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Metamath Proof Explorer

Theorem elpcliN

Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013) (New usage is discouraged.)

Ref Expression
Hypotheses elpcli.s ⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
elpcli.c ⊢ 𝑈 = ( PCl ‘ 𝐾 )
Assertion elpcliN ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) ∧ 𝑄 ∈ ( 𝑈 ‘ 𝑋 ) ) → 𝑄 ∈ 𝑌 )

Proof

Step Hyp Ref Expression
1 elpcli.s ⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
2 elpcli.c ⊢ 𝑈 = ( PCl ‘ 𝐾 )
3 simp1 ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → 𝐾 ∈ 𝑉 )
4 simp2 ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → 𝑋 ⊆ 𝑌 )
5 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
6 5 1 psubssat ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑌 ∈ 𝑆 ) → 𝑌 ⊆ ( Atoms ‘ 𝐾 ) )
7 6 3adant2 ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → 𝑌 ⊆ ( Atoms ‘ 𝐾 ) )
8 4 7 sstrd ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → 𝑋 ⊆ ( Atoms ‘ 𝐾 ) )
9 5 1 2 pclvalN ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ ( Atoms ‘ 𝐾 ) ) → ( 𝑈 ‘ 𝑋 ) = ∩ { 𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧 } )
10 3 8 9 syl2anc ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑈 ‘ 𝑋 ) = ∩ { 𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧 } )
11 10 eleq2d ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑄 ∈ ( 𝑈 ‘ 𝑋 ) ↔ 𝑄 ∈ ∩ { 𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧 } ) )
12 elintrabg ⊢ ( 𝑄 ∈ ∩ { 𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧 } → ( 𝑄 ∈ ∩ { 𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧 } ↔ ∀ 𝑧 ∈ 𝑆 ( 𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧 ) ) )
13 12 ibi ⊢ ( 𝑄 ∈ ∩ { 𝑧 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑧 } → ∀ 𝑧 ∈ 𝑆 ( 𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧 ) )
14 11 13 biimtrdi ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑄 ∈ ( 𝑈 ‘ 𝑋 ) → ∀ 𝑧 ∈ 𝑆 ( 𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧 ) ) )
15 sseq2 ⊢ ( 𝑧 = 𝑌 → ( 𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑌 ) )
16 eleq2 ⊢ ( 𝑧 = 𝑌 → ( 𝑄 ∈ 𝑧 ↔ 𝑄 ∈ 𝑌 ) )
17 15 16 imbi12d ⊢ ( 𝑧 = 𝑌 → ( ( 𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧 ) ↔ ( 𝑋 ⊆ 𝑌 → 𝑄 ∈ 𝑌 ) ) )
18 17 rspccv ⊢ ( ∀ 𝑧 ∈ 𝑆 ( 𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧 ) → ( 𝑌 ∈ 𝑆 → ( 𝑋 ⊆ 𝑌 → 𝑄 ∈ 𝑌 ) ) )
19 18 com13 ⊢ ( 𝑋 ⊆ 𝑌 → ( 𝑌 ∈ 𝑆 → ( ∀ 𝑧 ∈ 𝑆 ( 𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧 ) → 𝑄 ∈ 𝑌 ) ) )
20 19 imp ⊢ ( ( 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → ( ∀ 𝑧 ∈ 𝑆 ( 𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧 ) → 𝑄 ∈ 𝑌 ) )
21 20 3adant1 ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → ( ∀ 𝑧 ∈ 𝑆 ( 𝑋 ⊆ 𝑧 → 𝑄 ∈ 𝑧 ) → 𝑄 ∈ 𝑌 ) )
22 14 21 syld ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑄 ∈ ( 𝑈 ‘ 𝑋 ) → 𝑄 ∈ 𝑌 ) )
23 22 imp ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆 ) ∧ 𝑄 ∈ ( 𝑈 ‘ 𝑋 ) ) → 𝑄 ∈ 𝑌 )