Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  hpgcom Structured version   Visualization version   GIF version

Theorem hpgcom 25459
 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 3034 . . 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 25452 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
145, 6, 7, 8, 9, 10, 12, 11hpgbr 25452 . . 3 (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐𝑃 (𝐵𝑂𝑐𝐴𝑂𝑐)))
154, 13, 143bitr4d 299 . 2 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵𝐵((hpG‘𝐺)‘𝐷)𝐴))
161, 15mpbid 221 1 (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   class class class wbr 4583  {copab 4642  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136  hpGchpg 25449 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-hpg 25450 This theorem is referenced by:  trgcopyeulem  25497  tgasa1  25539
 Copyright terms: Public domain W3C validator