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Theorem hpgcom 25704
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 (𝜑𝐺 ∈ TarskiG)
hpgid.d (𝜑𝐷 ∈ ran 𝐿)
hpgid.a (𝜑𝐴𝑃)
hpgid.o 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
hpgcom.b (𝜑𝐵𝑃)
hpgcom.1 (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)
Assertion
Ref Expression
hpgcom (𝜑𝐵((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.
StepHypRef Expression
1 hpgcom.1 . 2 (𝜑𝐴((hpG‘𝐺)‘𝐷)𝐵)
2 ancom 465 . . . . 5 ((𝐴𝑂𝑐𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐𝐴𝑂𝑐))
32a1i 11 . . . 4 (𝜑 → ((𝐴𝑂𝑐𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐𝐴𝑂𝑐)))
43rexbidv 3081 . . 3 (𝜑 → (∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐) ↔ ∃𝑐𝑃 (𝐵𝑂𝑐𝐴𝑂𝑐)))
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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
9 hpgid.g . . . 4 (𝜑𝐺 ∈ TarskiG)
10 hpgid.d . . . 4 (𝜑𝐷 ∈ ran 𝐿)
11 hpgid.a . . . 4 (𝜑𝐴𝑃)
12 hpgcom.b . . . 4 (𝜑𝐵𝑃)
135, 6, 7, 8, 9, 10, 11, 12hpgbr 25697 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
145, 6, 7, 8, 9, 10, 12, 11hpgbr 25697 . . 3 (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐𝑃 (𝐵𝑂𝑐𝐴𝑂𝑐)))
154, 13, 143bitr4d 300 . 2 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵𝐵((hpG‘𝐺)‘𝐷)𝐴))
161, 15mpbid 222 1 (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  cdif 3604   class class class wbr 4685  {copab 4745  ran crn 5144  cfv 5926  (class class class)co 6690  Basecbs 15904  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381  hpGchpg 25694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-hpg 25695
This theorem is referenced by:  trgcopyeulem  25742  tgasa1  25784
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