Metamath Proof Explorer


Theorem lssatle

Description: The ordering of two subspaces is determined by the atoms under them. ( chrelat3 analog.) (Contributed by NM, 29-Oct-2014)

Ref Expression
Hypotheses lssatle.s 𝑆 = ( LSubSp ‘ 𝑊 )
lssatle.a 𝐴 = ( LSAtoms ‘ 𝑊 )
lssatle.w ( 𝜑𝑊 ∈ LMod )
lssatle.t ( 𝜑𝑇𝑆 )
lssatle.u ( 𝜑𝑈𝑆 )
Assertion lssatle ( 𝜑 → ( 𝑇𝑈 ↔ ∀ 𝑝𝐴 ( 𝑝𝑇𝑝𝑈 ) ) )

Proof

Step Hyp Ref Expression
1 lssatle.s 𝑆 = ( LSubSp ‘ 𝑊 )
2 lssatle.a 𝐴 = ( LSAtoms ‘ 𝑊 )
3 lssatle.w ( 𝜑𝑊 ∈ LMod )
4 lssatle.t ( 𝜑𝑇𝑆 )
5 lssatle.u ( 𝜑𝑈𝑆 )
6 sstr ( ( 𝑝𝑇𝑇𝑈 ) → 𝑝𝑈 )
7 6 expcom ( 𝑇𝑈 → ( 𝑝𝑇𝑝𝑈 ) )
8 7 ralrimivw ( 𝑇𝑈 → ∀ 𝑝𝐴 ( 𝑝𝑇𝑝𝑈 ) )
9 ss2rab ( { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } ↔ ∀ 𝑝𝐴 ( 𝑝𝑇𝑝𝑈 ) )
10 3 adantr ( ( 𝜑 ∧ { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } ) → 𝑊 ∈ LMod )
11 1 2 lsatlss ( 𝑊 ∈ LMod → 𝐴𝑆 )
12 rabss2 ( 𝐴𝑆 → { 𝑝𝐴𝑝𝑈 } ⊆ { 𝑝𝑆𝑝𝑈 } )
13 uniss ( { 𝑝𝐴𝑝𝑈 } ⊆ { 𝑝𝑆𝑝𝑈 } → { 𝑝𝐴𝑝𝑈 } ⊆ { 𝑝𝑆𝑝𝑈 } )
14 3 11 12 13 4syl ( 𝜑 { 𝑝𝐴𝑝𝑈 } ⊆ { 𝑝𝑆𝑝𝑈 } )
15 unimax ( 𝑈𝑆 { 𝑝𝑆𝑝𝑈 } = 𝑈 )
16 5 15 syl ( 𝜑 { 𝑝𝑆𝑝𝑈 } = 𝑈 )
17 eqid ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
18 17 1 lssss ( 𝑈𝑆𝑈 ⊆ ( Base ‘ 𝑊 ) )
19 5 18 syl ( 𝜑𝑈 ⊆ ( Base ‘ 𝑊 ) )
20 16 19 eqsstrd ( 𝜑 { 𝑝𝑆𝑝𝑈 } ⊆ ( Base ‘ 𝑊 ) )
21 14 20 sstrd ( 𝜑 { 𝑝𝐴𝑝𝑈 } ⊆ ( Base ‘ 𝑊 ) )
22 21 adantr ( ( 𝜑 ∧ { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } ) → { 𝑝𝐴𝑝𝑈 } ⊆ ( Base ‘ 𝑊 ) )
23 uniss ( { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } → { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } )
24 23 adantl ( ( 𝜑 ∧ { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } ) → { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } )
25 eqid ( LSpan ‘ 𝑊 ) = ( LSpan ‘ 𝑊 )
26 17 25 lspss ( ( 𝑊 ∈ LMod ∧ { 𝑝𝐴𝑝𝑈 } ⊆ ( Base ‘ 𝑊 ) ∧ { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } ) → ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑇 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑈 } ) )
27 10 22 24 26 syl3anc ( ( 𝜑 ∧ { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } ) → ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑇 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑈 } ) )
28 27 ex ( 𝜑 → ( { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } → ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑇 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑈 } ) ) )
29 1 25 2 lssats ( ( 𝑊 ∈ LMod ∧ 𝑇𝑆 ) → 𝑇 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑇 } ) )
30 3 4 29 syl2anc ( 𝜑𝑇 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑇 } ) )
31 1 25 2 lssats ( ( 𝑊 ∈ LMod ∧ 𝑈𝑆 ) → 𝑈 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑈 } ) )
32 3 5 31 syl2anc ( 𝜑𝑈 = ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑈 } ) )
33 30 32 sseq12d ( 𝜑 → ( 𝑇𝑈 ↔ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑇 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ { 𝑝𝐴𝑝𝑈 } ) ) )
34 28 33 sylibrd ( 𝜑 → ( { 𝑝𝐴𝑝𝑇 } ⊆ { 𝑝𝐴𝑝𝑈 } → 𝑇𝑈 ) )
35 9 34 syl5bir ( 𝜑 → ( ∀ 𝑝𝐴 ( 𝑝𝑇𝑝𝑈 ) → 𝑇𝑈 ) )
36 8 35 impbid2 ( 𝜑 → ( 𝑇𝑈 ↔ ∀ 𝑝𝐴 ( 𝑝𝑇𝑝𝑈 ) ) )