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Theorem islbs 20552
Description: The predicate "𝐡 is a basis for the left module or vector space π‘Š". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.)
Hypotheses
Ref Expression
islbs.v 𝑉 = (Baseβ€˜π‘Š)
islbs.f 𝐹 = (Scalarβ€˜π‘Š)
islbs.s Β· = ( ·𝑠 β€˜π‘Š)
islbs.k 𝐾 = (Baseβ€˜πΉ)
islbs.j 𝐽 = (LBasisβ€˜π‘Š)
islbs.n 𝑁 = (LSpanβ€˜π‘Š)
islbs.z 0 = (0gβ€˜πΉ)
Assertion
Ref Expression
islbs (π‘Š ∈ 𝑋 β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
Distinct variable groups:   π‘₯,𝑦,𝐡   𝑦,𝐾   π‘₯,𝑁,𝑦   π‘₯,π‘Š,𝑦   π‘₯,𝐹,𝑦   𝑦, 0
Allowed substitution hints:   Β· (π‘₯,𝑦)   𝐽(π‘₯,𝑦)   𝐾(π‘₯)   𝑉(π‘₯,𝑦)   𝑋(π‘₯,𝑦)   0 (π‘₯)

Proof of Theorem islbs
Dummy variables 𝑏 𝑓 𝑛 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3462 . . . 4 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
2 islbs.j . . . . 5 𝐽 = (LBasisβ€˜π‘Š)
3 fveq2 6843 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
4 islbs.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
53, 4eqtr4di 2791 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
65pweqd 4578 . . . . . . 7 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
7 fvexd 6858 . . . . . . . 8 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) ∈ V)
8 fveq2 6843 . . . . . . . . 9 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = (LSpanβ€˜π‘Š))
9 islbs.n . . . . . . . . 9 𝑁 = (LSpanβ€˜π‘Š)
108, 9eqtr4di 2791 . . . . . . . 8 (𝑀 = π‘Š β†’ (LSpanβ€˜π‘€) = 𝑁)
11 fvexd 6858 . . . . . . . . 9 ((𝑀 = π‘Š ∧ 𝑛 = 𝑁) β†’ (Scalarβ€˜π‘€) ∈ V)
12 fveq2 6843 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
1312adantr 482 . . . . . . . . . 10 ((𝑀 = π‘Š ∧ 𝑛 = 𝑁) β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
14 islbs.f . . . . . . . . . 10 𝐹 = (Scalarβ€˜π‘Š)
1513, 14eqtr4di 2791 . . . . . . . . 9 ((𝑀 = π‘Š ∧ 𝑛 = 𝑁) β†’ (Scalarβ€˜π‘€) = 𝐹)
16 simplr 768 . . . . . . . . . . . 12 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ 𝑛 = 𝑁)
1716fveq1d 6845 . . . . . . . . . . 11 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (π‘›β€˜π‘) = (π‘β€˜π‘))
185ad2antrr 725 . . . . . . . . . . 11 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (Baseβ€˜π‘€) = 𝑉)
1917, 18eqeq12d 2749 . . . . . . . . . 10 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ ((π‘›β€˜π‘) = (Baseβ€˜π‘€) ↔ (π‘β€˜π‘) = 𝑉))
20 simpr 486 . . . . . . . . . . . . . . 15 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
2120fveq2d 6847 . . . . . . . . . . . . . 14 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (Baseβ€˜π‘“) = (Baseβ€˜πΉ))
22 islbs.k . . . . . . . . . . . . . 14 𝐾 = (Baseβ€˜πΉ)
2321, 22eqtr4di 2791 . . . . . . . . . . . . 13 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (Baseβ€˜π‘“) = 𝐾)
2420fveq2d 6847 . . . . . . . . . . . . . . 15 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (0gβ€˜π‘“) = (0gβ€˜πΉ))
25 islbs.z . . . . . . . . . . . . . . 15 0 = (0gβ€˜πΉ)
2624, 25eqtr4di 2791 . . . . . . . . . . . . . 14 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (0gβ€˜π‘“) = 0 )
2726sneqd 4599 . . . . . . . . . . . . 13 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ {(0gβ€˜π‘“)} = { 0 })
2823, 27difeq12d 4084 . . . . . . . . . . . 12 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ ((Baseβ€˜π‘“) βˆ– {(0gβ€˜π‘“)}) = (𝐾 βˆ– { 0 }))
29 fveq2 6843 . . . . . . . . . . . . . . . . 17 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘Š))
30 islbs.s . . . . . . . . . . . . . . . . 17 Β· = ( ·𝑠 β€˜π‘Š)
3129, 30eqtr4di 2791 . . . . . . . . . . . . . . . 16 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = Β· )
3231ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ ( ·𝑠 β€˜π‘€) = Β· )
3332oveqd 7375 . . . . . . . . . . . . . 14 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (𝑦( ·𝑠 β€˜π‘€)π‘₯) = (𝑦 Β· π‘₯))
3416fveq1d 6845 . . . . . . . . . . . . . 14 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (π‘›β€˜(𝑏 βˆ– {π‘₯})) = (π‘β€˜(𝑏 βˆ– {π‘₯})))
3533, 34eleq12d 2828 . . . . . . . . . . . . 13 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ ((𝑦( ·𝑠 β€˜π‘€)π‘₯) ∈ (π‘›β€˜(𝑏 βˆ– {π‘₯})) ↔ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯}))))
3635notbid 318 . . . . . . . . . . . 12 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (Β¬ (𝑦( ·𝑠 β€˜π‘€)π‘₯) ∈ (π‘›β€˜(𝑏 βˆ– {π‘₯})) ↔ Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯}))))
3728, 36raleqbidv 3318 . . . . . . . . . . 11 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (βˆ€π‘¦ ∈ ((Baseβ€˜π‘“) βˆ– {(0gβ€˜π‘“)}) Β¬ (𝑦( ·𝑠 β€˜π‘€)π‘₯) ∈ (π‘›β€˜(𝑏 βˆ– {π‘₯})) ↔ βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯}))))
3837ralbidv 3171 . . . . . . . . . 10 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ ((Baseβ€˜π‘“) βˆ– {(0gβ€˜π‘“)}) Β¬ (𝑦( ·𝑠 β€˜π‘€)π‘₯) ∈ (π‘›β€˜(𝑏 βˆ– {π‘₯})) ↔ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯}))))
3919, 38anbi12d 632 . . . . . . . . 9 (((𝑀 = π‘Š ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) β†’ (((π‘›β€˜π‘) = (Baseβ€˜π‘€) ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ ((Baseβ€˜π‘“) βˆ– {(0gβ€˜π‘“)}) Β¬ (𝑦( ·𝑠 β€˜π‘€)π‘₯) ∈ (π‘›β€˜(𝑏 βˆ– {π‘₯}))) ↔ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))))
4011, 15, 39sbcied2 3787 . . . . . . . 8 ((𝑀 = π‘Š ∧ 𝑛 = 𝑁) β†’ ([(Scalarβ€˜π‘€) / 𝑓]((π‘›β€˜π‘) = (Baseβ€˜π‘€) ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ ((Baseβ€˜π‘“) βˆ– {(0gβ€˜π‘“)}) Β¬ (𝑦( ·𝑠 β€˜π‘€)π‘₯) ∈ (π‘›β€˜(𝑏 βˆ– {π‘₯}))) ↔ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))))
417, 10, 40sbcied2 3787 . . . . . . 7 (𝑀 = π‘Š β†’ ([(LSpanβ€˜π‘€) / 𝑛][(Scalarβ€˜π‘€) / 𝑓]((π‘›β€˜π‘) = (Baseβ€˜π‘€) ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ ((Baseβ€˜π‘“) βˆ– {(0gβ€˜π‘“)}) Β¬ (𝑦( ·𝑠 β€˜π‘€)π‘₯) ∈ (π‘›β€˜(𝑏 βˆ– {π‘₯}))) ↔ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))))
426, 41rabeqbidv 3423 . . . . . 6 (𝑀 = π‘Š β†’ {𝑏 ∈ 𝒫 (Baseβ€˜π‘€) ∣ [(LSpanβ€˜π‘€) / 𝑛][(Scalarβ€˜π‘€) / 𝑓]((π‘›β€˜π‘) = (Baseβ€˜π‘€) ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ ((Baseβ€˜π‘“) βˆ– {(0gβ€˜π‘“)}) Β¬ (𝑦( ·𝑠 β€˜π‘€)π‘₯) ∈ (π‘›β€˜(𝑏 βˆ– {π‘₯})))} = {𝑏 ∈ 𝒫 𝑉 ∣ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))})
43 df-lbs 20551 . . . . . 6 LBasis = (𝑀 ∈ V ↦ {𝑏 ∈ 𝒫 (Baseβ€˜π‘€) ∣ [(LSpanβ€˜π‘€) / 𝑛][(Scalarβ€˜π‘€) / 𝑓]((π‘›β€˜π‘) = (Baseβ€˜π‘€) ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ ((Baseβ€˜π‘“) βˆ– {(0gβ€˜π‘“)}) Β¬ (𝑦( ·𝑠 β€˜π‘€)π‘₯) ∈ (π‘›β€˜(𝑏 βˆ– {π‘₯})))})
444fvexi 6857 . . . . . . . 8 𝑉 ∈ V
4544pwex 5336 . . . . . . 7 𝒫 𝑉 ∈ V
4645rabex 5290 . . . . . 6 {𝑏 ∈ 𝒫 𝑉 ∣ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))} ∈ V
4742, 43, 46fvmpt 6949 . . . . 5 (π‘Š ∈ V β†’ (LBasisβ€˜π‘Š) = {𝑏 ∈ 𝒫 𝑉 ∣ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))})
482, 47eqtrid 2785 . . . 4 (π‘Š ∈ V β†’ 𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))})
491, 48syl 17 . . 3 (π‘Š ∈ 𝑋 β†’ 𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))})
5049eleq2d 2820 . 2 (π‘Š ∈ 𝑋 β†’ (𝐡 ∈ 𝐽 ↔ 𝐡 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))}))
5144elpw2 5303 . . . 4 (𝐡 ∈ 𝒫 𝑉 ↔ 𝐡 βŠ† 𝑉)
5251anbi1i 625 . . 3 ((𝐡 ∈ 𝒫 𝑉 ∧ ((π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))) ↔ (𝐡 βŠ† 𝑉 ∧ ((π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
53 fveqeq2 6852 . . . . 5 (𝑏 = 𝐡 β†’ ((π‘β€˜π‘) = 𝑉 ↔ (π‘β€˜π΅) = 𝑉))
54 difeq1 4076 . . . . . . . . . 10 (𝑏 = 𝐡 β†’ (𝑏 βˆ– {π‘₯}) = (𝐡 βˆ– {π‘₯}))
5554fveq2d 6847 . . . . . . . . 9 (𝑏 = 𝐡 β†’ (π‘β€˜(𝑏 βˆ– {π‘₯})) = (π‘β€˜(𝐡 βˆ– {π‘₯})))
5655eleq2d 2820 . . . . . . . 8 (𝑏 = 𝐡 β†’ ((𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})) ↔ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯}))))
5756notbid 318 . . . . . . 7 (𝑏 = 𝐡 β†’ (Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})) ↔ Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯}))))
5857ralbidv 3171 . . . . . 6 (𝑏 = 𝐡 β†’ (βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})) ↔ βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯}))))
5958raleqbi1dv 3306 . . . . 5 (𝑏 = 𝐡 β†’ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})) ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯}))))
6053, 59anbi12d 632 . . . 4 (𝑏 = 𝐡 β†’ (((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯}))) ↔ ((π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
6160elrab 3646 . . 3 (𝐡 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))} ↔ (𝐡 ∈ 𝒫 𝑉 ∧ ((π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
62 3anass 1096 . . 3 ((𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯}))) ↔ (𝐡 βŠ† 𝑉 ∧ ((π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
6352, 61, 623bitr4i 303 . 2 (𝐡 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((π‘β€˜π‘) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝑏 βˆ– {π‘₯})))} ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯}))))
6450, 63bitrdi 287 1 (π‘Š ∈ 𝑋 β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† 𝑉 ∧ (π‘β€˜π΅) = 𝑉 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ (𝐾 βˆ– { 0 }) Β¬ (𝑦 Β· π‘₯) ∈ (π‘β€˜(𝐡 βˆ– {π‘₯})))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  Vcvv 3444  [wsbc 3740   βˆ– cdif 3908   βŠ† wss 3911  π’« cpw 4561  {csn 4587  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326  LSpanclspn 20447  LBasisclbs 20550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-lbs 20551
This theorem is referenced by:  lbsss  20553  lbssp  20555  lbsind  20556  lbspropd  20575  islbs2  20631  frlmlbs  21219  islbs4  21254
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