Step |
Hyp |
Ref |
Expression |
1 |
|
mndpluscn.f |
⊢ 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) |
2 |
|
mndpluscn.p |
⊢ + : ( 𝐵 × 𝐵 ) ⟶ 𝐵 |
3 |
|
mndpluscn.t |
⊢ ∗ : ( 𝐶 × 𝐶 ) ⟶ 𝐶 |
4 |
|
mndpluscn.j |
⊢ 𝐽 ∈ ( TopOn ‘ 𝐵 ) |
5 |
|
mndpluscn.k |
⊢ 𝐾 ∈ ( TopOn ‘ 𝐶 ) |
6 |
|
mndpluscn.h |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) ) |
7 |
|
mndpluscn.o |
⊢ + ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) |
8 |
|
ffn |
⊢ ( ∗ : ( 𝐶 × 𝐶 ) ⟶ 𝐶 → ∗ Fn ( 𝐶 × 𝐶 ) ) |
9 |
|
fnov |
⊢ ( ∗ Fn ( 𝐶 × 𝐶 ) ↔ ∗ = ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( 𝑎 ∗ 𝑏 ) ) ) |
10 |
9
|
biimpi |
⊢ ( ∗ Fn ( 𝐶 × 𝐶 ) → ∗ = ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( 𝑎 ∗ 𝑏 ) ) ) |
11 |
3 8 10
|
mp2b |
⊢ ∗ = ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( 𝑎 ∗ 𝑏 ) ) |
12 |
4
|
toponunii |
⊢ 𝐵 = ∪ 𝐽 |
13 |
5
|
toponunii |
⊢ 𝐶 = ∪ 𝐾 |
14 |
12 13
|
hmeof1o |
⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) |
15 |
1 14
|
ax-mp |
⊢ 𝐹 : 𝐵 –1-1-onto→ 𝐶 |
16 |
|
f1ocnvdm |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ∧ 𝑎 ∈ 𝐶 ) → ( ◡ 𝐹 ‘ 𝑎 ) ∈ 𝐵 ) |
17 |
15 16
|
mpan |
⊢ ( 𝑎 ∈ 𝐶 → ( ◡ 𝐹 ‘ 𝑎 ) ∈ 𝐵 ) |
18 |
|
f1ocnvdm |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ( ◡ 𝐹 ‘ 𝑏 ) ∈ 𝐵 ) |
19 |
15 18
|
mpan |
⊢ ( 𝑏 ∈ 𝐶 → ( ◡ 𝐹 ‘ 𝑏 ) ∈ 𝐵 ) |
20 |
17 19
|
anim12i |
⊢ ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ( ( ◡ 𝐹 ‘ 𝑎 ) ∈ 𝐵 ∧ ( ◡ 𝐹 ‘ 𝑏 ) ∈ 𝐵 ) ) |
21 |
6
|
rgen2 |
⊢ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) |
22 |
|
fvoveq1 |
⊢ ( 𝑥 = ( ◡ 𝐹 ‘ 𝑎 ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + 𝑦 ) ) ) |
23 |
|
fveq2 |
⊢ ( 𝑥 = ( ◡ 𝐹 ‘ 𝑎 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ) |
24 |
23
|
oveq1d |
⊢ ( 𝑥 = ( ◡ 𝐹 ‘ 𝑎 ) → ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ∗ ( 𝐹 ‘ 𝑦 ) ) ) |
25 |
22 24
|
eqeq12d |
⊢ ( 𝑥 = ( ◡ 𝐹 ‘ 𝑎 ) → ( ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + 𝑦 ) ) = ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ∗ ( 𝐹 ‘ 𝑦 ) ) ) ) |
26 |
|
oveq2 |
⊢ ( 𝑦 = ( ◡ 𝐹 ‘ 𝑏 ) → ( ( ◡ 𝐹 ‘ 𝑎 ) + 𝑦 ) = ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) |
27 |
26
|
fveq2d |
⊢ ( 𝑦 = ( ◡ 𝐹 ‘ 𝑏 ) → ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + 𝑦 ) ) = ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) |
28 |
|
fveq2 |
⊢ ( 𝑦 = ( ◡ 𝐹 ‘ 𝑏 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑏 ) ) ) |
29 |
28
|
oveq2d |
⊢ ( 𝑦 = ( ◡ 𝐹 ‘ 𝑏 ) → ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ∗ ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ∗ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) |
30 |
27 29
|
eqeq12d |
⊢ ( 𝑦 = ( ◡ 𝐹 ‘ 𝑏 ) → ( ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + 𝑦 ) ) = ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ∗ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) = ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ∗ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) ) |
31 |
25 30
|
rspc2va |
⊢ ( ( ( ( ◡ 𝐹 ‘ 𝑎 ) ∈ 𝐵 ∧ ( ◡ 𝐹 ‘ 𝑏 ) ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ∗ ( 𝐹 ‘ 𝑦 ) ) ) → ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) = ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ∗ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) |
32 |
20 21 31
|
sylancl |
⊢ ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) = ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ∗ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) |
33 |
|
f1ocnvfv2 |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ∧ 𝑎 ∈ 𝐶 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) = 𝑎 ) |
34 |
15 33
|
mpan |
⊢ ( 𝑎 ∈ 𝐶 → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) = 𝑎 ) |
35 |
|
f1ocnvfv2 |
⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑏 ) ) = 𝑏 ) |
36 |
15 35
|
mpan |
⊢ ( 𝑏 ∈ 𝐶 → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑏 ) ) = 𝑏 ) |
37 |
34 36
|
oveqan12d |
⊢ ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑎 ) ) ∗ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑏 ) ) ) = ( 𝑎 ∗ 𝑏 ) ) |
38 |
32 37
|
eqtr2d |
⊢ ( ( 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) → ( 𝑎 ∗ 𝑏 ) = ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) |
39 |
38
|
mpoeq3ia |
⊢ ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( 𝑎 ∗ 𝑏 ) ) = ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) |
40 |
11 39
|
eqtri |
⊢ ∗ = ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) |
41 |
5
|
a1i |
⊢ ( ⊤ → 𝐾 ∈ ( TopOn ‘ 𝐶 ) ) |
42 |
41 41
|
cnmpt1st |
⊢ ( ⊤ → ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ 𝑎 ) ∈ ( ( 𝐾 ×t 𝐾 ) Cn 𝐾 ) ) |
43 |
|
hmeocnvcn |
⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → ◡ 𝐹 ∈ ( 𝐾 Cn 𝐽 ) ) |
44 |
1 43
|
mp1i |
⊢ ( ⊤ → ◡ 𝐹 ∈ ( 𝐾 Cn 𝐽 ) ) |
45 |
41 41 42 44
|
cnmpt21f |
⊢ ( ⊤ → ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( ◡ 𝐹 ‘ 𝑎 ) ) ∈ ( ( 𝐾 ×t 𝐾 ) Cn 𝐽 ) ) |
46 |
41 41
|
cnmpt2nd |
⊢ ( ⊤ → ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ 𝑏 ) ∈ ( ( 𝐾 ×t 𝐾 ) Cn 𝐾 ) ) |
47 |
41 41 46 44
|
cnmpt21f |
⊢ ( ⊤ → ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( ◡ 𝐹 ‘ 𝑏 ) ) ∈ ( ( 𝐾 ×t 𝐾 ) Cn 𝐽 ) ) |
48 |
7
|
a1i |
⊢ ( ⊤ → + ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) |
49 |
41 41 45 47 48
|
cnmpt22f |
⊢ ( ⊤ → ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) ∈ ( ( 𝐾 ×t 𝐾 ) Cn 𝐽 ) ) |
50 |
|
hmeocn |
⊢ ( 𝐹 ∈ ( 𝐽 Homeo 𝐾 ) → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) |
51 |
1 50
|
mp1i |
⊢ ( ⊤ → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) |
52 |
41 41 49 51
|
cnmpt21f |
⊢ ( ⊤ → ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) ∈ ( ( 𝐾 ×t 𝐾 ) Cn 𝐾 ) ) |
53 |
52
|
mptru |
⊢ ( 𝑎 ∈ 𝐶 , 𝑏 ∈ 𝐶 ↦ ( 𝐹 ‘ ( ( ◡ 𝐹 ‘ 𝑎 ) + ( ◡ 𝐹 ‘ 𝑏 ) ) ) ) ∈ ( ( 𝐾 ×t 𝐾 ) Cn 𝐾 ) |
54 |
40 53
|
eqeltri |
⊢ ∗ ∈ ( ( 𝐾 ×t 𝐾 ) Cn 𝐾 ) |