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Theorem islbs 19848
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 3498 . . . 4 (𝑊𝑋𝑊 ∈ V)
2 islbs.j . . . . 5 𝐽 = (LBasis‘𝑊)
3 fveq2 6661 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 islbs.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2877 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4541 . . . . . . 7 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7 fvexd 6676 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) ∈ V)
8 fveq2 6661 . . . . . . . . 9 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
9 islbs.n . . . . . . . . 9 𝑁 = (LSpan‘𝑊)
108, 9syl6eqr 2877 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
11 fvexd 6676 . . . . . . . . 9 ((𝑤 = 𝑊𝑛 = 𝑁) → (Scalar‘𝑤) ∈ V)
12 fveq2 6661 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
1312adantr 484 . . . . . . . . . 10 ((𝑤 = 𝑊𝑛 = 𝑁) → (Scalar‘𝑤) = (Scalar‘𝑊))
14 islbs.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
1513, 14syl6eqr 2877 . . . . . . . . 9 ((𝑤 = 𝑊𝑛 = 𝑁) → (Scalar‘𝑤) = 𝐹)
16 simplr 768 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑛 = 𝑁)
1716fveq1d 6663 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛𝑏) = (𝑁𝑏))
185ad2antrr 725 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑤) = 𝑉)
1917, 18eqeq12d 2840 . . . . . . . . . 10 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑛𝑏) = (Base‘𝑤) ↔ (𝑁𝑏) = 𝑉))
20 simpr 488 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
2120fveq2d 6665 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
22 islbs.k . . . . . . . . . . . . . 14 𝐾 = (Base‘𝐹)
2321, 22syl6eqr 2877 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
2420fveq2d 6665 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g𝑓) = (0g𝐹))
25 islbs.z . . . . . . . . . . . . . . 15 0 = (0g𝐹)
2624, 25syl6eqr 2877 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g𝑓) = 0 )
2726sneqd 4562 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → {(0g𝑓)} = { 0 })
2823, 27difeq12d 4086 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((Base‘𝑓) ∖ {(0g𝑓)}) = (𝐾 ∖ { 0 }))
29 fveq2 6661 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
30 islbs.s . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑊)
3129, 30syl6eqr 2877 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
3231ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ( ·𝑠𝑤) = · )
3332oveqd 7166 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑦( ·𝑠𝑤)𝑥) = (𝑦 · 𝑥))
3416fveq1d 6663 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝑏 ∖ {𝑥})))
3533, 34eleq12d 2910 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
3635notbid 321 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
3728, 36raleqbidv 3392 . . . . . . . . . . 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 3192 . . . . . . . . . 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 633 . . . . . . . . 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 3801 . . . . . . . 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 3801 . . . . . . 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 3471 . . . . . 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 19847 . . . . . 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 6675 . . . . . . . 8 𝑉 ∈ V
4544pwex 5268 . . . . . . 7 𝒫 𝑉 ∈ V
4645rabex 5221 . . . . . 6 {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ∈ V
4742, 43, 46fvmpt 6759 . . . . 5 (𝑊 ∈ V → (LBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
482, 47syl5eq 2871 . . . 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 2901 . 2 (𝑊𝑋 → (𝐵𝐽𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))}))
5144elpw2 5234 . . . 4 (𝐵 ∈ 𝒫 𝑉𝐵𝑉)
5251anbi1i 626 . . 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 6670 . . . . 5 (𝑏 = 𝐵 → ((𝑁𝑏) = 𝑉 ↔ (𝑁𝐵) = 𝑉))
54 difeq1 4078 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏 ∖ {𝑥}) = (𝐵 ∖ {𝑥}))
5554fveq2d 6665 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑁‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝑥})))
5655eleq2d 2901 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
5756notbid 321 . . . . . . 7 (𝑏 = 𝐵 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
5857ralbidv 3192 . . . . . 6 (𝑏 = 𝐵 → (∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
5958raleqbi1dv 3394 . . . . 5 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
6053, 59anbi12d 633 . . . 4 (𝑏 = 𝐵 → (((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
6160elrab 3666 . . 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 1092 . . 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 306 . 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, 63syl6bb 290 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  [wsbc 3758  cdif 3916  wss 3919  𝒫 cpw 4522  {csn 4550  cfv 6343  (class class class)co 7149  Basecbs 16483  Scalarcsca 16568   ·𝑠 cvsca 16569  0gc0g 16713  LSpanclspn 19743  LBasisclbs 19846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-lbs 19847
This theorem is referenced by:  lbsss  19849  lbssp  19851  lbsind  19852  lbspropd  19871  islbs2  19926  frlmlbs  20941  islbs4  20976
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