| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl 482 | . 2
⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (SubGrp‘𝑀)) | 
| 2 |  | simplr 769 | . . . . . . . . 9
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ 𝑍) | 
| 3 |  | simprr 773 | . . . . . . . . 9
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋) | 
| 4 | 2, 3 | sseldd 3984 | . . . . . . . 8
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑍) | 
| 5 |  | eqid 2737 | . . . . . . . . . 10
⊢
(Base‘𝑀) =
(Base‘𝑀) | 
| 6 |  | eqid 2737 | . . . . . . . . . 10
⊢
(Cntz‘𝑀) =
(Cntz‘𝑀) | 
| 7 | 5, 6 | cntrval 19337 | . . . . . . . . 9
⊢
((Cntz‘𝑀)‘(Base‘𝑀)) = (Cntr‘𝑀) | 
| 8 |  | cntrnsg.z | . . . . . . . . 9
⊢ 𝑍 = (Cntr‘𝑀) | 
| 9 | 7, 8 | eqtr4i 2768 | . . . . . . . 8
⊢
((Cntz‘𝑀)‘(Base‘𝑀)) = 𝑍 | 
| 10 | 4, 9 | eleqtrrdi 2852 | . . . . . . 7
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀))) | 
| 11 |  | simprl 771 | . . . . . . 7
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀)) | 
| 12 |  | eqid 2737 | . . . . . . . 8
⊢
(+g‘𝑀) = (+g‘𝑀) | 
| 13 | 12, 6 | cntzi 19347 | . . . . . . 7
⊢ ((𝑦 ∈ ((Cntz‘𝑀)‘(Base‘𝑀)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)) | 
| 14 | 10, 11, 13 | syl2anc 584 | . . . . . 6
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → (𝑦(+g‘𝑀)𝑥) = (𝑥(+g‘𝑀)𝑦)) | 
| 15 | 14 | oveq1d 7446 | . . . . 5
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥)) | 
| 16 |  | subgrcl 19149 | . . . . . . 7
⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑀 ∈ Grp) | 
| 17 | 16 | ad2antrr 726 | . . . . . 6
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑀 ∈ Grp) | 
| 18 | 5 | subgss 19145 | . . . . . . . 8
⊢ (𝑋 ∈ (SubGrp‘𝑀) → 𝑋 ⊆ (Base‘𝑀)) | 
| 19 | 18 | ad2antrr 726 | . . . . . . 7
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀)) | 
| 20 | 19, 3 | sseldd 3984 | . . . . . 6
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑀)) | 
| 21 |  | eqid 2737 | . . . . . . 7
⊢
(-g‘𝑀) = (-g‘𝑀) | 
| 22 | 5, 12, 21 | grppncan 19049 | . . . . . 6
⊢ ((𝑀 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦) | 
| 23 | 17, 20, 11, 22 | syl3anc 1373 | . . . . 5
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑦(+g‘𝑀)𝑥)(-g‘𝑀)𝑥) = 𝑦) | 
| 24 | 15, 23 | eqtr3d 2779 | . . . 4
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) = 𝑦) | 
| 25 | 24, 3 | eqeltrd 2841 | . . 3
⊢ (((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ 𝑋)) → ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋) | 
| 26 | 25 | ralrimivva 3202 | . 2
⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ 𝑋 ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋) | 
| 27 | 5, 12, 21 | isnsg3 19178 | . 2
⊢ (𝑋 ∈ (NrmSGrp‘𝑀) ↔ (𝑋 ∈ (SubGrp‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ 𝑋 ((𝑥(+g‘𝑀)𝑦)(-g‘𝑀)𝑥) ∈ 𝑋)) | 
| 28 | 1, 26, 27 | sylanbrc 583 | 1
⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀)) |