Step |
Hyp |
Ref |
Expression |
1 |
|
rhmdvdsr.x |
⊢ 𝑋 = ( Base ‘ 𝑅 ) |
2 |
|
rhmdvdsr.m |
⊢ ∥ = ( ∥r ‘ 𝑅 ) |
3 |
|
rhmdvdsr.n |
⊢ / = ( ∥r ‘ 𝑆 ) |
4 |
|
simpl1 |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
5 |
|
simpl2 |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → 𝐴 ∈ 𝑋 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
7 |
1 6
|
rhmf |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 : 𝑋 ⟶ ( Base ‘ 𝑆 ) ) |
8 |
7
|
ffvelrnda |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( Base ‘ 𝑆 ) ) |
9 |
4 5 8
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( Base ‘ 𝑆 ) ) |
10 |
|
simpll1 |
⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) ∧ 𝑐 ∈ 𝑋 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
11 |
|
simpr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) ∧ 𝑐 ∈ 𝑋 ) → 𝑐 ∈ 𝑋 ) |
12 |
7
|
ffvelrnda |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑐 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑐 ) ∈ ( Base ‘ 𝑆 ) ) |
13 |
10 11 12
|
syl2anc |
⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) ∧ 𝑐 ∈ 𝑋 ) → ( 𝐹 ‘ 𝑐 ) ∈ ( Base ‘ 𝑆 ) ) |
14 |
13
|
ralrimiva |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → ∀ 𝑐 ∈ 𝑋 ( 𝐹 ‘ 𝑐 ) ∈ ( Base ‘ 𝑆 ) ) |
15 |
5
|
adantr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) ∧ 𝑐 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 ) |
16 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
17 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
18 |
1 16 17
|
rhmmul |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑐 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) ) = ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) ) |
19 |
10 11 15 18
|
syl3anc |
⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) ∧ 𝑐 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) ) = ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) ) |
20 |
19
|
ralrimiva |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → ∀ 𝑐 ∈ 𝑋 ( 𝐹 ‘ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) ) = ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) ) |
21 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → 𝐴 ∥ 𝐵 ) |
22 |
1 2 16
|
dvdsr2 |
⊢ ( 𝐴 ∈ 𝑋 → ( 𝐴 ∥ 𝐵 ↔ ∃ 𝑐 ∈ 𝑋 ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) = 𝐵 ) ) |
23 |
22
|
biimpac |
⊢ ( ( 𝐴 ∥ 𝐵 ∧ 𝐴 ∈ 𝑋 ) → ∃ 𝑐 ∈ 𝑋 ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) = 𝐵 ) |
24 |
21 5 23
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → ∃ 𝑐 ∈ 𝑋 ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) = 𝐵 ) |
25 |
|
r19.29 |
⊢ ( ( ∀ 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 ) ) |
26 |
|
simpl |
⊢ ( ( ( 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 ) ) ) |
27 |
|
simpr |
⊢ ( ( ( 𝐹 ‘ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) ) = ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) ∧ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) = 𝐵 ) → ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) = 𝐵 ) |
28 |
27
|
fveq2d |
⊢ ( ( ( 𝐹 ‘ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) ) = ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) ∧ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) = 𝐵 ) → ( 𝐹 ‘ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) |
29 |
26 28
|
eqtr3d |
⊢ ( ( ( 𝐹 ‘ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) ) = ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) ∧ ( 𝑐 ( .r ‘ 𝑅 ) 𝐴 ) = 𝐵 ) → ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) |
30 |
29
|
reximi |
⊢ ( ∃ 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 ) ) |
31 |
25 30
|
syl |
⊢ ( ( ∀ 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 ) ) |
32 |
20 24 31
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → ∃ 𝑐 ∈ 𝑋 ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) |
33 |
|
r19.29 |
⊢ ( ( ∀ 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 ) ) ) |
34 |
14 32 33
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → ∃ 𝑐 ∈ 𝑋 ( ( 𝐹 ‘ 𝑐 ) ∈ ( Base ‘ 𝑆 ) ∧ ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) ) |
35 |
|
oveq1 |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑐 ) → ( 𝑦 ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) ) |
36 |
35
|
eqeq1d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑐 ) → ( ( 𝑦 ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ↔ ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) ) |
37 |
36
|
rspcev |
⊢ ( ( ( 𝐹 ‘ 𝑐 ) ∈ ( Base ‘ 𝑆 ) ∧ ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) → ∃ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) |
38 |
37
|
rexlimivw |
⊢ ( ∃ 𝑐 ∈ 𝑋 ( ( 𝐹 ‘ 𝑐 ) ∈ ( Base ‘ 𝑆 ) ∧ ( ( 𝐹 ‘ 𝑐 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) → ∃ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) |
39 |
34 38
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → ∃ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) |
40 |
6 3 17
|
dvdsr |
⊢ ( ( 𝐹 ‘ 𝐴 ) / ( 𝐹 ‘ 𝐵 ) ↔ ( ( 𝐹 ‘ 𝐴 ) ∈ ( Base ‘ 𝑆 ) ∧ ∃ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝑦 ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 𝐹 ‘ 𝐵 ) ) ) |
41 |
9 39 40
|
sylanbrc |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐴 ∥ 𝐵 ) → ( 𝐹 ‘ 𝐴 ) / ( 𝐹 ‘ 𝐵 ) ) |