) = 𝑉)
76adantl 480 . . 3 ((π‘Š ∈ LVec ∧ 𝐡 ∈ 𝐽) β†’ (π‘β€˜π΅) = 𝑉)
8 lveclmod 20996 . . . . . . 7 (π‘Š ∈ LVec β†’ π‘Š ∈ LMod)
9 eqid 2727 . . . . . . . . 9 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
109lvecdrng 20995 . . . . . . . 8 (π‘Š ∈ LVec β†’ (Scalarβ€˜π‘Š) ∈ DivRing)
11 eqid 2727 . . . . . . . . 9 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
12 eqid 2727 . . . . . . . . 9 (1rβ€˜(Scalarβ€˜π‘Š)) = (1rβ€˜(Scalarβ€˜π‘Š))
1311, 12drngunz 20648 . . . . . . . 8 ((Scalarβ€˜π‘Š) ∈ DivRing β†’ (1rβ€˜(Scalarβ€˜π‘Š)) β‰  (0gβ€˜(Scalarβ€˜π‘Š)))
1410, 13syl 17 . . . . . . 7 (π‘Š ∈ LVec β†’ (1rβ€˜(Scalarβ€˜π‘Š)) β‰  (0gβ€˜(Scalarβ€˜π‘Š)))
158, 14jca 510 . . . . . 6 (π‘Š ∈ LVec β†’ (π‘Š ∈ LMod ∧ (1rβ€˜(Scalarβ€˜π‘Š)) β‰  (0gβ€˜(Scalarβ€˜π‘Š))))
162, 5, 9, 12, 11lbsind2 20971 . . . . . 6 (((π‘Š ∈ LMod ∧ (1rβ€˜(Scalarβ€˜π‘Š)) β‰  (0gβ€˜(Scalarβ€˜π‘Š))) ∧ 𝐡 ∈ 𝐽 ∧ π‘₯ ∈ 𝐡) β†’ Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))
1715, 16syl3an1 1160 . . . . 5 ((π‘Š ∈ LVec ∧ 𝐡 ∈ 𝐽 ∧ π‘₯ ∈ 𝐡) β†’ Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))
18173expa 1115 . . . 4 (((π‘Š ∈ LVec ∧ 𝐡 ∈ 𝐽) ∧ π‘₯ ∈ 𝐡) β†’ Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))
1918ralrimiva 3142 . . 3 ((π‘Š ∈ LVec ∧ 𝐡 ∈ 𝐽) β†’ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))
204, 7, 193jca 1125 . 2 ((π‘Š ∈ LVec ∧ 𝐡 ∈ 𝐽) β†’ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯}))))
21 simpr1 1191 . . 3 ((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) β†’ 𝐡 βŠ† 𝑉)
22 simpr2 1192 . . 3 ((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) β†’ (π‘β€˜π΅) = 𝑉)
23 id 22 . . . . . . . 8 (π‘₯ = 𝑦 β†’ π‘₯ = 𝑦)
24 sneq 4640 . . . . . . . . . 10 (π‘₯ = 𝑦 β†’ {π‘₯} = {𝑦})
2524difeq2d 4120 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (𝐡 βˆ– {π‘₯}) = (𝐡 βˆ– {𝑦}))
2625fveq2d 6904 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (π‘β€˜(𝐡 βˆ– {π‘₯})) = (π‘β€˜(𝐡 βˆ– {𝑦})))
2723, 26eleq12d 2822 . . . . . . 7 (π‘₯ = 𝑦 β†’ (π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})) ↔ 𝑦 ∈ (π‘β€˜(𝐡 βˆ– {𝑦}))))
2827notbid 317 . . . . . 6 (π‘₯ = 𝑦 β†’ (Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})) ↔ Β¬ 𝑦 ∈ (π‘β€˜(𝐡 βˆ– {𝑦}))))
29 simplr3 1214 . . . . . 6 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))
30 simprl 769 . . . . . 6 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ 𝑦 ∈ 𝐡)
3128, 29, 30rspcdva 3610 . . . . 5 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ Β¬ 𝑦 ∈ (π‘β€˜(𝐡 βˆ– {𝑦})))
32 simpll 765 . . . . . . . 8 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ π‘Š ∈ LVec)
33 simprr 771 . . . . . . . . 9 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))
34 eldifsn 4793 . . . . . . . . 9 (𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) ↔ (𝑧 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑧 β‰  (0gβ€˜(Scalarβ€˜π‘Š))))
3533, 34sylib 217 . . . . . . . 8 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ (𝑧 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑧 β‰  (0gβ€˜(Scalarβ€˜π‘Š))))
3621adantr 479 . . . . . . . . 9 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ 𝐡 βŠ† 𝑉)
3736, 30sseldd 3981 . . . . . . . 8 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ 𝑦 ∈ 𝑉)
38 eqid 2727 . . . . . . . . 9 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
39 eqid 2727 . . . . . . . . 9 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
401, 9, 38, 39, 11, 5lspsnvs 21007 . . . . . . . 8 ((π‘Š ∈ LVec ∧ (𝑧 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑧 β‰  (0gβ€˜(Scalarβ€˜π‘Š))) ∧ 𝑦 ∈ 𝑉) β†’ (π‘β€˜{(𝑧( ·𝑠 β€˜π‘Š)𝑦)}) = (π‘β€˜{𝑦}))
4132, 35, 37, 40syl3anc 1368 . . . . . . 7 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ (π‘β€˜{(𝑧( ·𝑠 β€˜π‘Š)𝑦)}) = (π‘β€˜{𝑦}))
4241sseq1d 4011 . . . . . 6 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ ((π‘β€˜{(𝑧( ·𝑠 β€˜π‘Š)𝑦)}) βŠ† (π‘β€˜(𝐡 βˆ– {𝑦})) ↔ (π‘β€˜{𝑦}) βŠ† (π‘β€˜(𝐡 βˆ– {𝑦}))))
43 eqid 2727 . . . . . . 7 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
448adantr 479 . . . . . . . 8 ((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) β†’ π‘Š ∈ LMod)
4544adantr 479 . . . . . . 7 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ π‘Š ∈ LMod)
4636ssdifssd 4141 . . . . . . . 8 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ (𝐡 βˆ– {𝑦}) βŠ† 𝑉)
471, 43, 5lspcl 20865 . . . . . . . 8 ((π‘Š ∈ LMod ∧ (𝐡 βˆ– {𝑦}) βŠ† 𝑉) β†’ (π‘β€˜(𝐡 βˆ– {𝑦})) ∈ (LSubSpβ€˜π‘Š))
4845, 46, 47syl2anc 582 . . . . . . 7 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ (π‘β€˜(𝐡 βˆ– {𝑦})) ∈ (LSubSpβ€˜π‘Š))
4935simpld 493 . . . . . . . 8 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ 𝑧 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
501, 9, 38, 39lmodvscl 20766 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑧 ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑉) β†’ (𝑧( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑉)
5145, 49, 37, 50syl3anc 1368 . . . . . . 7 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ (𝑧( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑉)
521, 43, 5, 45, 48, 51lspsnel5 20884 . . . . . 6 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ ((𝑧( ·𝑠 β€˜π‘Š)𝑦) ∈ (π‘β€˜(𝐡 βˆ– {𝑦})) ↔ (π‘β€˜{(𝑧( ·𝑠 β€˜π‘Š)𝑦)}) βŠ† (π‘β€˜(𝐡 βˆ– {𝑦}))))
531, 43, 5, 45, 48, 37lspsnel5 20884 . . . . . 6 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ (𝑦 ∈ (π‘β€˜(𝐡 βˆ– {𝑦})) ↔ (π‘β€˜{𝑦}) βŠ† (π‘β€˜(𝐡 βˆ– {𝑦}))))
5442, 52, 533bitr4d 310 . . . . 5 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ ((𝑧( ·𝑠 β€˜π‘Š)𝑦) ∈ (π‘β€˜(𝐡 βˆ– {𝑦})) ↔ 𝑦 ∈ (π‘β€˜(𝐡 βˆ– {𝑦}))))
5531, 54mtbird 324 . . . 4 (((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ Β¬ (𝑧( ·𝑠 β€˜π‘Š)𝑦) ∈ (π‘β€˜(𝐡 βˆ– {𝑦})))
5655ralrimivva 3196 . . 3 ((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) β†’ βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (𝑧( ·𝑠 β€˜π‘Š)𝑦) ∈ (π‘β€˜(𝐡 βˆ– {𝑦})))
571, 9, 38, 39, 2, 5, 11islbs 20966 . . . 4 (π‘Š ∈ LVec β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (𝑧( ·𝑠 β€˜π‘Š)𝑦) ∈ (π‘β€˜(𝐡 βˆ– {𝑦})))))
5857adantr 479 . . 3 ((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (𝑧( ·𝑠 β€˜π‘Š)𝑦) ∈ (π‘β€˜(𝐡 βˆ– {𝑦})))))
5921, 22, 56, 58mpbir3and 1339 . 2 ((π‘Š ∈ LVec ∧ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) β†’ 𝐡 ∈ 𝐽)
6020, 59impbida 799 1 (π‘Š ∈ LVec β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 Β¬ π‘₯ ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2936  βˆ€wral 3057   βˆ– cdif 3944   βŠ† wss 3947  {csn 4630  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  Scalarcsca 17241   ·𝑠 cvsca 17242  0gc0g 17426  1rcur 20126  DivRingcdr 20629  LModclmod 20748  LSubSpclss 20820  LSpanclspn 20860  LBasisclbs 20964  LVecclvec 20992
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-tpos 8236  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-2 12311  df-3 12312  df-sets 17138  df-slot 17156  df-ndx 17168  df-base 17186  df-ress 17215  df-plusg 17251  df-mulr 17252  df-0g 17428  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-grp 18898  df-minusg 18899  df-sbg 18900  df-cmn 19742  df-abl 19743  df-mgp 20080  df-rng 20098  df-ur 20127  df-ring 20180  df-oppr 20278  df-dvdsr 20301  df-unit 20302  df-invr 20332  df-drng 20631  df-lmod 20750  df-lss 20821  df-lsp 20861  df-lbs 20965  df-lvec 20993
This theorem is referenced by:  islbs3  21048  lbsacsbs  21049  lbsextlem4  21054  lbslsat  33319
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