Step |
Hyp |
Ref |
Expression |
1 |
|
rhmimaidl.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
2 |
|
rhmimaidl.t |
⊢ 𝑇 = ( LIdeal ‘ 𝑅 ) |
3 |
|
rhmimaidl.u |
⊢ 𝑈 = ( LIdeal ‘ 𝑆 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
5 |
4 1
|
rhmf |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐵 ) |
6 |
|
fimass |
⊢ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐵 → ( 𝐹 “ 𝐼 ) ⊆ 𝐵 ) |
7 |
5 6
|
syl |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 “ 𝐼 ) ⊆ 𝐵 ) |
8 |
7
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → ( 𝐹 “ 𝐼 ) ⊆ 𝐵 ) |
9 |
5
|
ffnd |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 Fn ( Base ‘ 𝑅 ) ) |
10 |
9
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → 𝐹 Fn ( Base ‘ 𝑅 ) ) |
11 |
|
rhmrcl1 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑅 ∈ Ring ) |
12 |
11
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → 𝑅 ∈ Ring ) |
13 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
14 |
4 13
|
ring0cl |
⊢ ( 𝑅 ∈ Ring → ( 0g ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
15 |
12 14
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → ( 0g ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
16 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → 𝐼 ∈ 𝑇 ) |
17 |
2 13
|
lidl0cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑇 ) → ( 0g ‘ 𝑅 ) ∈ 𝐼 ) |
18 |
12 16 17
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → ( 0g ‘ 𝑅 ) ∈ 𝐼 ) |
19 |
10 15 18
|
fnfvimad |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → ( 𝐹 ‘ ( 0g ‘ 𝑅 ) ) ∈ ( 𝐹 “ 𝐼 ) ) |
20 |
19
|
ne0d |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → ( 𝐹 “ 𝐼 ) ≠ ∅ ) |
21 |
|
rhmghm |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
22 |
21
|
ad4antr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
23 |
11
|
ad4antr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → 𝑅 ∈ Ring ) |
24 |
|
simpr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → 𝑧 ∈ ( Base ‘ 𝑅 ) ) |
25 |
4 2
|
lidlss |
⊢ ( 𝐼 ∈ 𝑇 → 𝐼 ⊆ ( Base ‘ 𝑅 ) ) |
26 |
25
|
ad4antlr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → 𝐼 ⊆ ( Base ‘ 𝑅 ) ) |
27 |
|
simplr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → 𝑖 ∈ 𝐼 ) |
28 |
26 27
|
sseldd |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → 𝑖 ∈ ( Base ‘ 𝑅 ) ) |
29 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
30 |
4 29
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ∧ 𝑖 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ∈ ( Base ‘ 𝑅 ) ) |
31 |
23 24 28 30
|
syl3anc |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ∈ ( Base ‘ 𝑅 ) ) |
32 |
|
simpllr |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → 𝑗 ∈ 𝐼 ) |
33 |
26 32
|
sseldd |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → 𝑗 ∈ ( Base ‘ 𝑅 ) ) |
34 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
35 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
36 |
4 34 35
|
ghmlin |
⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ∈ ( Base ‘ 𝑅 ) ∧ 𝑗 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( 𝐹 ‘ ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
37 |
22 31 33 36
|
syl3anc |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( 𝐹 ‘ ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
38 |
|
simp-4l |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
39 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
40 |
4 29 39
|
rhmmul |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ∧ 𝑖 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ) |
41 |
38 24 28 40
|
syl3anc |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ) |
42 |
41
|
oveq1d |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝐹 ‘ ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) = ( ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
43 |
37 42
|
eqtrd |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
44 |
43
|
adantl4r |
⊢ ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
45 |
44
|
adantl3r |
⊢ ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
46 |
45
|
adantl3r |
⊢ ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
47 |
46
|
adantl3r |
⊢ ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
48 |
47
|
adantllr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
49 |
48
|
ad4ant13 |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) ) |
50 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( 𝐹 ‘ 𝑧 ) = 𝑥 ) |
51 |
|
simpllr |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( 𝐹 ‘ 𝑖 ) = 𝑎 ) |
52 |
50 51
|
oveq12d |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) = ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ) |
53 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( 𝐹 ‘ 𝑗 ) = 𝑏 ) |
54 |
52 53
|
oveq12d |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( ( ( 𝐹 ‘ 𝑧 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑖 ) ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑗 ) ) = ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ) |
55 |
49 54
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) = ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ) |
56 |
10
|
ad9antr |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → 𝐹 Fn ( Base ‘ 𝑅 ) ) |
57 |
16 25
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → 𝐼 ⊆ ( Base ‘ 𝑅 ) ) |
58 |
57
|
ad9antr |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → 𝐼 ⊆ ( Base ‘ 𝑅 ) ) |
59 |
16
|
ad9antr |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → 𝐼 ∈ 𝑇 ) |
60 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → 𝑧 ∈ ( Base ‘ 𝑅 ) ) |
61 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → 𝑖 ∈ 𝐼 ) |
62 |
|
simp-6r |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → 𝑗 ∈ 𝐼 ) |
63 |
2 4 34 29
|
islidl |
⊢ ( 𝐼 ∈ 𝑇 ↔ ( 𝐼 ⊆ ( Base ‘ 𝑅 ) ∧ 𝐼 ≠ ∅ ∧ ∀ 𝑧 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝐼 ∀ 𝑗 ∈ 𝐼 ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ∈ 𝐼 ) ) |
64 |
63
|
simp3bi |
⊢ ( 𝐼 ∈ 𝑇 → ∀ 𝑧 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝐼 ∀ 𝑗 ∈ 𝐼 ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ∈ 𝐼 ) |
65 |
64
|
r19.21bi |
⊢ ( ( 𝐼 ∈ 𝑇 ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) → ∀ 𝑖 ∈ 𝐼 ∀ 𝑗 ∈ 𝐼 ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ∈ 𝐼 ) |
66 |
65
|
r19.21bi |
⊢ ( ( ( 𝐼 ∈ 𝑇 ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ 𝑖 ∈ 𝐼 ) → ∀ 𝑗 ∈ 𝐼 ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ∈ 𝐼 ) |
67 |
66
|
r19.21bi |
⊢ ( ( ( ( 𝐼 ∈ 𝑇 ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑗 ∈ 𝐼 ) → ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ∈ 𝐼 ) |
68 |
59 60 61 62 67
|
syl1111anc |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ∈ 𝐼 ) |
69 |
58 68
|
sseldd |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ∈ ( Base ‘ 𝑅 ) ) |
70 |
56 69 68
|
fnfvimad |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( 𝐹 ‘ ( ( 𝑧 ( .r ‘ 𝑅 ) 𝑖 ) ( +g ‘ 𝑅 ) 𝑗 ) ) ∈ ( 𝐹 “ 𝐼 ) ) |
71 |
55 70
|
eqeltrrd |
⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ 𝑧 ) = 𝑥 ) → ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ∈ ( 𝐹 “ 𝐼 ) ) |
72 |
5
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐵 ) |
73 |
72
|
ffund |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → Fun 𝐹 ) |
74 |
73
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → Fun 𝐹 ) |
75 |
5
|
fdmd |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → dom 𝐹 = ( Base ‘ 𝑅 ) ) |
76 |
75
|
imaeq2d |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 “ dom 𝐹 ) = ( 𝐹 “ ( Base ‘ 𝑅 ) ) ) |
77 |
|
imadmrn |
⊢ ( 𝐹 “ dom 𝐹 ) = ran 𝐹 |
78 |
76 77
|
eqtr3di |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 “ ( Base ‘ 𝑅 ) ) = ran 𝐹 ) |
79 |
78
|
eqeq1d |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( ( 𝐹 “ ( Base ‘ 𝑅 ) ) = 𝐵 ↔ ran 𝐹 = 𝐵 ) ) |
80 |
79
|
biimpar |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) → ( 𝐹 “ ( Base ‘ 𝑅 ) ) = 𝐵 ) |
81 |
80
|
eleq2d |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) → ( 𝑥 ∈ ( 𝐹 “ ( Base ‘ 𝑅 ) ) ↔ 𝑥 ∈ 𝐵 ) ) |
82 |
81
|
biimpar |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ ( 𝐹 “ ( Base ‘ 𝑅 ) ) ) |
83 |
82
|
adantlr |
⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ ( 𝐹 “ ( Base ‘ 𝑅 ) ) ) |
84 |
83
|
ad6antr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → 𝑥 ∈ ( 𝐹 “ ( Base ‘ 𝑅 ) ) ) |
85 |
|
fvelima |
⊢ ( ( Fun 𝐹 ∧ 𝑥 ∈ ( 𝐹 “ ( Base ‘ 𝑅 ) ) ) → ∃ 𝑧 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ 𝑧 ) = 𝑥 ) |
86 |
74 84 85
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ∃ 𝑧 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ 𝑧 ) = 𝑥 ) |
87 |
71 86
|
r19.29a |
⊢ ( ( ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) ∧ 𝑖 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑖 ) = 𝑎 ) → ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ∈ ( 𝐹 “ 𝐼 ) ) |
88 |
73
|
ad5antr |
⊢ ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) → Fun 𝐹 ) |
89 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) → 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) |
90 |
|
fvelima |
⊢ ( ( Fun 𝐹 ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) → ∃ 𝑖 ∈ 𝐼 ( 𝐹 ‘ 𝑖 ) = 𝑎 ) |
91 |
88 89 90
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) → ∃ 𝑖 ∈ 𝐼 ( 𝐹 ‘ 𝑖 ) = 𝑎 ) |
92 |
87 91
|
r19.29a |
⊢ ( ( ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑗 ∈ 𝐼 ) ∧ ( 𝐹 ‘ 𝑗 ) = 𝑏 ) → ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ∈ ( 𝐹 “ 𝐼 ) ) |
93 |
73
|
ad3antrrr |
⊢ ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) → Fun 𝐹 ) |
94 |
|
simpr |
⊢ ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) → 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) |
95 |
|
fvelima |
⊢ ( ( Fun 𝐹 ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) → ∃ 𝑗 ∈ 𝐼 ( 𝐹 ‘ 𝑗 ) = 𝑏 ) |
96 |
93 94 95
|
syl2anc |
⊢ ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) → ∃ 𝑗 ∈ 𝐼 ( 𝐹 ‘ 𝑗 ) = 𝑏 ) |
97 |
92 96
|
r19.29a |
⊢ ( ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) → ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ∈ ( 𝐹 “ 𝐼 ) ) |
98 |
97
|
anasss |
⊢ ( ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑎 ∈ ( 𝐹 “ 𝐼 ) ∧ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ) ) → ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ∈ ( 𝐹 “ 𝐼 ) ) |
99 |
98
|
ralrimivva |
⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) ∧ 𝑥 ∈ 𝐵 ) → ∀ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ∀ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ∈ ( 𝐹 “ 𝐼 ) ) |
100 |
99
|
ralrimiva |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ∀ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ∈ ( 𝐹 “ 𝐼 ) ) |
101 |
3 1 35 39
|
islidl |
⊢ ( ( 𝐹 “ 𝐼 ) ∈ 𝑈 ↔ ( ( 𝐹 “ 𝐼 ) ⊆ 𝐵 ∧ ( 𝐹 “ 𝐼 ) ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑎 ∈ ( 𝐹 “ 𝐼 ) ∀ 𝑏 ∈ ( 𝐹 “ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝑆 ) 𝑎 ) ( +g ‘ 𝑆 ) 𝑏 ) ∈ ( 𝐹 “ 𝐼 ) ) ) |
102 |
8 20 100 101
|
syl3anbrc |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ) ∧ 𝐼 ∈ 𝑇 ) → ( 𝐹 “ 𝐼 ) ∈ 𝑈 ) |
103 |
102
|
3impa |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ran 𝐹 = 𝐵 ∧ 𝐼 ∈ 𝑇 ) → ( 𝐹 “ 𝐼 ) ∈ 𝑈 ) |