Step |
Hyp |
Ref |
Expression |
1 |
|
ismot.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
ismot.m |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
motgrp.1 |
⊢ ( 𝜑 → 𝐺 ∈ 𝑉 ) |
4 |
|
motco.2 |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐺 Ismt 𝐺 ) ) |
5 |
|
motco.3 |
⊢ ( 𝜑 → 𝐻 ∈ ( 𝐺 Ismt 𝐺 ) ) |
6 |
1 2 3 4
|
motf1o |
⊢ ( 𝜑 → 𝐹 : 𝑃 –1-1-onto→ 𝑃 ) |
7 |
1 2 3 5
|
motf1o |
⊢ ( 𝜑 → 𝐻 : 𝑃 –1-1-onto→ 𝑃 ) |
8 |
|
f1oco |
⊢ ( ( 𝐹 : 𝑃 –1-1-onto→ 𝑃 ∧ 𝐻 : 𝑃 –1-1-onto→ 𝑃 ) → ( 𝐹 ∘ 𝐻 ) : 𝑃 –1-1-onto→ 𝑃 ) |
9 |
6 7 8
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐻 ) : 𝑃 –1-1-onto→ 𝑃 ) |
10 |
|
f1of |
⊢ ( 𝐻 : 𝑃 –1-1-onto→ 𝑃 → 𝐻 : 𝑃 ⟶ 𝑃 ) |
11 |
7 10
|
syl |
⊢ ( 𝜑 → 𝐻 : 𝑃 ⟶ 𝑃 ) |
12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝐻 : 𝑃 ⟶ 𝑃 ) |
13 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝑎 ∈ 𝑃 ) |
14 |
|
fvco3 |
⊢ ( ( 𝐻 : 𝑃 ⟶ 𝑃 ∧ 𝑎 ∈ 𝑃 ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑎 ) = ( 𝐹 ‘ ( 𝐻 ‘ 𝑎 ) ) ) |
15 |
12 13 14
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑎 ) = ( 𝐹 ‘ ( 𝐻 ‘ 𝑎 ) ) ) |
16 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝑏 ∈ 𝑃 ) |
17 |
|
fvco3 |
⊢ ( ( 𝐻 : 𝑃 ⟶ 𝑃 ∧ 𝑏 ∈ 𝑃 ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑏 ) = ( 𝐹 ‘ ( 𝐻 ‘ 𝑏 ) ) ) |
18 |
12 16 17
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑏 ) = ( 𝐹 ‘ ( 𝐻 ‘ 𝑏 ) ) ) |
19 |
15 18
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑎 ) − ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑏 ) ) = ( ( 𝐹 ‘ ( 𝐻 ‘ 𝑎 ) ) − ( 𝐹 ‘ ( 𝐻 ‘ 𝑏 ) ) ) ) |
20 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝐺 ∈ 𝑉 ) |
21 |
12 13
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( 𝐻 ‘ 𝑎 ) ∈ 𝑃 ) |
22 |
12 16
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( 𝐻 ‘ 𝑏 ) ∈ 𝑃 ) |
23 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝐹 ∈ ( 𝐺 Ismt 𝐺 ) ) |
24 |
1 2 20 21 22 23
|
motcgr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( ( 𝐹 ‘ ( 𝐻 ‘ 𝑎 ) ) − ( 𝐹 ‘ ( 𝐻 ‘ 𝑏 ) ) ) = ( ( 𝐻 ‘ 𝑎 ) − ( 𝐻 ‘ 𝑏 ) ) ) |
25 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → 𝐻 ∈ ( 𝐺 Ismt 𝐺 ) ) |
26 |
1 2 20 13 16 25
|
motcgr |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( ( 𝐻 ‘ 𝑎 ) − ( 𝐻 ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) ) |
27 |
19 24 26
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃 ) ) → ( ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑎 ) − ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) ) |
28 |
27
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝑃 ∀ 𝑏 ∈ 𝑃 ( ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑎 ) − ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) ) |
29 |
1 2
|
ismot |
⊢ ( 𝐺 ∈ 𝑉 → ( ( 𝐹 ∘ 𝐻 ) ∈ ( 𝐺 Ismt 𝐺 ) ↔ ( ( 𝐹 ∘ 𝐻 ) : 𝑃 –1-1-onto→ 𝑃 ∧ ∀ 𝑎 ∈ 𝑃 ∀ 𝑏 ∈ 𝑃 ( ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑎 ) − ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) ) ) ) |
30 |
3 29
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 ∘ 𝐻 ) ∈ ( 𝐺 Ismt 𝐺 ) ↔ ( ( 𝐹 ∘ 𝐻 ) : 𝑃 –1-1-onto→ 𝑃 ∧ ∀ 𝑎 ∈ 𝑃 ∀ 𝑏 ∈ 𝑃 ( ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑎 ) − ( ( 𝐹 ∘ 𝐻 ) ‘ 𝑏 ) ) = ( 𝑎 − 𝑏 ) ) ) ) |
31 |
9 28 30
|
mpbir2and |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐻 ) ∈ ( 𝐺 Ismt 𝐺 ) ) |