| Step | Hyp | Ref
| Expression |
| 1 | | nsgsubg 19176 |
. . 3
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) |
| 2 | | isnsg3.1 |
. . . . . 6
⊢ 𝑋 = (Base‘𝐺) |
| 3 | | isnsg3.2 |
. . . . . 6
⊢ + =
(+g‘𝐺) |
| 4 | | isnsg3.3 |
. . . . . 6
⊢ − =
(-g‘𝐺) |
| 5 | 2, 3, 4 | nsgconj 19177 |
. . . . 5
⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) |
| 6 | 5 | 3expb 1121 |
. . . 4
⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑆)) → ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) |
| 7 | 6 | ralrimivva 3202 |
. . 3
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) |
| 8 | 1, 7 | jca 511 |
. 2
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)) |
| 9 | | simpl 482 |
. . 3
⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺)) |
| 10 | | subgrcl 19149 |
. . . . . . . . . . . 12
⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
| 11 | 10 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝐺 ∈ Grp) |
| 12 | | simprll 779 |
. . . . . . . . . . 11
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑧 ∈ 𝑋) |
| 13 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 14 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(invg‘𝐺) = (invg‘𝐺) |
| 15 | 2, 3, 13, 14 | grplinv 19007 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺)) |
| 16 | 11, 12, 15 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺)) |
| 17 | 16 | oveq1d 7446 |
. . . . . . . . 9
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g‘𝐺) + 𝑤)) |
| 18 | 2, 14 | grpinvcl 19005 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → ((invg‘𝐺)‘𝑧) ∈ 𝑋) |
| 19 | 11, 12, 18 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((invg‘𝐺)‘𝑧) ∈ 𝑋) |
| 20 | | simprlr 780 |
. . . . . . . . . 10
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → 𝑤 ∈ 𝑋) |
| 21 | 2, 3 | grpass 18960 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧
(((invg‘𝐺)‘𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤))) |
| 22 | 11, 19, 12, 20, 21 | syl13anc 1374 |
. . . . . . . . 9
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤))) |
| 23 | 2, 3, 13 | grplid 18985 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺) + 𝑤) = 𝑤) |
| 24 | 11, 20, 23 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((0g‘𝐺) + 𝑤) = 𝑤) |
| 25 | 17, 22, 24 | 3eqtr3d 2785 |
. . . . . . . 8
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) = 𝑤) |
| 26 | 25 | oveq1d 7446 |
. . . . . . 7
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) −
((invg‘𝐺)‘𝑧)) = (𝑤 −
((invg‘𝐺)‘𝑧))) |
| 27 | 2, 3, 4, 14, 11, 20, 12 | grpsubinv 19030 |
. . . . . . 7
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 −
((invg‘𝐺)‘𝑧)) = (𝑤 + 𝑧)) |
| 28 | 26, 27 | eqtrd 2777 |
. . . . . 6
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) −
((invg‘𝐺)‘𝑧)) = (𝑤 + 𝑧)) |
| 29 | | simprr 773 |
. . . . . . 7
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑧 + 𝑤) ∈ 𝑆) |
| 30 | | simplr 769 |
. . . . . . 7
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) |
| 31 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (𝑥 + 𝑦) = (((invg‘𝐺)‘𝑧) + 𝑦)) |
| 32 | | id 22 |
. . . . . . . . . 10
⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → 𝑥 = ((invg‘𝐺)‘𝑧)) |
| 33 | 31, 32 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → ((𝑥 + 𝑦) − 𝑥) = ((((invg‘𝐺)‘𝑧) + 𝑦) −
((invg‘𝐺)‘𝑧))) |
| 34 | 33 | eleq1d 2826 |
. . . . . . . 8
⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (((𝑥 + 𝑦) − 𝑥) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝑧) + 𝑦) −
((invg‘𝐺)‘𝑧)) ∈ 𝑆)) |
| 35 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑦 = (𝑧 + 𝑤) → (((invg‘𝐺)‘𝑧) + 𝑦) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤))) |
| 36 | 35 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝑦 = (𝑧 + 𝑤) → ((((invg‘𝐺)‘𝑧) + 𝑦) −
((invg‘𝐺)‘𝑧)) = ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) −
((invg‘𝐺)‘𝑧))) |
| 37 | 36 | eleq1d 2826 |
. . . . . . . 8
⊢ (𝑦 = (𝑧 + 𝑤) → (((((invg‘𝐺)‘𝑧) + 𝑦) −
((invg‘𝐺)‘𝑧)) ∈ 𝑆 ↔ ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) −
((invg‘𝐺)‘𝑧)) ∈ 𝑆)) |
| 38 | 34, 37 | rspc2va 3634 |
. . . . . . 7
⊢
(((((invg‘𝐺)‘𝑧) ∈ 𝑋 ∧ (𝑧 + 𝑤) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) −
((invg‘𝐺)‘𝑧)) ∈ 𝑆) |
| 39 | 19, 29, 30, 38 | syl21anc 838 |
. . . . . 6
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → ((((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) −
((invg‘𝐺)‘𝑧)) ∈ 𝑆) |
| 40 | 28, 39 | eqeltrrd 2842 |
. . . . 5
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 + 𝑤) ∈ 𝑆)) → (𝑤 + 𝑧) ∈ 𝑆) |
| 41 | 40 | expr 456 |
. . . 4
⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆)) |
| 42 | 41 | ralrimivva 3202 |
. . 3
⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆)) |
| 43 | 2, 3 | isnsg2 19174 |
. . 3
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑧 + 𝑤) ∈ 𝑆 → (𝑤 + 𝑧) ∈ 𝑆))) |
| 44 | 9, 42, 43 | sylanbrc 583 |
. 2
⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆) → 𝑆 ∈ (NrmSGrp‘𝐺)) |
| 45 | 8, 44 | impbii 209 |
1
⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑆 ((𝑥 + 𝑦) − 𝑥) ∈ 𝑆)) |