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Theorem hpgcom 26565
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 464 . . . . 5 ((𝐴𝑂𝑐𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐𝐴𝑂𝑐))
32a1i 11 . . . 4 (𝜑 → ((𝐴𝑂𝑐𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐𝐴𝑂𝑐)))
43rexbidv 3259 . . 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 26558 . . 3 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐𝑃 (𝐴𝑂𝑐𝐵𝑂𝑐)))
145, 6, 7, 8, 9, 10, 12, 11hpgbr 26558 . . 3 (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐𝑃 (𝐵𝑂𝑐𝐴𝑂𝑐)))
154, 13, 143bitr4d 314 . 2 (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵𝐵((hpG‘𝐺)‘𝐷)𝐴))
161, 15mpbid 235 1 (𝜑𝐵((hpG‘𝐺)‘𝐷)𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wrex 3110  cdif 3881   class class class wbr 5033  {copab 5095  ran crn 5524  cfv 6328  (class class class)co 7139  Basecbs 16479  TarskiGcstrkg 26228  Itvcitv 26234  LineGclng 26235  hpGchpg 26555
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-hpg 26556
This theorem is referenced by:  trgcopyeulem  26603  tgasa1  26656
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