Theorem hpgcom 28822

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 460	. . . . 5 ⊢ ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐))
3	2	a1i 11	. . . 4 ⊢ (𝜑 → ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐)))
4	3	rexbidv 3161	. . 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 28815	. . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)))
14	5, 6, 7, 8, 9, 10, 12, 11	hpgbr 28815	. . 3 ⊢ (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐 ∈ 𝑃 (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐)))
15	4, 13, 14	3bitr4d 311	. 2 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ 𝐵((hpG‘𝐺)‘𝐷)𝐴))
16	1, 15	mpbid 232	1 ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ∖ cdif 3899   class class class wbr 5099  {copab 5161  ran crn 5626  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  TarskiGcstrkg 28482  Itvcitv 28488  LineGclng 28489  hpGchpg 28812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-hpg 28813

This theorem is referenced by:  trgcopyeulem  28860  tgasa1  28913