Metamath Proof Explorer


Theorem motco

Description: The composition of two motions is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019)

Ref Expression
Hypotheses ismot.p 𝑃 = ( Base ‘ 𝐺 )
ismot.m = ( dist ‘ 𝐺 )
motgrp.1 ( 𝜑𝐺𝑉 )
motco.2 ( 𝜑𝐹 ∈ ( 𝐺 Ismt 𝐺 ) )
motco.3 ( 𝜑𝐻 ∈ ( 𝐺 Ismt 𝐺 ) )
Assertion motco ( 𝜑 → ( 𝐹𝐻 ) ∈ ( 𝐺 Ismt 𝐺 ) )

Proof

Step Hyp Ref Expression
1 ismot.p 𝑃 = ( Base ‘ 𝐺 )
2 ismot.m = ( dist ‘ 𝐺 )
3 motgrp.1 ( 𝜑𝐺𝑉 )
4 motco.2 ( 𝜑𝐹 ∈ ( 𝐺 Ismt 𝐺 ) )
5 motco.3 ( 𝜑𝐻 ∈ ( 𝐺 Ismt 𝐺 ) )
6 1 2 3 4 motf1o ( 𝜑𝐹 : 𝑃1-1-onto𝑃 )
7 1 2 3 5 motf1o ( 𝜑𝐻 : 𝑃1-1-onto𝑃 )
8 f1oco ( ( 𝐹 : 𝑃1-1-onto𝑃𝐻 : 𝑃1-1-onto𝑃 ) → ( 𝐹𝐻 ) : 𝑃1-1-onto𝑃 )
9 6 7 8 syl2anc ( 𝜑 → ( 𝐹𝐻 ) : 𝑃1-1-onto𝑃 )
10 f1of ( 𝐻 : 𝑃1-1-onto𝑃𝐻 : 𝑃𝑃 )
11 7 10 syl ( 𝜑𝐻 : 𝑃𝑃 )
12 11 adantr ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → 𝐻 : 𝑃𝑃 )
13 simprl ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → 𝑎𝑃 )
14 fvco3 ( ( 𝐻 : 𝑃𝑃𝑎𝑃 ) → ( ( 𝐹𝐻 ) ‘ 𝑎 ) = ( 𝐹 ‘ ( 𝐻𝑎 ) ) )
15 12 13 14 syl2anc ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → ( ( 𝐹𝐻 ) ‘ 𝑎 ) = ( 𝐹 ‘ ( 𝐻𝑎 ) ) )
16 simprr ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → 𝑏𝑃 )
17 fvco3 ( ( 𝐻 : 𝑃𝑃𝑏𝑃 ) → ( ( 𝐹𝐻 ) ‘ 𝑏 ) = ( 𝐹 ‘ ( 𝐻𝑏 ) ) )
18 12 16 17 syl2anc ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → ( ( 𝐹𝐻 ) ‘ 𝑏 ) = ( 𝐹 ‘ ( 𝐻𝑏 ) ) )
19 15 18 oveq12d ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → ( ( ( 𝐹𝐻 ) ‘ 𝑎 ) ( ( 𝐹𝐻 ) ‘ 𝑏 ) ) = ( ( 𝐹 ‘ ( 𝐻𝑎 ) ) ( 𝐹 ‘ ( 𝐻𝑏 ) ) ) )
20 3 adantr ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → 𝐺𝑉 )
21 12 13 ffvelrnd ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → ( 𝐻𝑎 ) ∈ 𝑃 )
22 12 16 ffvelrnd ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → ( 𝐻𝑏 ) ∈ 𝑃 )
23 4 adantr ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → 𝐹 ∈ ( 𝐺 Ismt 𝐺 ) )
24 1 2 20 21 22 23 motcgr ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → ( ( 𝐹 ‘ ( 𝐻𝑎 ) ) ( 𝐹 ‘ ( 𝐻𝑏 ) ) ) = ( ( 𝐻𝑎 ) ( 𝐻𝑏 ) ) )
25 5 adantr ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → 𝐻 ∈ ( 𝐺 Ismt 𝐺 ) )
26 1 2 20 13 16 25 motcgr ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → ( ( 𝐻𝑎 ) ( 𝐻𝑏 ) ) = ( 𝑎 𝑏 ) )
27 19 24 26 3eqtrd ( ( 𝜑 ∧ ( 𝑎𝑃𝑏𝑃 ) ) → ( ( ( 𝐹𝐻 ) ‘ 𝑎 ) ( ( 𝐹𝐻 ) ‘ 𝑏 ) ) = ( 𝑎 𝑏 ) )
28 27 ralrimivva ( 𝜑 → ∀ 𝑎𝑃𝑏𝑃 ( ( ( 𝐹𝐻 ) ‘ 𝑎 ) ( ( 𝐹𝐻 ) ‘ 𝑏 ) ) = ( 𝑎 𝑏 ) )
29 1 2 ismot ( 𝐺𝑉 → ( ( 𝐹𝐻 ) ∈ ( 𝐺 Ismt 𝐺 ) ↔ ( ( 𝐹𝐻 ) : 𝑃1-1-onto𝑃 ∧ ∀ 𝑎𝑃𝑏𝑃 ( ( ( 𝐹𝐻 ) ‘ 𝑎 ) ( ( 𝐹𝐻 ) ‘ 𝑏 ) ) = ( 𝑎 𝑏 ) ) ) )
30 3 29 syl ( 𝜑 → ( ( 𝐹𝐻 ) ∈ ( 𝐺 Ismt 𝐺 ) ↔ ( ( 𝐹𝐻 ) : 𝑃1-1-onto𝑃 ∧ ∀ 𝑎𝑃𝑏𝑃 ( ( ( 𝐹𝐻 ) ‘ 𝑎 ) ( ( 𝐹𝐻 ) ‘ 𝑏 ) ) = ( 𝑎 𝑏 ) ) ) )
31 9 28 30 mpbir2and ( 𝜑 → ( 𝐹𝐻 ) ∈ ( 𝐺 Ismt 𝐺 ) )