Step | Hyp | Ref
| Expression |
1 | | eqid 2171 |
. . . . . 6
⊢
(Base‘𝑆) =
(Base‘𝑆) |
2 | | eqid 2171 |
. . . . . 6
⊢
(Base‘𝑇) =
(Base‘𝑇) |
3 | 1, 2 | mhmf 12692 |
. . . . 5
⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
4 | 3 | adantr 274 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
5 | 4 | ffnd 5350 |
. . 3
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆)) |
6 | 1, 2 | mhmf 12692 |
. . . . 5
⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) |
7 | 6 | adantl 275 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) |
8 | 7 | ffnd 5350 |
. . 3
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺 Fn (Base‘𝑆)) |
9 | | fndmin 5607 |
. . 3
⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}) |
10 | 5, 8, 9 | syl2anc 409 |
. 2
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}) |
11 | | ssrab2 3233 |
. . . 4
⊢ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆) |
12 | 11 | a1i 9 |
. . 3
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆)) |
13 | | fveq2 5499 |
. . . . 5
⊢ (𝑧 = (0g‘𝑆) → (𝐹‘𝑧) = (𝐹‘(0g‘𝑆))) |
14 | | fveq2 5499 |
. . . . 5
⊢ (𝑧 = (0g‘𝑆) → (𝐺‘𝑧) = (𝐺‘(0g‘𝑆))) |
15 | 13, 14 | eqeq12d 2186 |
. . . 4
⊢ (𝑧 = (0g‘𝑆) → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘(0g‘𝑆)) = (𝐺‘(0g‘𝑆)))) |
16 | | mhmrcl1 12690 |
. . . . . 6
⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd) |
17 | 16 | adantr 274 |
. . . . 5
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd) |
18 | | eqid 2171 |
. . . . . 6
⊢
(0g‘𝑆) = (0g‘𝑆) |
19 | 1, 18 | mndidcl 12670 |
. . . . 5
⊢ (𝑆 ∈ Mnd →
(0g‘𝑆)
∈ (Base‘𝑆)) |
20 | 17, 19 | syl 14 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑆) ∈ (Base‘𝑆)) |
21 | | eqid 2171 |
. . . . . . 7
⊢
(0g‘𝑇) = (0g‘𝑇) |
22 | 18, 21 | mhm0 12695 |
. . . . . 6
⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇)) |
23 | 22 | adantr 274 |
. . . . 5
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇)) |
24 | 18, 21 | mhm0 12695 |
. . . . . 6
⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇)) |
25 | 24 | adantl 275 |
. . . . 5
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇)) |
26 | 23, 25 | eqtr4d 2207 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (𝐺‘(0g‘𝑆))) |
27 | 15, 20, 26 | elrabd 2889 |
. . 3
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}) |
28 | | fveq2 5499 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → (𝐹‘𝑧) = (𝐹‘(𝑥(+g‘𝑆)𝑦))) |
29 | | fveq2 5499 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → (𝐺‘𝑧) = (𝐺‘(𝑥(+g‘𝑆)𝑦))) |
30 | 28, 29 | eqeq12d 2186 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑥(+g‘𝑆)𝑦) → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘(𝑥(+g‘𝑆)𝑦)) = (𝐺‘(𝑥(+g‘𝑆)𝑦)))) |
31 | 17 | ad2antrr 486 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑆 ∈ Mnd) |
32 | | simplrl 531 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑥 ∈ (Base‘𝑆)) |
33 | | simprl 527 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝑦 ∈ (Base‘𝑆)) |
34 | | eqid 2171 |
. . . . . . . . . . . 12
⊢
(+g‘𝑆) = (+g‘𝑆) |
35 | 1, 34 | mndcl 12663 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
36 | 31, 32, 33, 35 | syl3anc 1234 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
37 | | simplll 529 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝐹 ∈ (𝑆 MndHom 𝑇)) |
38 | | eqid 2171 |
. . . . . . . . . . . . 13
⊢
(+g‘𝑇) = (+g‘𝑇) |
39 | 1, 34, 38 | mhmlin 12694 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
40 | 37, 32, 33, 39 | syl3anc 1234 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
41 | | simpllr 530 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → 𝐺 ∈ (𝑆 MndHom 𝑇)) |
42 | 1, 34, 38 | mhmlin 12694 |
. . . . . . . . . . . . 13
⊢ ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) |
43 | 41, 32, 33, 42 | syl3anc 1234 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) |
44 | | simplrr 532 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
45 | | simprr 528 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘𝑦) = (𝐺‘𝑦)) |
46 | 44, 45 | oveq12d 5875 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) |
47 | 43, 46 | eqtr4d 2207 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦))) |
48 | 40, 47 | eqtr4d 2207 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = (𝐺‘(𝑥(+g‘𝑆)𝑦))) |
49 | 30, 36, 48 | elrabd 2889 |
. . . . . . . . 9
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹‘𝑦) = (𝐺‘𝑦))) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}) |
50 | 49 | expr 373 |
. . . . . . . 8
⊢ ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})) |
51 | 50 | ralrimiva 2544 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})) |
52 | | fveq2 5499 |
. . . . . . . . 9
⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦)) |
53 | | fveq2 5499 |
. . . . . . . . 9
⊢ (𝑧 = 𝑦 → (𝐺‘𝑧) = (𝐺‘𝑦)) |
54 | 52, 53 | eqeq12d 2186 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑦) = (𝐺‘𝑦))) |
55 | 54 | ralrab 2892 |
. . . . . . 7
⊢
(∀𝑦 ∈
{𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹‘𝑦) = (𝐺‘𝑦) → (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})) |
56 | 51, 55 | sylibr 133 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹‘𝑥) = (𝐺‘𝑥))) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}) |
57 | 56 | expr 373 |
. . . . 5
⊢ (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹‘𝑥) = (𝐺‘𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})) |
58 | 57 | ralrimiva 2544 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})) |
59 | | fveq2 5499 |
. . . . . 6
⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) |
60 | | fveq2 5499 |
. . . . . 6
⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥)) |
61 | 59, 60 | eqeq12d 2186 |
. . . . 5
⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
62 | 61 | ralrab 2892 |
. . . 4
⊢
(∀𝑥 ∈
{𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ↔ ∀𝑥 ∈ (Base‘𝑆)((𝐹‘𝑥) = (𝐺‘𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)})) |
63 | 58, 62 | sylibr 133 |
. . 3
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}) |
64 | 1, 18, 34 | issubm 12699 |
. . . 4
⊢ (𝑆 ∈ Mnd → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMnd‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆) ∧ (0g‘𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))) |
65 | 17, 64 | syl 14 |
. . 3
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMnd‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ⊆ (Base‘𝑆) ∧ (0g‘𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} (𝑥(+g‘𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)}))) |
66 | 12, 27, 63, 65 | mpbir3and 1176 |
. 2
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹‘𝑧) = (𝐺‘𝑧)} ∈ (SubMnd‘𝑆)) |
67 | 10, 66 | eqeltrd 2248 |
1
⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubMnd‘𝑆)) |