Step | Hyp | Ref
| Expression |
1 | | rnghommul.1 |
. . . . . . 7
⊢ 𝐺 = (1st ‘𝑅) |
2 | | rnghommul.3 |
. . . . . . 7
⊢ 𝐻 = (2nd ‘𝑅) |
3 | | rnghommul.2 |
. . . . . . 7
⊢ 𝑋 = ran 𝐺 |
4 | | eqid 2738 |
. . . . . . 7
⊢
(GId‘𝐻) =
(GId‘𝐻) |
5 | | eqid 2738 |
. . . . . . 7
⊢
(1st ‘𝑆) = (1st ‘𝑆) |
6 | | rnghommul.4 |
. . . . . . 7
⊢ 𝐾 = (2nd ‘𝑆) |
7 | | eqid 2738 |
. . . . . . 7
⊢ ran
(1st ‘𝑆) =
ran (1st ‘𝑆) |
8 | | eqid 2738 |
. . . . . . 7
⊢
(GId‘𝐾) =
(GId‘𝐾) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | isrngohom 36123 |
. . . . . 6
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋⟶ran (1st ‘𝑆) ∧ (𝐹‘(GId‘𝐻)) = (GId‘𝐾) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))) |
10 | 9 | biimpa 477 |
. . . . 5
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋⟶ran (1st ‘𝑆) ∧ (𝐹‘(GId‘𝐻)) = (GId‘𝐾) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))) |
11 | 10 | simp3d 1143 |
. . . 4
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))) |
12 | 11 | 3impa 1109 |
. . 3
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))) |
13 | | simpr 485 |
. . . 4
⊢ (((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) → (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) |
14 | 13 | 2ralimi 3088 |
. . 3
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) |
15 | 12, 14 | syl 17 |
. 2
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) |
16 | | fvoveq1 7298 |
. . . 4
⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥𝐻𝑦)) = (𝐹‘(𝐴𝐻𝑦))) |
17 | | fveq2 6774 |
. . . . 5
⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) |
18 | 17 | oveq1d 7290 |
. . . 4
⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦))) |
19 | 16, 18 | eqeq12d 2754 |
. . 3
⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)))) |
20 | | oveq2 7283 |
. . . . 5
⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵)) |
21 | 20 | fveq2d 6778 |
. . . 4
⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴𝐻𝑦)) = (𝐹‘(𝐴𝐻𝐵))) |
22 | | fveq2 6774 |
. . . . 5
⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) |
23 | 22 | oveq2d 7291 |
. . . 4
⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))) |
24 | 21, 23 | eqeq12d 2754 |
. . 3
⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))) |
25 | 19, 24 | rspc2v 3570 |
. 2
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))) |
26 | 15, 25 | mpan9 507 |
1
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))) |