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Metamath Proof Explorer

Theorem lsmdisj2r

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016)

Ref Expression
Hypotheses lsmcntz.p ⊢ ⊕ = ( LSSum ‘ 𝐺 )
lsmcntz.s ⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
lsmcntz.t ⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
lsmcntz.u ⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
lsmdisj.o ⊢ 0 = ( 0g ‘ 𝐺 )
lsmdisjr.i ⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } )
lsmdisj2r.i ⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = { 0 } )
Assertion lsmdisj2r ( 𝜑 → ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 ) = { 0 } )

Proof

Step Hyp Ref Expression
1 lsmcntz.p ⊢ ⊕ = ( LSSum ‘ 𝐺 )
2 lsmcntz.s ⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
3 lsmcntz.t ⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
4 lsmcntz.u ⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
5 lsmdisj.o ⊢ 0 = ( 0g ‘ 𝐺 )
6 lsmdisjr.i ⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } )
7 lsmdisj2r.i ⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = { 0 } )
8 eqid ⊢ ( oppg ‘ 𝐺 ) = ( oppg ‘ 𝐺 )
9 8 1 oppglsm ⊢ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑆 ) = ( 𝑆 ⊕ 𝑈 )
10 9 ineq2i ⊢ ( 𝑇 ∩ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑆 ) ) = ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) )
11 incom ⊢ ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 )
12 10 11 eqtri ⊢ ( 𝑇 ∩ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑆 ) ) = ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 )
13 eqid ⊢ ( LSSum ‘ ( oppg ‘ 𝐺 ) ) = ( LSSum ‘ ( oppg ‘ 𝐺 ) )
14 8 oppgsubg ⊢ ( SubGrp ‘ 𝐺 ) = ( SubGrp ‘ ( oppg ‘ 𝐺 ) )
15 4 14 eleqtrdi ⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ ( oppg ‘ 𝐺 ) ) )
16 3 14 eleqtrdi ⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ ( oppg ‘ 𝐺 ) ) )
17 2 14 eleqtrdi ⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ ( oppg ‘ 𝐺 ) ) )
18 8 5 oppgid ⊢ 0 = ( 0g ‘ ( oppg ‘ 𝐺 ) )
19 8 1 oppglsm ⊢ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑇 ) = ( 𝑇 ⊕ 𝑈 )
20 19 ineq1i ⊢ ( ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑇 ) ∩ 𝑆 ) = ( ( 𝑇 ⊕ 𝑈 ) ∩ 𝑆 )
21 incom ⊢ ( ( 𝑇 ⊕ 𝑈 ) ∩ 𝑆 ) = ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) )
22 20 21 eqtri ⊢ ( ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑇 ) ∩ 𝑆 ) = ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) )
23 22 6 eqtrid ⊢ ( 𝜑 → ( ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑇 ) ∩ 𝑆 ) = { 0 } )
24 incom ⊢ ( 𝑇 ∩ 𝑈 ) = ( 𝑈 ∩ 𝑇 )
25 24 7 eqtr3id ⊢ ( 𝜑 → ( 𝑈 ∩ 𝑇 ) = { 0 } )
26 13 15 16 17 18 23 25 lsmdisj2 ⊢ ( 𝜑 → ( 𝑇 ∩ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑆 ) ) = { 0 } )
27 12 26 eqtr3id ⊢ ( 𝜑 → ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 ) = { 0 } )