Theorem hpgcom

Description: The half-plane relation commutes. Theorem 9.12 of Schwabhauser p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020)

	Ref	Expression
Hypotheses	hpgid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	hpgid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	hpgid.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	hpgid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	hpgid.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
	hpgid.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	hpgid.o	⊢ 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
	hpgcom.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	hpgcom.1	⊢ ( 𝜑 → 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 )
Assertion	hpgcom	⊢ ( 𝜑 → 𝐵 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		hpgid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		hpgid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		hpgid.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
4		hpgid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		hpgid.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
6		hpgid.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
7		hpgid.o	⊢ 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
8		hpgcom.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
9		hpgcom.1	⊢ ( 𝜑 → 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 )
10		ancom	⊢ ( ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ↔ ( 𝐵 𝑂 𝑐 ∧ 𝐴 𝑂 𝑐 ) )
11	10	a1i	⊢ ( 𝜑 → ( ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ↔ ( 𝐵 𝑂 𝑐 ∧ 𝐴 𝑂 𝑐 ) ) )
12	11	rexbidv	⊢ ( 𝜑 → ( ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐵 𝑂 𝑐 ∧ 𝐴 𝑂 𝑐 ) ) )
13	1 2 3 7 4 5 6 8	hpgbr	⊢ ( 𝜑 → ( 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐴 𝑂 𝑐 ∧ 𝐵 𝑂 𝑐 ) ) )
14	1 2 3 7 4 5 8 6	hpgbr	⊢ ( 𝜑 → ( 𝐵 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐴 ↔ ∃ 𝑐 ∈ 𝑃 ( 𝐵 𝑂 𝑐 ∧ 𝐴 𝑂 𝑐 ) ) )
15	12 13 14	3bitr4d	⊢ ( 𝜑 → ( 𝐴 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐵 ↔ 𝐵 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐴 ) )
16	9 15	mpbid	⊢ ( 𝜑 → 𝐵 ( ( hpG ‘ 𝐺 ) ‘ 𝐷 ) 𝐴 )