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Theorem hpgcom 28790
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 460 . . . . 5 ((𝐴𝑂𝑐𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐𝐴𝑂𝑐))
32a1i 11 . . . 4 (𝜑 → ((𝐴𝑂𝑐𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐𝐴𝑂𝑐)))
43rexbidv 3177 . . 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 28783 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
145, 6, 7, 8, 9, 10, 12, 11hpgbr 28783 . . 3 (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐𝑃 (𝐵𝑂𝑐𝐴𝑂𝑐)))
154, 13, 143bitr4d 311 . 2 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵𝐵((hpG‘𝐺)‘𝐷)𝐴))
161, 15mpbid 232 1 (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  cdif 3960   class class class wbr 5148  {copab 5210  ran crn 5690  cfv 6563  (class class class)co 7431  Basecbs 17245  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457  hpGchpg 28780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-hpg 28781
This theorem is referenced by:  trgcopyeulem  28828  tgasa1  28881
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