Theorem hpgcom 26812

Description: The half-plane relation commutes. 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 464	. . . . 5 ⊢ ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐))
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐)))
4	3	rexbidv 3206	. . 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 26805	. . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
14	5, 6, 7, 8, 9, 10, 12, 11	hpgbr 26805	. . 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)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∃wrex 3052   ∖ cdif 3850   class class class wbr 5039  {copab 5101  ran crn 5537  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  TarskiGcstrkg 26475  Itvcitv 26481  LineGclng 26482  hpGchpg 26802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-hpg 26803

This theorem is referenced by:  trgcopyeulem  26850  tgasa1  26903