Step | Hyp | Ref
| Expression |
1 | | simpl1 1190 |
. . 3
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
2 | | simpl2 1191 |
. . 3
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝐴 ∈ 𝑋) |
3 | | rhmdvdsr.x |
. . . . 5
⊢ 𝑋 = (Base‘𝑅) |
4 | | eqid 2740 |
. . . . 5
⊢
(Base‘𝑆) =
(Base‘𝑆) |
5 | 3, 4 | rhmf 19966 |
. . . 4
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝑋⟶(Base‘𝑆)) |
6 | 5 | ffvelrnda 6956 |
. . 3
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ (Base‘𝑆)) |
7 | 1, 2, 6 | syl2anc 584 |
. 2
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐹‘𝐴) ∈ (Base‘𝑆)) |
8 | | simpll1 1211 |
. . . . . 6
⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
9 | | simpr 485 |
. . . . . 6
⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → 𝑐 ∈ 𝑋) |
10 | 5 | ffvelrnda 6956 |
. . . . . 6
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑐 ∈ 𝑋) → (𝐹‘𝑐) ∈ (Base‘𝑆)) |
11 | 8, 9, 10 | syl2anc 584 |
. . . . 5
⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → (𝐹‘𝑐) ∈ (Base‘𝑆)) |
12 | 11 | ralrimiva 3110 |
. . . 4
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∀𝑐 ∈ 𝑋 (𝐹‘𝑐) ∈ (Base‘𝑆)) |
13 | 2 | adantr 481 |
. . . . . . 7
⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → 𝐴 ∈ 𝑋) |
14 | | eqid 2740 |
. . . . . . . 8
⊢
(.r‘𝑅) = (.r‘𝑅) |
15 | | eqid 2740 |
. . . . . . . 8
⊢
(.r‘𝑆) = (.r‘𝑆) |
16 | 3, 14, 15 | rhmmul 19967 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑐 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴))) |
17 | 8, 9, 13, 16 | syl3anc 1370 |
. . . . . 6
⊢ ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) ∧ 𝑐 ∈ 𝑋) → (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴))) |
18 | 17 | ralrimiva 3110 |
. . . . 5
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∀𝑐 ∈ 𝑋 (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴))) |
19 | | simpr 485 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → 𝐴 ∥ 𝐵) |
20 | | rhmdvdsr.m |
. . . . . . . 8
⊢ ∥ =
(∥r‘𝑅) |
21 | 3, 20, 14 | dvdsr2 19885 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑋 → (𝐴 ∥ 𝐵 ↔ ∃𝑐 ∈ 𝑋 (𝑐(.r‘𝑅)𝐴) = 𝐵)) |
22 | 21 | biimpac 479 |
. . . . . 6
⊢ ((𝐴 ∥ 𝐵 ∧ 𝐴 ∈ 𝑋) → ∃𝑐 ∈ 𝑋 (𝑐(.r‘𝑅)𝐴) = 𝐵) |
23 | 19, 2, 22 | syl2anc 584 |
. . . . 5
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∃𝑐 ∈ 𝑋 (𝑐(.r‘𝑅)𝐴) = 𝐵) |
24 | | r19.29 3186 |
. . . . . 6
⊢
((∀𝑐 ∈
𝑋 (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ ∃𝑐 ∈ 𝑋 (𝑐(.r‘𝑅)𝐴) = 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵)) |
25 | | simpl 483 |
. . . . . . . 8
⊢ (((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴))) |
26 | | simpr 485 |
. . . . . . . . 9
⊢ (((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → (𝑐(.r‘𝑅)𝐴) = 𝐵) |
27 | 26 | fveq2d 6773 |
. . . . . . . 8
⊢ (((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → (𝐹‘(𝑐(.r‘𝑅)𝐴)) = (𝐹‘𝐵)) |
28 | 25, 27 | eqtr3d 2782 |
. . . . . . 7
⊢ (((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) |
29 | 28 | reximi 3177 |
. . . . . 6
⊢
(∃𝑐 ∈
𝑋 ((𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ (𝑐(.r‘𝑅)𝐴) = 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) |
30 | 24, 29 | syl 17 |
. . . . 5
⊢
((∀𝑐 ∈
𝑋 (𝐹‘(𝑐(.r‘𝑅)𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) ∧ ∃𝑐 ∈ 𝑋 (𝑐(.r‘𝑅)𝐴) = 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) |
31 | 18, 23, 30 | syl2anc 584 |
. . . 4
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) |
32 | | r19.29 3186 |
. . . 4
⊢
((∀𝑐 ∈
𝑋 (𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))) |
33 | 12, 31, 32 | syl2anc 584 |
. . 3
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∃𝑐 ∈ 𝑋 ((𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))) |
34 | | oveq1 7276 |
. . . . . 6
⊢ (𝑦 = (𝐹‘𝑐) → (𝑦(.r‘𝑆)(𝐹‘𝐴)) = ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴))) |
35 | 34 | eqeq1d 2742 |
. . . . 5
⊢ (𝑦 = (𝐹‘𝑐) → ((𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵) ↔ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))) |
36 | 35 | rspcev 3561 |
. . . 4
⊢ (((𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) → ∃𝑦 ∈ (Base‘𝑆)(𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) |
37 | 36 | rexlimivw 3213 |
. . 3
⊢
(∃𝑐 ∈
𝑋 ((𝐹‘𝑐) ∈ (Base‘𝑆) ∧ ((𝐹‘𝑐)(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) → ∃𝑦 ∈ (Base‘𝑆)(𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) |
38 | 33, 37 | syl 17 |
. 2
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → ∃𝑦 ∈ (Base‘𝑆)(𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵)) |
39 | | rhmdvdsr.n |
. . 3
⊢ / =
(∥r‘𝑆) |
40 | 4, 39, 15 | dvdsr 19884 |
. 2
⊢ ((𝐹‘𝐴) / (𝐹‘𝐵) ↔ ((𝐹‘𝐴) ∈ (Base‘𝑆) ∧ ∃𝑦 ∈ (Base‘𝑆)(𝑦(.r‘𝑆)(𝐹‘𝐴)) = (𝐹‘𝐵))) |
41 | 7, 38, 40 | sylanbrc 583 |
1
⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐹‘𝐴) / (𝐹‘𝐵)) |