Metamath Proof Explorer

Theorem ragcom

Description: Commutative rule for right angles. Theorem 8.2 of Schwabhauser p. 57. (Contributed by Thierry Arnoux, 25-Aug-2019)

Ref Expression
Hypotheses israg.p ⊢ 𝑃 = ( Base ‘ 𝐺 )
israg.d ⊢ − = ( dist ‘ 𝐺 )
israg.i ⊢ 𝐼 = ( Itv ‘ 𝐺 )
israg.l ⊢ 𝐿 = ( LineG ‘ 𝐺 )
israg.s ⊢ 𝑆 = ( pInvG ‘ 𝐺 )
israg.g ⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
israg.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
israg.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
israg.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
ragcom.1 ⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
Assertion ragcom ( 𝜑 → ⟨“ 𝐶 𝐵 𝐴 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )

Proof

Step Hyp Ref Expression
1 israg.p ⊢ 𝑃 = ( Base ‘ 𝐺 )
2 israg.d ⊢ − = ( dist ‘ 𝐺 )
3 israg.i ⊢ 𝐼 = ( Itv ‘ 𝐺 )
4 israg.l ⊢ 𝐿 = ( LineG ‘ 𝐺 )
5 israg.s ⊢ 𝑆 = ( pInvG ‘ 𝐺 )
6 israg.g ⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
7 israg.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
8 israg.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
9 israg.c ⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
10 ragcom.1 ⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )
11 eqid ⊢ ( 𝑆 ‘ 𝐵 ) = ( 𝑆 ‘ 𝐵 )
12 1 2 3 4 5 6 8 11 9 mircl ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ∈ 𝑃 )
13 1 2 3 4 5 6 7 8 9 israg ⊢ ( 𝜑 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) ) )
14 10 13 mpbid ⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = ( 𝐴 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) )
15 1 2 3 6 7 9 7 12 14 tgcgrcomlr ⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) − 𝐴 ) )
16 1 2 3 4 5 6 8 11 12 7 miriso ⊢ ( 𝜑 → ( ( ( 𝑆 ‘ 𝐵 ) ‘ ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐴 ) ) = ( ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) − 𝐴 ) )
17 1 2 3 4 5 6 8 11 9 mirmir ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐵 ) ‘ ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) = 𝐶 )
18 17 oveq1d ⊢ ( 𝜑 → ( ( ( 𝑆 ‘ 𝐵 ) ‘ ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐶 ) ) − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐴 ) ) = ( 𝐶 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐴 ) ) )
19 15 16 18 3eqtr2d ⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐶 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐴 ) ) )
20 1 2 3 4 5 6 9 8 7 israg ⊢ ( 𝜑 → ( ⟨“ 𝐶 𝐵 𝐴 ”⟩ ∈ ( ∟G ‘ 𝐺 ) ↔ ( 𝐶 − 𝐴 ) = ( 𝐶 − ( ( 𝑆 ‘ 𝐵 ) ‘ 𝐴 ) ) ) )
21 19 20 mpbird ⊢ ( 𝜑 → ⟨“ 𝐶 𝐵 𝐴 ”⟩ ∈ ( ∟G ‘ 𝐺 ) )