Step | Hyp | Ref
| Expression |
1 | | df-ghm 17481 |
. . 3
⊢ GrpHom =
(𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑓 ∣
[(Base‘𝑠) /
𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))}) |
2 | 1 | elmpt2cl 6774 |
. 2
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp)) |
3 | | fvex 6113 |
. . . . . . . 8
⊢
(Base‘𝑠)
∈ V |
4 | | feq2 5940 |
. . . . . . . . 9
⊢ (𝑤 = (Base‘𝑠) → (𝑓:𝑤⟶(Base‘𝑡) ↔ 𝑓:(Base‘𝑠)⟶(Base‘𝑡))) |
5 | | raleq 3115 |
. . . . . . . . . 10
⊢ (𝑤 = (Base‘𝑠) → (∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))) |
6 | 5 | raleqbi1dv 3123 |
. . . . . . . . 9
⊢ (𝑤 = (Base‘𝑠) → (∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))) |
7 | 4, 6 | anbi12d 743 |
. . . . . . . 8
⊢ (𝑤 = (Base‘𝑠) → ((𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))) |
8 | 3, 7 | sbcie 3437 |
. . . . . . 7
⊢
([(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))) |
9 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆)) |
10 | | isghm.w |
. . . . . . . . . 10
⊢ 𝑋 = (Base‘𝑆) |
11 | 9, 10 | syl6eqr 2662 |
. . . . . . . . 9
⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑋) |
12 | 11 | feq2d 5944 |
. . . . . . . 8
⊢ (𝑠 = 𝑆 → (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ↔ 𝑓:𝑋⟶(Base‘𝑡))) |
13 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆)) |
14 | | isghm.a |
. . . . . . . . . . . . . 14
⊢ + =
(+g‘𝑆) |
15 | 13, 14 | syl6eqr 2662 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑆 → (+g‘𝑠) = + ) |
16 | 15 | oveqd 6566 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑆 → (𝑢(+g‘𝑠)𝑣) = (𝑢 + 𝑣)) |
17 | 16 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑆 → (𝑓‘(𝑢(+g‘𝑠)𝑣)) = (𝑓‘(𝑢 + 𝑣))) |
18 | 17 | eqeq1d 2612 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → ((𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))) |
19 | 11, 18 | raleqbidv 3129 |
. . . . . . . . 9
⊢ (𝑠 = 𝑆 → (∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))) |
20 | 11, 19 | raleqbidv 3129 |
. . . . . . . 8
⊢ (𝑠 = 𝑆 → (∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))) |
21 | 12, 20 | anbi12d 743 |
. . . . . . 7
⊢ (𝑠 = 𝑆 → ((𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))) |
22 | 8, 21 | syl5bb 271 |
. . . . . 6
⊢ (𝑠 = 𝑆 → ([(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))) |
23 | 22 | abbidv 2728 |
. . . . 5
⊢ (𝑠 = 𝑆 → {𝑓 ∣ [(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))} = {𝑓 ∣ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))}) |
24 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇)) |
25 | | isghm.x |
. . . . . . . . 9
⊢ 𝑌 = (Base‘𝑇) |
26 | 24, 25 | syl6eqr 2662 |
. . . . . . . 8
⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝑌) |
27 | 26 | feq3d 5945 |
. . . . . . 7
⊢ (𝑡 = 𝑇 → (𝑓:𝑋⟶(Base‘𝑡) ↔ 𝑓:𝑋⟶𝑌)) |
28 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇)) |
29 | | isghm.b |
. . . . . . . . . . 11
⊢ ⨣ =
(+g‘𝑇) |
30 | 28, 29 | syl6eqr 2662 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑇 → (+g‘𝑡) = ⨣ ) |
31 | 30 | oveqd 6566 |
. . . . . . . . 9
⊢ (𝑡 = 𝑇 → ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))) |
32 | 31 | eqeq2d 2620 |
. . . . . . . 8
⊢ (𝑡 = 𝑇 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))) |
33 | 32 | 2ralbidv 2972 |
. . . . . . 7
⊢ (𝑡 = 𝑇 → (∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))) |
34 | 27, 33 | anbi12d 743 |
. . . . . 6
⊢ (𝑡 = 𝑇 → ((𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))))) |
35 | 34 | abbidv 2728 |
. . . . 5
⊢ (𝑡 = 𝑇 → {𝑓 ∣ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))} = {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))}) |
36 | | fvex 6113 |
. . . . . . . 8
⊢
(Base‘𝑆)
∈ V |
37 | 10, 36 | eqeltri 2684 |
. . . . . . 7
⊢ 𝑋 ∈ V |
38 | | fvex 6113 |
. . . . . . . 8
⊢
(Base‘𝑇)
∈ V |
39 | 25, 38 | eqeltri 2684 |
. . . . . . 7
⊢ 𝑌 ∈ V |
40 | | mapex 7750 |
. . . . . . 7
⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑓 ∣ 𝑓:𝑋⟶𝑌} ∈ V) |
41 | 37, 39, 40 | mp2an 704 |
. . . . . 6
⊢ {𝑓 ∣ 𝑓:𝑋⟶𝑌} ∈ V |
42 | | simpl 472 |
. . . . . . 7
⊢ ((𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))) → 𝑓:𝑋⟶𝑌) |
43 | 42 | ss2abi 3637 |
. . . . . 6
⊢ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ⊆ {𝑓 ∣ 𝑓:𝑋⟶𝑌} |
44 | 41, 43 | ssexi 4731 |
. . . . 5
⊢ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ∈ V |
45 | 23, 35, 1, 44 | ovmpt2 6694 |
. . . 4
⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))}) |
46 | 45 | eleq2d 2673 |
. . 3
⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))})) |
47 | | fex 6394 |
. . . . . 6
⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ∈ V) → 𝐹 ∈ V) |
48 | 37, 47 | mpan2 703 |
. . . . 5
⊢ (𝐹:𝑋⟶𝑌 → 𝐹 ∈ V) |
49 | 48 | adantr 480 |
. . . 4
⊢ ((𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))) → 𝐹 ∈ V) |
50 | | feq1 5939 |
. . . . 5
⊢ (𝑓 = 𝐹 → (𝑓:𝑋⟶𝑌 ↔ 𝐹:𝑋⟶𝑌)) |
51 | | fveq1 6102 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → (𝑓‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝑣))) |
52 | | fveq1 6102 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → (𝑓‘𝑢) = (𝐹‘𝑢)) |
53 | | fveq1 6102 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → (𝑓‘𝑣) = (𝐹‘𝑣)) |
54 | 52, 53 | oveq12d 6567 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))) |
55 | 51, 54 | eqeq12d 2625 |
. . . . . 6
⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) ↔ (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))) |
56 | 55 | 2ralbidv 2972 |
. . . . 5
⊢ (𝑓 = 𝐹 → (∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))) |
57 | 50, 56 | anbi12d 743 |
. . . 4
⊢ (𝑓 = 𝐹 → ((𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))) |
58 | 49, 57 | elab3 3327 |
. . 3
⊢ (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))) |
59 | 46, 58 | syl6bb 275 |
. 2
⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))) |
60 | 2, 59 | biadan2 672 |
1
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))) |