Metamath Proof Explorer

Theorem lsppropd

Description: If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015) (Revised by Mario Carneiro, 14-Jun-2015) (Revised by AV, 24-Apr-2024)

Ref Expression
Hypotheses lsspropd.b1 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
lsspropd.b2 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
lsspropd.w ⊢ ( 𝜑 → 𝐵 ⊆ 𝑊 )
lsspropd.p ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) )
lsspropd.s1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐾 ) 𝑦 ) ∈ 𝑊 )
lsspropd.s2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝐿 ) 𝑦 ) )
lsspropd.p1 ⊢ ( 𝜑 → 𝑃 = ( Base ‘ ( Scalar ‘ 𝐾 ) ) )
lsspropd.p2 ⊢ ( 𝜑 → 𝑃 = ( Base ‘ ( Scalar ‘ 𝐿 ) ) )
lsppropd.v1 ⊢ ( 𝜑 → 𝐾 ∈ 𝑋 )
lsppropd.v2 ⊢ ( 𝜑 → 𝐿 ∈ 𝑌 )
Assertion lsppropd ( 𝜑 → ( LSpan ‘ 𝐾 ) = ( LSpan ‘ 𝐿 ) )

Proof

Step Hyp Ref Expression
1 lsspropd.b1 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
2 lsspropd.b2 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
3 lsspropd.w ⊢ ( 𝜑 → 𝐵 ⊆ 𝑊 )
4 lsspropd.p ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) )
5 lsspropd.s1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐾 ) 𝑦 ) ∈ 𝑊 )
6 lsspropd.s2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝐿 ) 𝑦 ) )
7 lsspropd.p1 ⊢ ( 𝜑 → 𝑃 = ( Base ‘ ( Scalar ‘ 𝐾 ) ) )
8 lsspropd.p2 ⊢ ( 𝜑 → 𝑃 = ( Base ‘ ( Scalar ‘ 𝐿 ) ) )
9 lsppropd.v1 ⊢ ( 𝜑 → 𝐾 ∈ 𝑋 )
10 lsppropd.v2 ⊢ ( 𝜑 → 𝐿 ∈ 𝑌 )
11 1 2 eqtr3d ⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) )
12 11 pweqd ⊢ ( 𝜑 → 𝒫 ( Base ‘ 𝐾 ) = 𝒫 ( Base ‘ 𝐿 ) )
13 1 2 3 4 5 6 7 8 lsspropd ⊢ ( 𝜑 → ( LSubSp ‘ 𝐾 ) = ( LSubSp ‘ 𝐿 ) )
14 13 rabeqdv ⊢ ( 𝜑 → { 𝑡 ∈ ( LSubSp ‘ 𝐾 ) ∣ 𝑠 ⊆ 𝑡 } = { 𝑡 ∈ ( LSubSp ‘ 𝐿 ) ∣ 𝑠 ⊆ 𝑡 } )
15 14 inteqd ⊢ ( 𝜑 → ∩ { 𝑡 ∈ ( LSubSp ‘ 𝐾 ) ∣ 𝑠 ⊆ 𝑡 } = ∩ { 𝑡 ∈ ( LSubSp ‘ 𝐿 ) ∣ 𝑠 ⊆ 𝑡 } )
16 12 15 mpteq12dv ⊢ ( 𝜑 → ( 𝑠 ∈ 𝒫 ( Base ‘ 𝐾 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝐾 ) ∣ 𝑠 ⊆ 𝑡 } ) = ( 𝑠 ∈ 𝒫 ( Base ‘ 𝐿 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝐿 ) ∣ 𝑠 ⊆ 𝑡 } ) )
17 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
18 eqid ⊢ ( LSubSp ‘ 𝐾 ) = ( LSubSp ‘ 𝐾 )
19 eqid ⊢ ( LSpan ‘ 𝐾 ) = ( LSpan ‘ 𝐾 )
20 17 18 19 lspfval ⊢ ( 𝐾 ∈ 𝑋 → ( LSpan ‘ 𝐾 ) = ( 𝑠 ∈ 𝒫 ( Base ‘ 𝐾 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝐾 ) ∣ 𝑠 ⊆ 𝑡 } ) )
21 9 20 syl ⊢ ( 𝜑 → ( LSpan ‘ 𝐾 ) = ( 𝑠 ∈ 𝒫 ( Base ‘ 𝐾 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝐾 ) ∣ 𝑠 ⊆ 𝑡 } ) )
22 eqid ⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 )
23 eqid ⊢ ( LSubSp ‘ 𝐿 ) = ( LSubSp ‘ 𝐿 )
24 eqid ⊢ ( LSpan ‘ 𝐿 ) = ( LSpan ‘ 𝐿 )
25 22 23 24 lspfval ⊢ ( 𝐿 ∈ 𝑌 → ( LSpan ‘ 𝐿 ) = ( 𝑠 ∈ 𝒫 ( Base ‘ 𝐿 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝐿 ) ∣ 𝑠 ⊆ 𝑡 } ) )
26 10 25 syl ⊢ ( 𝜑 → ( LSpan ‘ 𝐿 ) = ( 𝑠 ∈ 𝒫 ( Base ‘ 𝐿 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝐿 ) ∣ 𝑠 ⊆ 𝑡 } ) )
27 16 21 26 3eqtr4d ⊢ ( 𝜑 → ( LSpan ‘ 𝐾 ) = ( LSpan ‘ 𝐿 ) )