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Theorem islbs 21093
Description: The predicate "𝐵 is a basis for the left module or vector space 𝑊". 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 = (Base‘𝑊)
islbs.f 𝐹 = (Scalar‘𝑊)
islbs.s · = ( ·𝑠𝑊)
islbs.k 𝐾 = (Base‘𝐹)
islbs.j 𝐽 = (LBasis‘𝑊)
islbs.n 𝑁 = (LSpan‘𝑊)
islbs.z 0 = (0g𝐹)
Assertion
Ref Expression
islbs (𝑊𝑋 → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑦,𝐾   𝑥,𝑁,𝑦   𝑥,𝑊,𝑦   𝑥,𝐹,𝑦   𝑦, 0
Allowed substitution hints:   · (𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥)   𝑉(𝑥,𝑦)   𝑋(𝑥,𝑦)   0 (𝑥)

Proof of Theorem islbs
Dummy variables 𝑏 𝑓 𝑛 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3499 . . . 4 (𝑊𝑋𝑊 ∈ V)
2 islbs.j . . . . 5 𝐽 = (LBasis‘𝑊)
3 fveq2 6907 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 islbs.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
53, 4eqtr4di 2793 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4622 . . . . . . 7 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7 fvexd 6922 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) ∈ V)
8 fveq2 6907 . . . . . . . . 9 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
9 islbs.n . . . . . . . . 9 𝑁 = (LSpan‘𝑊)
108, 9eqtr4di 2793 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
11 fvexd 6922 . . . . . . . . 9 ((𝑤 = 𝑊𝑛 = 𝑁) → (Scalar‘𝑤) ∈ V)
12 fveq2 6907 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
1312adantr 480 . . . . . . . . . 10 ((𝑤 = 𝑊𝑛 = 𝑁) → (Scalar‘𝑤) = (Scalar‘𝑊))
14 islbs.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
1513, 14eqtr4di 2793 . . . . . . . . 9 ((𝑤 = 𝑊𝑛 = 𝑁) → (Scalar‘𝑤) = 𝐹)
16 simplr 769 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑛 = 𝑁)
1716fveq1d 6909 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛𝑏) = (𝑁𝑏))
185ad2antrr 726 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑤) = 𝑉)
1917, 18eqeq12d 2751 . . . . . . . . . 10 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑛𝑏) = (Base‘𝑤) ↔ (𝑁𝑏) = 𝑉))
20 simpr 484 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
2120fveq2d 6911 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
22 islbs.k . . . . . . . . . . . . . 14 𝐾 = (Base‘𝐹)
2321, 22eqtr4di 2793 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
2420fveq2d 6911 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g𝑓) = (0g𝐹))
25 islbs.z . . . . . . . . . . . . . . 15 0 = (0g𝐹)
2624, 25eqtr4di 2793 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g𝑓) = 0 )
2726sneqd 4643 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → {(0g𝑓)} = { 0 })
2823, 27difeq12d 4137 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((Base‘𝑓) ∖ {(0g𝑓)}) = (𝐾 ∖ { 0 }))
29 fveq2 6907 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
30 islbs.s . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑊)
3129, 30eqtr4di 2793 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
3231ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ( ·𝑠𝑤) = · )
3332oveqd 7448 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑦( ·𝑠𝑤)𝑥) = (𝑦 · 𝑥))
3416fveq1d 6909 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝑏 ∖ {𝑥})))
3533, 34eleq12d 2833 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
3635notbid 318 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
3728, 36raleqbidv 3344 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ ((Base‘𝑓) ∖ {(0g𝑓)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
3837ralbidv 3176 . . . . . . . . . 10 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (∀𝑥𝑏𝑦 ∈ ((Base‘𝑓) ∖ {(0g𝑓)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
3919, 38anbi12d 632 . . . . . . . . 9 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (((𝑛𝑏) = (Base‘𝑤) ∧ ∀𝑥𝑏𝑦 ∈ ((Base‘𝑓) ∖ {(0g𝑓)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))))
4011, 15, 39sbcied2 3839 . . . . . . . 8 ((𝑤 = 𝑊𝑛 = 𝑁) → ([(Scalar‘𝑤) / 𝑓]((𝑛𝑏) = (Base‘𝑤) ∧ ∀𝑥𝑏𝑦 ∈ ((Base‘𝑓) ∖ {(0g𝑓)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))))
417, 10, 40sbcied2 3839 . . . . . . 7 (𝑤 = 𝑊 → ([(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑓]((𝑛𝑏) = (Base‘𝑤) ∧ ∀𝑥𝑏𝑦 ∈ ((Base‘𝑓) ∖ {(0g𝑓)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))))
426, 41rabeqbidv 3452 . . . . . 6 (𝑤 = 𝑊 → {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑓]((𝑛𝑏) = (Base‘𝑤) ∧ ∀𝑥𝑏𝑦 ∈ ((Base‘𝑓) ∖ {(0g𝑓)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))} = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
43 df-lbs 21092 . . . . . 6 LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑓]((𝑛𝑏) = (Base‘𝑤) ∧ ∀𝑥𝑏𝑦 ∈ ((Base‘𝑓) ∖ {(0g𝑓)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))})
444fvexi 6921 . . . . . . . 8 𝑉 ∈ V
4544pwex 5386 . . . . . . 7 𝒫 𝑉 ∈ V
4645rabex 5345 . . . . . 6 {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ∈ V
4742, 43, 46fvmpt 7016 . . . . 5 (𝑊 ∈ V → (LBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
482, 47eqtrid 2787 . . . 4 (𝑊 ∈ V → 𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
491, 48syl 17 . . 3 (𝑊𝑋𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
5049eleq2d 2825 . 2 (𝑊𝑋 → (𝐵𝐽𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))}))
5144elpw2 5340 . . . 4 (𝐵 ∈ 𝒫 𝑉𝐵𝑉)
5251anbi1i 624 . . 3 ((𝐵 ∈ 𝒫 𝑉 ∧ ((𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ↔ (𝐵𝑉 ∧ ((𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
53 fveqeq2 6916 . . . . 5 (𝑏 = 𝐵 → ((𝑁𝑏) = 𝑉 ↔ (𝑁𝐵) = 𝑉))
54 difeq1 4129 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏 ∖ {𝑥}) = (𝐵 ∖ {𝑥}))
5554fveq2d 6911 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑁‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝑥})))
5655eleq2d 2825 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
5756notbid 318 . . . . . . 7 (𝑏 = 𝐵 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
5857ralbidv 3176 . . . . . 6 (𝑏 = 𝐵 → (∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
5958raleqbi1dv 3336 . . . . 5 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
6053, 59anbi12d 632 . . . 4 (𝑏 = 𝐵 → (((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
6160elrab 3695 . . 3 (𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ↔ (𝐵 ∈ 𝒫 𝑉 ∧ ((𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
62 3anass 1094 . . 3 ((𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))) ↔ (𝐵𝑉 ∧ ((𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
6352, 61, 623bitr4i 303 . 2 (𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
6450, 63bitrdi 287 1 (𝑊𝑋 → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  [wsbc 3791  cdif 3960  wss 3963  𝒫 cpw 4605  {csn 4631  cfv 6563  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486  LSpanclspn 20987  LBasisclbs 21091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-lbs 21092
This theorem is referenced by:  lbsss  21094  lbssp  21096  lbsind  21097  lbspropd  21116  islbs2  21174  frlmlbs  21835  islbs4  21870
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