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Theorem islbs 19270
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 3344 . . . 4 (𝑊𝑋𝑊 ∈ V)
2 islbs.j . . . . 5 𝐽 = (LBasis‘𝑊)
3 fveq2 6344 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 islbs.v . . . . . . . . 9 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2804 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4299 . . . . . . 7 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7 fvexd 6356 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) ∈ V)
8 fveq2 6344 . . . . . . . . 9 (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
9 islbs.n . . . . . . . . 9 𝑁 = (LSpan‘𝑊)
108, 9syl6eqr 2804 . . . . . . . 8 (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
11 fvexd 6356 . . . . . . . . 9 ((𝑤 = 𝑊𝑛 = 𝑁) → (Scalar‘𝑤) ∈ V)
12 fveq2 6344 . . . . . . . . . . 11 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
1312adantr 472 . . . . . . . . . 10 ((𝑤 = 𝑊𝑛 = 𝑁) → (Scalar‘𝑤) = (Scalar‘𝑊))
14 islbs.f . . . . . . . . . 10 𝐹 = (Scalar‘𝑊)
1513, 14syl6eqr 2804 . . . . . . . . 9 ((𝑤 = 𝑊𝑛 = 𝑁) → (Scalar‘𝑤) = 𝐹)
16 simplr 809 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑛 = 𝑁)
1716fveq1d 6346 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛𝑏) = (𝑁𝑏))
185ad2antrr 764 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑤) = 𝑉)
1917, 18eqeq12d 2767 . . . . . . . . . 10 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑛𝑏) = (Base‘𝑤) ↔ (𝑁𝑏) = 𝑉))
20 simpr 479 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
2120fveq2d 6348 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
22 islbs.k . . . . . . . . . . . . . 14 𝐾 = (Base‘𝐹)
2321, 22syl6eqr 2804 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
2420fveq2d 6348 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g𝑓) = (0g𝐹))
25 islbs.z . . . . . . . . . . . . . . 15 0 = (0g𝐹)
2624, 25syl6eqr 2804 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g𝑓) = 0 )
2726sneqd 4325 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → {(0g𝑓)} = { 0 })
2823, 27difeq12d 3864 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((Base‘𝑓) ∖ {(0g𝑓)}) = (𝐾 ∖ { 0 }))
29 fveq2 6344 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
30 islbs.s . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑊)
3129, 30syl6eqr 2804 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
3231ad2antrr 764 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ( ·𝑠𝑤) = · )
3332oveqd 6822 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑦( ·𝑠𝑤)𝑥) = (𝑦 · 𝑥))
3416fveq1d 6346 . . . . . . . . . . . . . 14 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝑏 ∖ {𝑥})))
3533, 34eleq12d 2825 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
3635notbid 307 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
3728, 36raleqbidv 3283 . . . . . . . . . . 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 3116 . . . . . . . . . 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 749 . . . . . . . . 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 3606 . . . . . . . 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 3606 . . . . . . 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 3327 . . . . . 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 19269 . . . . . 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})))})
44 fvex 6354 . . . . . . . . 9 (Base‘𝑊) ∈ V
454, 44eqeltri 2827 . . . . . . . 8 𝑉 ∈ V
4645pwex 4989 . . . . . . 7 𝒫 𝑉 ∈ V
4746rabex 4956 . . . . . 6 {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ∈ V
4842, 43, 47fvmpt 6436 . . . . 5 (𝑊 ∈ V → (LBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
492, 48syl5eq 2798 . . . 4 (𝑊 ∈ V → 𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
501, 49syl 17 . . 3 (𝑊𝑋𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
5150eleq2d 2817 . 2 (𝑊𝑋 → (𝐵𝐽𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))}))
5245elpw2 4969 . . . 4 (𝐵 ∈ 𝒫 𝑉𝐵𝑉)
5352anbi1i 733 . . 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})))))
54 fveq2 6344 . . . . . 6 (𝑏 = 𝐵 → (𝑁𝑏) = (𝑁𝐵))
5554eqeq1d 2754 . . . . 5 (𝑏 = 𝐵 → ((𝑁𝑏) = 𝑉 ↔ (𝑁𝐵) = 𝑉))
56 difeq1 3856 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏 ∖ {𝑥}) = (𝐵 ∖ {𝑥}))
5756fveq2d 6348 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑁‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝑥})))
5857eleq2d 2817 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
5958notbid 307 . . . . . . 7 (𝑏 = 𝐵 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
6059ralbidv 3116 . . . . . 6 (𝑏 = 𝐵 → (∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
6160raleqbi1dv 3277 . . . . 5 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
6255, 61anbi12d 749 . . . 4 (𝑏 = 𝐵 → (((𝑁𝑏) = 𝑉 ∧ ∀𝑥𝑏𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
6362elrab 3496 . . 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})))))
64 3anass 1081 . . 3 ((𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))) ↔ (𝐵𝑉 ∧ ((𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
6553, 63, 643bitr4i 292 . 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}))))
6651, 65syl6bb 276 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 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wral 3042  {crab 3046  Vcvv 3332  [wsbc 3568  cdif 3704  wss 3707  𝒫 cpw 4294  {csn 4313  cfv 6041  (class class class)co 6805  Basecbs 16051  Scalarcsca 16138   ·𝑠 cvsca 16139  0gc0g 16294  LSpanclspn 19165  LBasisclbs 19268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-lbs 19269
This theorem is referenced by:  lbsss  19271  lbssp  19273  lbsind  19274  lbspropd  19293  islbs2  19348  frlmlbs  20330  islbs4  20365
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