Step | Hyp | Ref
| Expression |
1 | | mhmrcl2 18222 |
. . 3
⊢ (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝑈 ∈ Mnd) |
2 | | mhmrcl1 18221 |
. . 3
⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd) |
3 | 1, 2 | anim12ci 617 |
. 2
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd)) |
4 | | eqid 2737 |
. . . . 5
⊢
(Base‘𝑇) =
(Base‘𝑇) |
5 | | eqid 2737 |
. . . . 5
⊢
(Base‘𝑈) =
(Base‘𝑈) |
6 | 4, 5 | mhmf 18223 |
. . . 4
⊢ (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈)) |
7 | | eqid 2737 |
. . . . 5
⊢
(Base‘𝑆) =
(Base‘𝑆) |
8 | 7, 4 | mhmf 18223 |
. . . 4
⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) |
9 | | fco 6569 |
. . . 4
⊢ ((𝐹:(Base‘𝑇)⟶(Base‘𝑈) ∧ 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈)) |
10 | 6, 8, 9 | syl2an 599 |
. . 3
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈)) |
11 | | eqid 2737 |
. . . . . . . . . 10
⊢
(+g‘𝑆) = (+g‘𝑆) |
12 | | eqid 2737 |
. . . . . . . . . 10
⊢
(+g‘𝑇) = (+g‘𝑇) |
13 | 7, 11, 12 | mhmlin 18225 |
. . . . . . . . 9
⊢ ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) |
14 | 13 | 3expb 1122 |
. . . . . . . 8
⊢ ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) |
15 | 14 | adantll 714 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g‘𝑆)𝑦)) = ((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) |
16 | 15 | fveq2d 6721 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))) = (𝐹‘((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦)))) |
17 | | simpll 767 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐹 ∈ (𝑇 MndHom 𝑈)) |
18 | 8 | ad2antlr 727 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) |
19 | | simprl 771 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆)) |
20 | 18, 19 | ffvelrnd 6905 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑥) ∈ (Base‘𝑇)) |
21 | | simprr 773 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆)) |
22 | 18, 21 | ffvelrnd 6905 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘𝑦) ∈ (Base‘𝑇)) |
23 | | eqid 2737 |
. . . . . . . 8
⊢
(+g‘𝑈) = (+g‘𝑈) |
24 | 4, 12, 23 | mhmlin 18225 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ (𝐺‘𝑥) ∈ (Base‘𝑇) ∧ (𝐺‘𝑦) ∈ (Base‘𝑇)) → (𝐹‘((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦)))) |
25 | 17, 20, 22, 24 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘((𝐺‘𝑥)(+g‘𝑇)(𝐺‘𝑦))) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦)))) |
26 | 16, 25 | eqtrd 2777 |
. . . . 5
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦))) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦)))) |
27 | 2 | adantl 485 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd) |
28 | 7, 11 | mndcl 18181 |
. . . . . . . 8
⊢ ((𝑆 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
29 | 28 | 3expb 1122 |
. . . . . . 7
⊢ ((𝑆 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
30 | 27, 29 | sylan 583 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) |
31 | | fvco3 6810 |
. . . . . 6
⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦)))) |
32 | 18, 30, 31 | syl2anc 587 |
. . . . 5
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g‘𝑆)𝑦)))) |
33 | | fvco3 6810 |
. . . . . . 7
⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥))) |
34 | 18, 19, 33 | syl2anc 587 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥))) |
35 | | fvco3 6810 |
. . . . . . 7
⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦))) |
36 | 18, 21, 35 | syl2anc 587 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦))) |
37 | 34, 36 | oveq12d 7231 |
. . . . 5
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)) = ((𝐹‘(𝐺‘𝑥))(+g‘𝑈)(𝐹‘(𝐺‘𝑦)))) |
38 | 26, 32, 37 | 3eqtr4d 2787 |
. . . 4
⊢ (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦))) |
39 | 38 | ralrimivva 3112 |
. . 3
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦))) |
40 | 8 | adantl 485 |
. . . . 5
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) |
41 | | eqid 2737 |
. . . . . . 7
⊢
(0g‘𝑆) = (0g‘𝑆) |
42 | 7, 41 | mndidcl 18188 |
. . . . . 6
⊢ (𝑆 ∈ Mnd →
(0g‘𝑆)
∈ (Base‘𝑆)) |
43 | 27, 42 | syl 17 |
. . . . 5
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑆) ∈ (Base‘𝑆)) |
44 | | fvco3 6810 |
. . . . 5
⊢ ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (0g‘𝑆) ∈ (Base‘𝑆)) → ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (𝐹‘(𝐺‘(0g‘𝑆)))) |
45 | 40, 43, 44 | syl2anc 587 |
. . . 4
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (𝐹‘(𝐺‘(0g‘𝑆)))) |
46 | | eqid 2737 |
. . . . . . 7
⊢
(0g‘𝑇) = (0g‘𝑇) |
47 | 41, 46 | mhm0 18226 |
. . . . . 6
⊢ (𝐺 ∈ (𝑆 MndHom 𝑇) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇)) |
48 | 47 | adantl 485 |
. . . . 5
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐺‘(0g‘𝑆)) = (0g‘𝑇)) |
49 | 48 | fveq2d 6721 |
. . . 4
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(𝐺‘(0g‘𝑆))) = (𝐹‘(0g‘𝑇))) |
50 | | eqid 2737 |
. . . . . 6
⊢
(0g‘𝑈) = (0g‘𝑈) |
51 | 46, 50 | mhm0 18226 |
. . . . 5
⊢ (𝐹 ∈ (𝑇 MndHom 𝑈) → (𝐹‘(0g‘𝑇)) = (0g‘𝑈)) |
52 | 51 | adantr 484 |
. . . 4
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑇)) = (0g‘𝑈)) |
53 | 45, 49, 52 | 3eqtrd 2781 |
. . 3
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (0g‘𝑈)) |
54 | 10, 39, 53 | 3jca 1130 |
. 2
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)) ∧ ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (0g‘𝑈))) |
55 | 7, 5, 11, 23, 41, 50 | ismhm 18220 |
. 2
⊢ ((𝐹 ∘ 𝐺) ∈ (𝑆 MndHom 𝑈) ↔ ((𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd) ∧ ((𝐹 ∘ 𝐺):(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹 ∘ 𝐺)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥)(+g‘𝑈)((𝐹 ∘ 𝐺)‘𝑦)) ∧ ((𝐹 ∘ 𝐺)‘(0g‘𝑆)) = (0g‘𝑈)))) |
56 | 3, 54, 55 | sylanbrc 586 |
1
⊢ ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 MndHom 𝑈)) |