Step |
Hyp |
Ref |
Expression |
1 |
|
imasmnd.u |
⊢ ( 𝜑 → 𝑈 = ( 𝐹 “s 𝑅 ) ) |
2 |
|
imasmnd.v |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) ) |
3 |
|
imasmnd.p |
⊢ + = ( +g ‘ 𝑅 ) |
4 |
|
imasmnd.f |
⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 ) |
5 |
|
imasmnd.e |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑝 ) ∧ ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑞 ) ) → ( 𝐹 ‘ ( 𝑎 + 𝑏 ) ) = ( 𝐹 ‘ ( 𝑝 + 𝑞 ) ) ) ) |
6 |
|
imasmnd2.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑊 ) |
7 |
|
imasmnd2.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑥 + 𝑦 ) ∈ 𝑉 ) |
8 |
|
imasmnd2.2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐹 ‘ ( ( 𝑥 + 𝑦 ) + 𝑧 ) ) = ( 𝐹 ‘ ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) ) |
9 |
|
imasmnd2.3 |
⊢ ( 𝜑 → 0 ∈ 𝑉 ) |
10 |
|
imasmnd2.4 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝐹 ‘ ( 0 + 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
11 |
|
imasmnd2.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝐹 ‘ ( 𝑥 + 0 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
12 |
1 2 4 6
|
imasbas |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑈 ) ) |
13 |
|
eqidd |
⊢ ( 𝜑 → ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) ) |
14 |
|
eqid |
⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 ) |
15 |
7
|
3expb |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑉 ) |
16 |
15
|
caovclg |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( 𝑝 + 𝑞 ) ∈ 𝑉 ) |
17 |
4 5 1 2 6 3 14 16
|
imasaddf |
⊢ ( 𝜑 → ( +g ‘ 𝑈 ) : ( 𝐵 × 𝐵 ) ⟶ 𝐵 ) |
18 |
|
fovrn |
⊢ ( ( ( +g ‘ 𝑈 ) : ( 𝐵 × 𝐵 ) ⟶ 𝐵 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) → ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ∈ 𝐵 ) |
19 |
17 18
|
syl3an1 |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) → ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ∈ 𝐵 ) |
20 |
|
forn |
⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → ran 𝐹 = 𝐵 ) |
21 |
4 20
|
syl |
⊢ ( 𝜑 → ran 𝐹 = 𝐵 ) |
22 |
21
|
eleq2d |
⊢ ( 𝜑 → ( 𝑢 ∈ ran 𝐹 ↔ 𝑢 ∈ 𝐵 ) ) |
23 |
21
|
eleq2d |
⊢ ( 𝜑 → ( 𝑣 ∈ ran 𝐹 ↔ 𝑣 ∈ 𝐵 ) ) |
24 |
21
|
eleq2d |
⊢ ( 𝜑 → ( 𝑤 ∈ ran 𝐹 ↔ 𝑤 ∈ 𝐵 ) ) |
25 |
22 23 24
|
3anbi123d |
⊢ ( 𝜑 → ( ( 𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹 ) ↔ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) ) |
26 |
|
fofn |
⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → 𝐹 Fn 𝑉 ) |
27 |
4 26
|
syl |
⊢ ( 𝜑 → 𝐹 Fn 𝑉 ) |
28 |
|
fvelrnb |
⊢ ( 𝐹 Fn 𝑉 → ( 𝑢 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ) ) |
29 |
|
fvelrnb |
⊢ ( 𝐹 Fn 𝑉 → ( 𝑣 ∈ ran 𝐹 ↔ ∃ 𝑦 ∈ 𝑉 ( 𝐹 ‘ 𝑦 ) = 𝑣 ) ) |
30 |
|
fvelrnb |
⊢ ( 𝐹 Fn 𝑉 → ( 𝑤 ∈ ran 𝐹 ↔ ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ) |
31 |
28 29 30
|
3anbi123d |
⊢ ( 𝐹 Fn 𝑉 → ( ( 𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹 ) ↔ ( ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ∃ 𝑦 ∈ 𝑉 ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ) ) |
32 |
27 31
|
syl |
⊢ ( 𝜑 → ( ( 𝑢 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ∧ 𝑤 ∈ ran 𝐹 ) ↔ ( ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ∃ 𝑦 ∈ 𝑉 ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ) ) |
33 |
25 32
|
bitr3d |
⊢ ( 𝜑 → ( ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ↔ ( ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ∃ 𝑦 ∈ 𝑉 ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ) ) |
34 |
|
3reeanv |
⊢ ( ∃ 𝑥 ∈ 𝑉 ∃ 𝑦 ∈ 𝑉 ∃ 𝑧 ∈ 𝑉 ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ↔ ( ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ∃ 𝑦 ∈ 𝑉 ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ∃ 𝑧 ∈ 𝑉 ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ) |
35 |
33 34
|
bitr4di |
⊢ ( 𝜑 → ( ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝑉 ∃ 𝑦 ∈ 𝑉 ∃ 𝑧 ∈ 𝑉 ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) ) ) |
36 |
|
simpl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝜑 ) |
37 |
7
|
3adant3r3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑉 ) |
38 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑧 ∈ 𝑉 ) |
39 |
4 5 1 2 6 3 14
|
imasaddval |
⊢ ( ( 𝜑 ∧ ( 𝑥 + 𝑦 ) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) → ( ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( ( 𝑥 + 𝑦 ) + 𝑧 ) ) ) |
40 |
36 37 38 39
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( ( 𝑥 + 𝑦 ) + 𝑧 ) ) ) |
41 |
|
simpr1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑥 ∈ 𝑉 ) |
42 |
16
|
caovclg |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑦 + 𝑧 ) ∈ 𝑉 ) |
43 |
42
|
3adantr1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑦 + 𝑧 ) ∈ 𝑉 ) |
44 |
4 5 1 2 6 3 14
|
imasaddval |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ ( 𝑦 + 𝑧 ) ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) ) |
45 |
36 41 43 44
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) ) |
46 |
8 40 45
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) ) |
47 |
4 5 1 2 6 3 14
|
imasaddval |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ) |
48 |
47
|
3adant3r3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ) |
49 |
48
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) ) |
50 |
4 5 1 2 6 3 14
|
imasaddval |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) |
51 |
50
|
3adant3r1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) |
52 |
51
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ ( 𝑦 + 𝑧 ) ) ) ) |
53 |
46 49 52
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) ) ) |
54 |
|
simp1 |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ 𝑥 ) = 𝑢 ) |
55 |
|
simp2 |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ 𝑦 ) = 𝑣 ) |
56 |
54 55
|
oveq12d |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) = ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ) |
57 |
|
simp3 |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( 𝐹 ‘ 𝑧 ) = 𝑤 ) |
58 |
56 57
|
oveq12d |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ( +g ‘ 𝑈 ) 𝑤 ) ) |
59 |
55 57
|
oveq12d |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) |
60 |
54 59
|
oveq12d |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) ) |
61 |
58 60
|
eqeq12d |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑦 ) ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑧 ) ) ) ↔ ( ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ( +g ‘ 𝑈 ) 𝑤 ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) ) ) |
62 |
53 61
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ( +g ‘ 𝑈 ) 𝑤 ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) ) ) |
63 |
62
|
3exp2 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑉 → ( 𝑦 ∈ 𝑉 → ( 𝑧 ∈ 𝑉 → ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ( +g ‘ 𝑈 ) 𝑤 ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) ) ) ) ) ) |
64 |
63
|
imp32 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( 𝑧 ∈ 𝑉 → ( ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ( +g ‘ 𝑈 ) 𝑤 ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) ) ) ) |
65 |
64
|
rexlimdv |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ) → ( ∃ 𝑧 ∈ 𝑉 ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ( +g ‘ 𝑈 ) 𝑤 ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) ) ) |
66 |
65
|
rexlimdvva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝑉 ∃ 𝑦 ∈ 𝑉 ∃ 𝑧 ∈ 𝑉 ( ( 𝐹 ‘ 𝑥 ) = 𝑢 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑣 ∧ ( 𝐹 ‘ 𝑧 ) = 𝑤 ) → ( ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ( +g ‘ 𝑈 ) 𝑤 ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) ) ) |
67 |
35 66
|
sylbid |
⊢ ( 𝜑 → ( ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) → ( ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ( +g ‘ 𝑈 ) 𝑤 ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) ) ) |
68 |
67
|
imp |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 ( +g ‘ 𝑈 ) 𝑣 ) ( +g ‘ 𝑈 ) 𝑤 ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝑣 ( +g ‘ 𝑈 ) 𝑤 ) ) ) |
69 |
|
fof |
⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → 𝐹 : 𝑉 ⟶ 𝐵 ) |
70 |
4 69
|
syl |
⊢ ( 𝜑 → 𝐹 : 𝑉 ⟶ 𝐵 ) |
71 |
70 9
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) ∈ 𝐵 ) |
72 |
27 28
|
syl |
⊢ ( 𝜑 → ( 𝑢 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ) ) |
73 |
22 72
|
bitr3d |
⊢ ( 𝜑 → ( 𝑢 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 ) ) |
74 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝜑 ) |
75 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 0 ∈ 𝑉 ) |
76 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝑥 ∈ 𝑉 ) |
77 |
4 5 1 2 6 3 14
|
imasaddval |
⊢ ( ( 𝜑 ∧ 0 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 0 + 𝑥 ) ) ) |
78 |
74 75 76 77
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 0 + 𝑥 ) ) ) |
79 |
78 10
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
80 |
|
oveq2 |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑢 → ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) 𝑢 ) ) |
81 |
|
id |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑢 → ( 𝐹 ‘ 𝑥 ) = 𝑢 ) |
82 |
80 81
|
eqeq12d |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑢 → ( ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) 𝑢 ) = 𝑢 ) ) |
83 |
79 82
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝑢 → ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) 𝑢 ) = 𝑢 ) ) |
84 |
83
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 → ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) 𝑢 ) = 𝑢 ) ) |
85 |
73 84
|
sylbid |
⊢ ( 𝜑 → ( 𝑢 ∈ 𝐵 → ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) 𝑢 ) = 𝑢 ) ) |
86 |
85
|
imp |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) → ( ( 𝐹 ‘ 0 ) ( +g ‘ 𝑈 ) 𝑢 ) = 𝑢 ) |
87 |
4 5 1 2 6 3 14
|
imasaddval |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 0 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = ( 𝐹 ‘ ( 𝑥 + 0 ) ) ) |
88 |
75 87
|
mpd3an3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = ( 𝐹 ‘ ( 𝑥 + 0 ) ) ) |
89 |
88 11
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = ( 𝐹 ‘ 𝑥 ) ) |
90 |
|
oveq1 |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑢 → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = ( 𝑢 ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) ) |
91 |
90 81
|
eqeq12d |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑢 → ( ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝑢 ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = 𝑢 ) ) |
92 |
89 91
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑥 ) = 𝑢 → ( 𝑢 ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = 𝑢 ) ) |
93 |
92
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝑉 ( 𝐹 ‘ 𝑥 ) = 𝑢 → ( 𝑢 ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = 𝑢 ) ) |
94 |
73 93
|
sylbid |
⊢ ( 𝜑 → ( 𝑢 ∈ 𝐵 → ( 𝑢 ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = 𝑢 ) ) |
95 |
94
|
imp |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ 𝐵 ) → ( 𝑢 ( +g ‘ 𝑈 ) ( 𝐹 ‘ 0 ) ) = 𝑢 ) |
96 |
12 13 19 68 71 86 95
|
ismndd |
⊢ ( 𝜑 → 𝑈 ∈ Mnd ) |
97 |
12 13 71 86 95
|
grpidd |
⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) = ( 0g ‘ 𝑈 ) ) |
98 |
96 97
|
jca |
⊢ ( 𝜑 → ( 𝑈 ∈ Mnd ∧ ( 𝐹 ‘ 0 ) = ( 0g ‘ 𝑈 ) ) ) |