| Step | Hyp | Ref
| Expression |
| 1 | | elex 3501 |
. . 3
⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) |
| 2 | 1 | anim2i 617 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (𝐺 ∈ Grp ∧ 𝑆 ∈ V)) |
| 3 | | gaid.1 |
. . . . . . . 8
⊢ 𝑋 = (Base‘𝐺) |
| 4 | | eqid 2737 |
. . . . . . . 8
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 5 | 3, 4 | grpidcl 18983 |
. . . . . . 7
⊢ (𝐺 ∈ Grp →
(0g‘𝐺)
∈ 𝑋) |
| 6 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (0g‘𝐺) ∈ 𝑋) |
| 7 | | ovres 7599 |
. . . . . . 7
⊢
(((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((0g‘𝐺)2nd 𝑥)) |
| 8 | | df-ov 7434 |
. . . . . . . 8
⊢
((0g‘𝐺)2nd 𝑥) = (2nd
‘〈(0g‘𝐺), 𝑥〉) |
| 9 | | fvex 6919 |
. . . . . . . . 9
⊢
(0g‘𝐺) ∈ V |
| 10 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
| 11 | 9, 10 | op2nd 8023 |
. . . . . . . 8
⊢
(2nd ‘〈(0g‘𝐺), 𝑥〉) = 𝑥 |
| 12 | 8, 11 | eqtri 2765 |
. . . . . . 7
⊢
((0g‘𝐺)2nd 𝑥) = 𝑥 |
| 13 | 7, 12 | eqtrdi 2793 |
. . . . . 6
⊢
(((0g‘𝐺) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
| 14 | 6, 13 | sylan 580 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
| 15 | | simprl 771 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋) |
| 16 | | simplr 769 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑥 ∈ 𝑆) |
| 17 | | ovres 7599 |
. . . . . . . . 9
⊢ ((𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦2nd 𝑥)) |
| 18 | | df-ov 7434 |
. . . . . . . . . 10
⊢ (𝑦2nd 𝑥) = (2nd
‘〈𝑦, 𝑥〉) |
| 19 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
| 20 | 19, 10 | op2nd 8023 |
. . . . . . . . . 10
⊢
(2nd ‘〈𝑦, 𝑥〉) = 𝑥 |
| 21 | 18, 20 | eqtri 2765 |
. . . . . . . . 9
⊢ (𝑦2nd 𝑥) = 𝑥 |
| 22 | 17, 21 | eqtrdi 2793 |
. . . . . . . 8
⊢ ((𝑦 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
| 23 | 15, 16, 22 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
| 24 | | simprr 773 |
. . . . . . . . 9
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋) |
| 25 | | ovres 7599 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑧2nd 𝑥)) |
| 26 | | df-ov 7434 |
. . . . . . . . . . 11
⊢ (𝑧2nd 𝑥) = (2nd
‘〈𝑧, 𝑥〉) |
| 27 | | vex 3484 |
. . . . . . . . . . . 12
⊢ 𝑧 ∈ V |
| 28 | 27, 10 | op2nd 8023 |
. . . . . . . . . . 11
⊢
(2nd ‘〈𝑧, 𝑥〉) = 𝑥 |
| 29 | 26, 28 | eqtri 2765 |
. . . . . . . . . 10
⊢ (𝑧2nd 𝑥) = 𝑥 |
| 30 | 25, 29 | eqtrdi 2793 |
. . . . . . . . 9
⊢ ((𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
| 31 | 24, 16, 30 | syl2anc 584 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑧(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
| 32 | 31 | oveq2d 7447 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)) = (𝑦(2nd ↾ (𝑋 × 𝑆))𝑥)) |
| 33 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 34 | 3, 33 | grpcl 18959 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋) |
| 35 | 34 | 3expb 1121 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋) |
| 36 | 35 | ad4ant14 752 |
. . . . . . . 8
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋) |
| 37 | | ovres 7599 |
. . . . . . . . 9
⊢ (((𝑦(+g‘𝐺)𝑧) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = ((𝑦(+g‘𝐺)𝑧)2nd 𝑥)) |
| 38 | | df-ov 7434 |
. . . . . . . . . 10
⊢ ((𝑦(+g‘𝐺)𝑧)2nd 𝑥) = (2nd ‘〈(𝑦(+g‘𝐺)𝑧), 𝑥〉) |
| 39 | | ovex 7464 |
. . . . . . . . . . 11
⊢ (𝑦(+g‘𝐺)𝑧) ∈ V |
| 40 | 39, 10 | op2nd 8023 |
. . . . . . . . . 10
⊢
(2nd ‘〈(𝑦(+g‘𝐺)𝑧), 𝑥〉) = 𝑥 |
| 41 | 38, 40 | eqtri 2765 |
. . . . . . . . 9
⊢ ((𝑦(+g‘𝐺)𝑧)2nd 𝑥) = 𝑥 |
| 42 | 37, 41 | eqtrdi 2793 |
. . . . . . . 8
⊢ (((𝑦(+g‘𝐺)𝑧) ∈ 𝑋 ∧ 𝑥 ∈ 𝑆) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
| 43 | 36, 16, 42 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥) |
| 44 | 23, 32, 43 | 3eqtr4rd 2788 |
. . . . . 6
⊢ ((((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))) |
| 45 | 44 | ralrimivva 3202 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))) |
| 46 | 14, 45 | jca 511 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) ∧ 𝑥 ∈ 𝑆) → (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))) |
| 47 | 46 | ralrimiva 3146 |
. . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → ∀𝑥 ∈ 𝑆 (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))) |
| 48 | | f2ndres 8039 |
. . 3
⊢
(2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 |
| 49 | 47, 48 | jctil 519 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → ((2nd ↾ (𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑥 ∈ 𝑆 (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥))))) |
| 50 | 3, 33, 4 | isga 19309 |
. 2
⊢
((2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆) ↔ ((𝐺 ∈ Grp ∧ 𝑆 ∈ V) ∧ ((2nd ↾
(𝑋 × 𝑆)):(𝑋 × 𝑆)⟶𝑆 ∧ ∀𝑥 ∈ 𝑆 (((0g‘𝐺)(2nd ↾ (𝑋 × 𝑆))𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦(+g‘𝐺)𝑧)(2nd ↾ (𝑋 × 𝑆))𝑥) = (𝑦(2nd ↾ (𝑋 × 𝑆))(𝑧(2nd ↾ (𝑋 × 𝑆))𝑥)))))) |
| 51 | 2, 49, 50 | sylanbrc 583 |
1
⊢ ((𝐺 ∈ Grp ∧ 𝑆 ∈ 𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆)) |