Step | Hyp | Ref
| Expression |
1 | | rnghmrcl 45447 |
. 2
⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng)) |
2 | | isrnghm.b |
. . . . 5
⊢ 𝐵 = (Base‘𝑅) |
3 | | isrnghm.t |
. . . . 5
⊢ · =
(.r‘𝑅) |
4 | | isrnghm.m |
. . . . 5
⊢ ∗ =
(.r‘𝑆) |
5 | | eqid 2738 |
. . . . 5
⊢
(Base‘𝑆) =
(Base‘𝑆) |
6 | | eqid 2738 |
. . . . 5
⊢
(+g‘𝑅) = (+g‘𝑅) |
7 | | eqid 2738 |
. . . . 5
⊢
(+g‘𝑆) = (+g‘𝑆) |
8 | 2, 3, 4, 5, 6, 7 | rnghmval 45449 |
. . . 4
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 𝑆) = {𝑓 ∈ ((Base‘𝑆) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))}) |
9 | 8 | eleq2d 2824 |
. . 3
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ 𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))})) |
10 | | fveq1 6773 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘(𝑥(+g‘𝑅)𝑦))) |
11 | | fveq1 6773 |
. . . . . . . . 9
⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) |
12 | | fveq1 6773 |
. . . . . . . . 9
⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦)) |
13 | 11, 12 | oveq12d 7293 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) |
14 | 10, 13 | eqeq12d 2754 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ↔ (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))) |
15 | | fveq1 6773 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦))) |
16 | 11, 12 | oveq12d 7293 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ∗ (𝑓‘𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) |
17 | 15, 16 | eqeq12d 2754 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))) |
18 | 14, 17 | anbi12d 631 |
. . . . . 6
⊢ (𝑓 = 𝐹 → (((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦))) ↔ ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |
19 | 18 | 2ralbidv 3129 |
. . . . 5
⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |
20 | 19 | elrab 3624 |
. . . 4
⊢ (𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))} ↔ (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |
21 | | r19.26-2 3096 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))) |
22 | 21 | anbi2i 623 |
. . . . . 6
⊢ ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))) ↔ (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |
23 | | anass 469 |
. . . . . 6
⊢ (((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) ↔ (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |
24 | 22, 23 | bitr4i 277 |
. . . . 5
⊢ ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))) ↔ ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))) |
25 | 2, 5, 6, 7 | isghm 18834 |
. . . . . . 7
⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))))) |
26 | | fvex 6787 |
. . . . . . . . . . 11
⊢
(Base‘𝑆)
∈ V |
27 | 2 | fvexi 6788 |
. . . . . . . . . . 11
⊢ 𝐵 ∈ V |
28 | 26, 27 | pm3.2i 471 |
. . . . . . . . . 10
⊢
((Base‘𝑆)
∈ V ∧ 𝐵 ∈
V) |
29 | | elmapg 8628 |
. . . . . . . . . 10
⊢
(((Base‘𝑆)
∈ V ∧ 𝐵 ∈ V)
→ (𝐹 ∈
((Base‘𝑆)
↑m 𝐵)
↔ 𝐹:𝐵⟶(Base‘𝑆))) |
30 | 28, 29 | mp1i 13 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ↔ 𝐹:𝐵⟶(Base‘𝑆))) |
31 | 30 | anbi1d 630 |
. . . . . . . 8
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ↔ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))))) |
32 | | rngabl 45435 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) |
33 | | ablgrp 19391 |
. . . . . . . . . 10
⊢ (𝑅 ∈ Abel → 𝑅 ∈ Grp) |
34 | 32, 33 | syl 17 |
. . . . . . . . 9
⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) |
35 | | rngabl 45435 |
. . . . . . . . . 10
⊢ (𝑆 ∈ Rng → 𝑆 ∈ Abel) |
36 | | ablgrp 19391 |
. . . . . . . . . 10
⊢ (𝑆 ∈ Abel → 𝑆 ∈ Grp) |
37 | 35, 36 | syl 17 |
. . . . . . . . 9
⊢ (𝑆 ∈ Rng → 𝑆 ∈ Grp) |
38 | | ibar 529 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → ((𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))) |
39 | 34, 37, 38 | syl2an 596 |
. . . . . . . 8
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ↔ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))))) |
40 | 31, 39 | bitr2d 279 |
. . . . . . 7
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) ∧ (𝐹:𝐵⟶(Base‘𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)))) ↔ (𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))))) |
41 | 25, 40 | bitr2id 284 |
. . . . . 6
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ↔ 𝐹 ∈ (𝑅 GrpHom 𝑆))) |
42 | 41 | anbi1d 630 |
. . . . 5
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦))) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |
43 | 24, 42 | syl5bb 283 |
. . . 4
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ ((Base‘𝑆) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(+g‘𝑅)𝑦)) = ((𝐹‘𝑥)(+g‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |
44 | 20, 43 | syl5bb 283 |
. . 3
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ {𝑓 ∈ ((Base‘𝑆) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥(+g‘𝑅)𝑦)) = ((𝑓‘𝑥)(+g‘𝑆)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))} ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |
45 | 9, 44 | bitrd 278 |
. 2
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |
46 | 1, 45 | biadanii 819 |
1
⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))))) |