Step |
Hyp |
Ref |
Expression |
1 |
|
rhmdvd.u |
⊢ 𝑈 = ( Unit ‘ 𝑆 ) |
2 |
|
rhmdvd.x |
⊢ 𝑋 = ( Base ‘ 𝑅 ) |
3 |
|
rhmdvd.d |
⊢ / = ( /r ‘ 𝑆 ) |
4 |
|
rhmdvd.m |
⊢ · = ( .r ‘ 𝑅 ) |
5 |
|
simp1 |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
6 |
|
simp21 |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → 𝐴 ∈ 𝑋 ) |
7 |
|
simp23 |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → 𝐶 ∈ 𝑋 ) |
8 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
9 |
2 4 8
|
rhmmul |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐴 · 𝐶 ) ) = ( ( 𝐹 ‘ 𝐴 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) ) |
10 |
5 6 7 9
|
syl3anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → ( 𝐹 ‘ ( 𝐴 · 𝐶 ) ) = ( ( 𝐹 ‘ 𝐴 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) ) |
11 |
|
simp22 |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → 𝐵 ∈ 𝑋 ) |
12 |
2 4 8
|
rhmmul |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐵 · 𝐶 ) ) = ( ( 𝐹 ‘ 𝐵 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) ) |
13 |
5 11 7 12
|
syl3anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → ( 𝐹 ‘ ( 𝐵 · 𝐶 ) ) = ( ( 𝐹 ‘ 𝐵 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) ) |
14 |
10 13
|
oveq12d |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → ( ( 𝐹 ‘ ( 𝐴 · 𝐶 ) ) / ( 𝐹 ‘ ( 𝐵 · 𝐶 ) ) ) = ( ( ( 𝐹 ‘ 𝐴 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) / ( ( 𝐹 ‘ 𝐵 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) ) ) |
15 |
|
rhmrcl2 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑆 ∈ Ring ) |
16 |
15
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → 𝑆 ∈ Ring ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
18 |
2 17
|
rhmf |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 : 𝑋 ⟶ ( Base ‘ 𝑆 ) ) |
19 |
18
|
3ad2ant1 |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → 𝐹 : 𝑋 ⟶ ( Base ‘ 𝑆 ) ) |
20 |
19 6
|
ffvelrnd |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( Base ‘ 𝑆 ) ) |
21 |
|
simp3l |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ) |
22 |
|
simp3r |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) |
23 |
17 1 3 8
|
dvrcan5 |
⊢ ( ( 𝑆 ∈ Ring ∧ ( ( 𝐹 ‘ 𝐴 ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → ( ( ( 𝐹 ‘ 𝐴 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) / ( ( 𝐹 ‘ 𝐵 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) ) = ( ( 𝐹 ‘ 𝐴 ) / ( 𝐹 ‘ 𝐵 ) ) ) |
24 |
16 20 21 22 23
|
syl13anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → ( ( ( 𝐹 ‘ 𝐴 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) / ( ( 𝐹 ‘ 𝐵 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐶 ) ) ) = ( ( 𝐹 ‘ 𝐴 ) / ( 𝐹 ‘ 𝐵 ) ) ) |
25 |
14 24
|
eqtr2d |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ∧ ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑈 ∧ ( 𝐹 ‘ 𝐶 ) ∈ 𝑈 ) ) → ( ( 𝐹 ‘ 𝐴 ) / ( 𝐹 ‘ 𝐵 ) ) = ( ( 𝐹 ‘ ( 𝐴 · 𝐶 ) ) / ( 𝐹 ‘ ( 𝐵 · 𝐶 ) ) ) ) |