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Theorem hpgcom 27538
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 462 . . . . 5 ((𝐴𝑂𝑐𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐𝐴𝑂𝑐))
32a1i 11 . . . 4 (𝜑 → ((𝐴𝑂𝑐𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐𝐴𝑂𝑐)))
43rexbidv 3174 . . 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 27531 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
145, 6, 7, 8, 9, 10, 12, 11hpgbr 27531 . . 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 231 1 (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wrex 3072  cdif 3906   class class class wbr 5104  {copab 5166  ran crn 5633  cfv 6494  (class class class)co 7352  Basecbs 17043  TarskiGcstrkg 27198  Itvcitv 27204  LineGclng 27205  hpGchpg 27528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7355  df-hpg 27529
This theorem is referenced by:  trgcopyeulem  27576  tgasa1  27629
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