Step |
Hyp |
Ref |
Expression |
1 |
|
dprdff.w |
⊢ 𝑊 = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp 0 } |
2 |
|
dprdff.1 |
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 ) |
3 |
|
dprdff.2 |
⊢ ( 𝜑 → dom 𝑆 = 𝐼 ) |
4 |
|
dprdff.3 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑊 ) |
5 |
|
dprdfcntz.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
7 |
1 2 3 4 6
|
dprdff |
⊢ ( 𝜑 → 𝐹 : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
8 |
7
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐼 ) |
9 |
7
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
10 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 = 𝑧 ) → 𝑦 = 𝑧 ) |
11 |
10
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 = 𝑧 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ) |
12 |
10
|
equcomd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 = 𝑧 ) → 𝑧 = 𝑦 ) |
13 |
12
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 = 𝑧 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑦 ) ) |
14 |
11 13
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 = 𝑧 ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) |
15 |
2
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → 𝐺 dom DProd 𝑆 ) |
16 |
3
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → dom 𝑆 = 𝐼 ) |
17 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → 𝑦 ∈ 𝐼 ) |
18 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → 𝑧 ∈ 𝐼 ) |
19 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → 𝑦 ≠ 𝑧 ) |
20 |
15 16 17 18 19 5
|
dprdcntz |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → ( 𝑆 ‘ 𝑦 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑧 ) ) ) |
21 |
1 2 3 4
|
dprdfcl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑆 ‘ 𝑦 ) ) |
22 |
21
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑆 ‘ 𝑦 ) ) |
23 |
20 22
|
sseldd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑍 ‘ ( 𝑆 ‘ 𝑧 ) ) ) |
24 |
1 2 3 4
|
dprdfcl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑆 ‘ 𝑧 ) ) |
25 |
24
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑆 ‘ 𝑧 ) ) |
26 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
27 |
26 5
|
cntzi |
⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑍 ‘ ( 𝑆 ‘ 𝑧 ) ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ ( 𝑆 ‘ 𝑧 ) ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) |
28 |
23 25 27
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) ∧ 𝑦 ≠ 𝑧 ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) |
29 |
14 28
|
pm2.61dane |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) ∧ 𝑧 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) |
30 |
29
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ∀ 𝑧 ∈ 𝐼 ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) |
31 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → 𝐹 Fn 𝐼 ) |
32 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑧 ) → ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑥 ) = ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑧 ) ) ) |
33 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑧 ) → ( 𝑥 ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) |
34 |
32 33
|
eqeq12d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑧 ) → ( ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑥 ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
35 |
34
|
ralrn |
⊢ ( 𝐹 Fn 𝐼 → ( ∀ 𝑥 ∈ ran 𝐹 ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑥 ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑧 ∈ 𝐼 ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
36 |
31 35
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( ∀ 𝑥 ∈ ran 𝐹 ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑥 ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ↔ ∀ 𝑧 ∈ 𝐼 ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
37 |
30 36
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ∀ 𝑥 ∈ ran 𝐹 ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑥 ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) |
38 |
7
|
frnd |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( Base ‘ 𝐺 ) ) |
39 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ran 𝐹 ⊆ ( Base ‘ 𝐺 ) ) |
40 |
6 26 5
|
elcntz |
⊢ ( ran 𝐹 ⊆ ( Base ‘ 𝐺 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑍 ‘ ran 𝐹 ) ↔ ( ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ ran 𝐹 ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑥 ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
41 |
39 40
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑍 ‘ ran 𝐹 ) ↔ ( ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ ran 𝐹 ( ( 𝐹 ‘ 𝑦 ) ( +g ‘ 𝐺 ) 𝑥 ) = ( 𝑥 ( +g ‘ 𝐺 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
42 |
9 37 41
|
mpbir2and |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑍 ‘ ran 𝐹 ) ) |
43 |
42
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐼 ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑍 ‘ ran 𝐹 ) ) |
44 |
|
ffnfv |
⊢ ( 𝐹 : 𝐼 ⟶ ( 𝑍 ‘ ran 𝐹 ) ↔ ( 𝐹 Fn 𝐼 ∧ ∀ 𝑦 ∈ 𝐼 ( 𝐹 ‘ 𝑦 ) ∈ ( 𝑍 ‘ ran 𝐹 ) ) ) |
45 |
8 43 44
|
sylanbrc |
⊢ ( 𝜑 → 𝐹 : 𝐼 ⟶ ( 𝑍 ‘ ran 𝐹 ) ) |
46 |
45
|
frnd |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |