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Metamath Proof Explorer

Theorem cntzrcl

Description: Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015)

Ref Expression
Hypotheses cntzrcl.b ⊢ 𝐵 = ( Base ‘ 𝑀 )
cntzrcl.z ⊢ 𝑍 = ( Cntz ‘ 𝑀 )
Assertion cntzrcl ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) → ( 𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 cntzrcl.b ⊢ 𝐵 = ( Base ‘ 𝑀 )
2 cntzrcl.z ⊢ 𝑍 = ( Cntz ‘ 𝑀 )
3 noel ⊢ ¬ 𝑋 ∈ ∅
4 fvprc ⊢ ( ¬ 𝑀 ∈ V → ( Cntz ‘ 𝑀 ) = ∅ )
5 2 4 eqtrid ⊢ ( ¬ 𝑀 ∈ V → 𝑍 = ∅ )
6 5 fveq1d ⊢ ( ¬ 𝑀 ∈ V → ( 𝑍 ‘ 𝑆 ) = ( ∅ ‘ 𝑆 ) )
7 0fv ⊢ ( ∅ ‘ 𝑆 ) = ∅
8 6 7 eqtrdi ⊢ ( ¬ 𝑀 ∈ V → ( 𝑍 ‘ 𝑆 ) = ∅ )
9 8 eleq2d ⊢ ( ¬ 𝑀 ∈ V → ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) ↔ 𝑋 ∈ ∅ ) )
10 3 9 mtbiri ⊢ ( ¬ 𝑀 ∈ V → ¬ 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) )
11 10 con4i ⊢ ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) → 𝑀 ∈ V )
12 eqid ⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 )
13 1 12 2 cntzfval ⊢ ( 𝑀 ∈ V → 𝑍 = ( 𝑥 ∈ 𝒫 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ( +g ‘ 𝑀 ) 𝑧 ) = ( 𝑧 ( +g ‘ 𝑀 ) 𝑦 ) } ) )
14 11 13 syl ⊢ ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) → 𝑍 = ( 𝑥 ∈ 𝒫 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ( +g ‘ 𝑀 ) 𝑧 ) = ( 𝑧 ( +g ‘ 𝑀 ) 𝑦 ) } ) )
15 14 dmeqd ⊢ ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) → dom 𝑍 = dom ( 𝑥 ∈ 𝒫 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ( +g ‘ 𝑀 ) 𝑧 ) = ( 𝑧 ( +g ‘ 𝑀 ) 𝑦 ) } ) )
16 eqid ⊢ ( 𝑥 ∈ 𝒫 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ( +g ‘ 𝑀 ) 𝑧 ) = ( 𝑧 ( +g ‘ 𝑀 ) 𝑦 ) } ) = ( 𝑥 ∈ 𝒫 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ( +g ‘ 𝑀 ) 𝑧 ) = ( 𝑧 ( +g ‘ 𝑀 ) 𝑦 ) } )
17 16 dmmptss ⊢ dom ( 𝑥 ∈ 𝒫 𝐵 ↦ { 𝑦 ∈ 𝐵 ∣ ∀ 𝑧 ∈ 𝑥 ( 𝑦 ( +g ‘ 𝑀 ) 𝑧 ) = ( 𝑧 ( +g ‘ 𝑀 ) 𝑦 ) } ) ⊆ 𝒫 𝐵
18 15 17 eqsstrdi ⊢ ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) → dom 𝑍 ⊆ 𝒫 𝐵 )
19 elfvdm ⊢ ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) → 𝑆 ∈ dom 𝑍 )
20 18 19 sseldd ⊢ ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) → 𝑆 ∈ 𝒫 𝐵 )
21 20 elpwid ⊢ ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) → 𝑆 ⊆ 𝐵 )
22 11 21 jca ⊢ ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) → ( 𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵 ) )