Theorem hpgcom 29302

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 3190	. . 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 29295	. . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
14	5, 6, 7, 8, 9, 10, 12, 11	hpgbr 29295	. . 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 2147  ∃wrex 3090   ∖ cdif 4048   class class class wbr 5108  {copab 5173  ran crn 5659  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  TarskiGcstrkg 28946  Itvcitv 28952  LineGclng 28953  hpGchpg 29292

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-hpg 29293

This theorem is used by:  plngcplem  29320  plngrotlem1  29322  plngmiropp  29329  nhpmirhp  29333  trgcopyeulem  29369  angmgmaddeu1  29436  tgasa1  29460  prlnghpg  29481  tgextahpl  33126