Theorem hpgcom 29080

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

Hypotheses

Ref	Expression
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 ∈ (𝑎𝐼𝑏))}
hpgcom.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
hpgcom.1	⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵)

Assertion

Ref	Expression
hpgcom	⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐴)

Distinct variable groups:   𝑡,𝐴   𝑡,𝐵   𝐷,𝑎,𝑏,𝑡   𝐺,𝑎,𝑏,𝑡   𝐼,𝑎,𝑏,𝑡   𝑂,𝑎,𝑏,𝑡   𝑃,𝑎,𝑏,𝑡   𝜑,𝑡

Allowed substitution hints:   𝜑(𝑎, 𝑏)   𝐴(𝑎, 𝑏)   𝐵(𝑎, 𝑏)   𝐿(𝑡, 𝑎, 𝑏)

Proof of Theorem hpgcom

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hpgcom.1	. 2 ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵)
2		ancom 466	. . . . 5 ⊢ ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐))
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐)))
4	3	rexbidv 3191	. . 3 ⊢ (𝜑 → (∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐)))
5		hpgid.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
6		hpgid.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
7		hpgid.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
8		hpgid.o	. . . 4 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
9		hpgid.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
10		hpgid.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
11		hpgid.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
12		hpgcom.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	5, 6, 7, 8, 9, 10, 11, 12	hpgbr 29073	. . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
14	5, 6, 7, 8, 9, 10, 12, 11	hpgbr 29073	. . 3 ⊢ (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐 ∈ 𝑃 (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐)))
15	4, 13, 14	3bitr4d 314	. 2 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ 𝐵((hpG‘𝐺)‘𝐷)𝐴))
16	1, 15	mpbid 235	1 ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ∖ cdif 3903   class class class wbr 5111  {copab 5175  ran crn 5664  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  TarskiGcstrkg 28727  Itvcitv 28733  LineGclng 28734  hpGchpg 29070

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-hpg 29071

This theorem is used by:  plngcplem  29098  plngrotlem1  29100  plngmiropp  29107  nhpmirhp  29111  trgcopyeulem  29147  tgasa1  29206  prlnghpg  29227