Step | Hyp | Ref
| Expression |
1 | | mgmhmpropd.e |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
2 | 1 | fveq2d 6742 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦))) |
3 | 2 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓‘(𝑥(+g‘𝐽)𝑦)) = (𝑓‘(𝑥(+g‘𝐿)𝑦))) |
4 | | ffvelrn 6923 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑥 ∈ 𝐵) → (𝑓‘𝑥) ∈ 𝐶) |
5 | | ffvelrn 6923 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:𝐵⟶𝐶 ∧ 𝑦 ∈ 𝐵) → (𝑓‘𝑦) ∈ 𝐶) |
6 | 4, 5 | anim12dan 622 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:𝐵⟶𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘𝑥) ∈ 𝐶 ∧ (𝑓‘𝑦) ∈ 𝐶)) |
7 | | mgmhmpropd.f |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) |
8 | 7 | ralrimivva 3114 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) |
9 | | oveq1 7241 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑦)) |
10 | | oveq1 7241 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑤 → (𝑥(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑦)) |
11 | 9, 10 | eqeq12d 2755 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑤 → ((𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦))) |
12 | | oveq2 7242 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝐾)𝑧)) |
13 | | oveq2 7242 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑧 → (𝑤(+g‘𝑀)𝑦) = (𝑤(+g‘𝑀)𝑧)) |
14 | 12, 13 | eqeq12d 2755 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑧 → ((𝑤(+g‘𝐾)𝑦) = (𝑤(+g‘𝑀)𝑦) ↔ (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧))) |
15 | 11, 14 | cbvral2vw 3385 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑥 ∈
𝐶 ∀𝑦 ∈ 𝐶 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦) ↔ ∀𝑤 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧)) |
16 | 8, 15 | sylib 221 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑤 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧)) |
17 | | oveq1 7241 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)𝑧)) |
18 | | oveq1 7241 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = (𝑓‘𝑥) → (𝑤(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧)) |
19 | 17, 18 | eqeq12d 2755 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = (𝑓‘𝑥) → ((𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧) ↔ ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧))) |
20 | | oveq2 7242 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))) |
21 | | oveq2 7242 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (𝑓‘𝑦) → ((𝑓‘𝑥)(+g‘𝑀)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))) |
22 | 20, 21 | eqeq12d 2755 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (𝑓‘𝑦) → (((𝑓‘𝑥)(+g‘𝐾)𝑧) = ((𝑓‘𝑥)(+g‘𝑀)𝑧) ↔ ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
23 | 19, 22 | rspc2va 3562 |
. . . . . . . . . . . . . 14
⊢ ((((𝑓‘𝑥) ∈ 𝐶 ∧ (𝑓‘𝑦) ∈ 𝐶) ∧ ∀𝑤 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑤(+g‘𝐾)𝑧) = (𝑤(+g‘𝑀)𝑧)) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))) |
24 | 6, 16, 23 | syl2anr 600 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓:𝐵⟶𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))) |
25 | 24 | anassrs 471 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))) |
26 | 3, 25 | eqeq12d 2755 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓:𝐵⟶𝐶) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
27 | 26 | 2ralbidva 3121 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓:𝐵⟶𝐶) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
28 | 27 | adantrl 716 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵⟶𝐶)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
29 | | mgmhmpropd.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 = (Base‘𝐽)) |
30 | | raleq 3333 |
. . . . . . . . . . . 12
⊢ (𝐵 = (Base‘𝐽) → (∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))) |
31 | 30 | raleqbi1dv 3331 |
. . . . . . . . . . 11
⊢ (𝐵 = (Base‘𝐽) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))) |
32 | 29, 31 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))) |
33 | 32 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵⟶𝐶)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))) |
34 | | mgmhmpropd.c |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
35 | | raleq 3333 |
. . . . . . . . . . . 12
⊢ (𝐵 = (Base‘𝐿) → (∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
36 | 35 | raleqbi1dv 3331 |
. . . . . . . . . . 11
⊢ (𝐵 = (Base‘𝐿) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
37 | 34, 36 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
38 | 37 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵⟶𝐶)) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
39 | 28, 33, 38 | 3bitr3d 312 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵⟶𝐶)) → (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
40 | 39 | anassrs 471 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) ∧ 𝑓:𝐵⟶𝐶) → (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) |
41 | 40 | pm5.32da 582 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:𝐵⟶𝐶 ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))) ↔ (𝑓:𝐵⟶𝐶 ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))) |
42 | | mgmhmpropd.b |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 = (Base‘𝐾)) |
43 | 29, 42 | feq23d 6561 |
. . . . . . . 8
⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾))) |
44 | 43 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐽)⟶(Base‘𝐾))) |
45 | 44 | anbi1d 633 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:𝐵⟶𝐶 ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))) ↔ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))))) |
46 | | mgmhmpropd.d |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 = (Base‘𝑀)) |
47 | 34, 46 | feq23d 6561 |
. . . . . . . 8
⊢ (𝜑 → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀))) |
48 | 47 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → (𝑓:𝐵⟶𝐶 ↔ 𝑓:(Base‘𝐿)⟶(Base‘𝑀))) |
49 | 48 | anbi1d 633 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:𝐵⟶𝐶 ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))) |
50 | 41, 45, 49 | 3bitr3d 312 |
. . . . 5
⊢ ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))) |
51 | 50 | pm5.32da 582 |
. . . 4
⊢ (𝜑 → (((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))) ↔ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))))) |
52 | | mgmhmpropd.0 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ≠ ∅) |
53 | 29, 34, 52, 1 | mgmpropd 45035 |
. . . . . 6
⊢ (𝜑 → (𝐽 ∈ Mgm ↔ 𝐿 ∈ Mgm)) |
54 | | mgmhmpropd.C |
. . . . . . 7
⊢ (𝜑 → 𝐶 ≠ ∅) |
55 | 42, 46, 54, 7 | mgmpropd 45035 |
. . . . . 6
⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝑀 ∈ Mgm)) |
56 | 53, 55 | anbi12d 634 |
. . . . 5
⊢ (𝜑 → ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ↔ (𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm))) |
57 | 56 | anbi1d 633 |
. . . 4
⊢ (𝜑 → (((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))) ↔ ((𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))))) |
58 | 51, 57 | bitrd 282 |
. . 3
⊢ (𝜑 → (((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦)))) ↔ ((𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦)))))) |
59 | | eqid 2739 |
. . . 4
⊢
(Base‘𝐽) =
(Base‘𝐽) |
60 | | eqid 2739 |
. . . 4
⊢
(Base‘𝐾) =
(Base‘𝐾) |
61 | | eqid 2739 |
. . . 4
⊢
(+g‘𝐽) = (+g‘𝐽) |
62 | | eqid 2739 |
. . . 4
⊢
(+g‘𝐾) = (+g‘𝐾) |
63 | 59, 60, 61, 62 | ismgmhm 45043 |
. . 3
⊢ (𝑓 ∈ (𝐽 MgmHom 𝐾) ↔ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g‘𝐽)𝑦)) = ((𝑓‘𝑥)(+g‘𝐾)(𝑓‘𝑦))))) |
64 | | eqid 2739 |
. . . 4
⊢
(Base‘𝐿) =
(Base‘𝐿) |
65 | | eqid 2739 |
. . . 4
⊢
(Base‘𝑀) =
(Base‘𝑀) |
66 | | eqid 2739 |
. . . 4
⊢
(+g‘𝐿) = (+g‘𝐿) |
67 | | eqid 2739 |
. . . 4
⊢
(+g‘𝑀) = (+g‘𝑀) |
68 | 64, 65, 66, 67 | ismgmhm 45043 |
. . 3
⊢ (𝑓 ∈ (𝐿 MgmHom 𝑀) ↔ ((𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g‘𝐿)𝑦)) = ((𝑓‘𝑥)(+g‘𝑀)(𝑓‘𝑦))))) |
69 | 58, 63, 68 | 3bitr4g 317 |
. 2
⊢ (𝜑 → (𝑓 ∈ (𝐽 MgmHom 𝐾) ↔ 𝑓 ∈ (𝐿 MgmHom 𝑀))) |
70 | 69 | eqrdv 2737 |
1
⊢ (𝜑 → (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀)) |