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Theorem hpgcom 27806
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 = (Baseβ€˜πΊ)
hpgid.i 𝐼 = (Itvβ€˜πΊ)
hpgid.l 𝐿 = (LineGβ€˜πΊ)
hpgid.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
hpgid.d (πœ‘ β†’ 𝐷 ∈ ran 𝐿)
hpgid.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
hpgid.o 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
hpgcom.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
hpgcom.1 (πœ‘ β†’ 𝐴((hpGβ€˜πΊ)β€˜π·)𝐡)
Assertion
Ref Expression
hpgcom (πœ‘ β†’ 𝐡((hpGβ€˜πΊ)β€˜π·)𝐴)
Distinct variable groups:   𝑑,𝐴   𝑑,𝐡   𝐷,π‘Ž,𝑏,𝑑   𝐺,π‘Ž,𝑏,𝑑   𝐼,π‘Ž,𝑏,𝑑   𝑂,π‘Ž,𝑏,𝑑   𝑃,π‘Ž,𝑏,𝑑   πœ‘,𝑑
Allowed substitution hints:   πœ‘(π‘Ž,𝑏)   𝐴(π‘Ž,𝑏)   𝐡(π‘Ž,𝑏)   𝐿(𝑑,π‘Ž,𝑏)

Proof of Theorem hpgcom
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 hpgcom.1 . 2 (πœ‘ β†’ 𝐴((hpGβ€˜πΊ)β€˜π·)𝐡)
2 ancom 461 . . . . 5 ((𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐) ↔ (𝐡𝑂𝑐 ∧ 𝐴𝑂𝑐))
32a1i 11 . . . 4 (πœ‘ β†’ ((𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐) ↔ (𝐡𝑂𝑐 ∧ 𝐴𝑂𝑐)))
43rexbidv 3177 . . 3 (πœ‘ β†’ (βˆƒπ‘ ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐) ↔ βˆƒπ‘ ∈ 𝑃 (𝐡𝑂𝑐 ∧ 𝐴𝑂𝑐)))
5 hpgid.p . . . 4 𝑃 = (Baseβ€˜πΊ)
6 hpgid.i . . . 4 𝐼 = (Itvβ€˜πΊ)
7 hpgid.l . . . 4 𝐿 = (LineGβ€˜πΊ)
8 hpgid.o . . . 4 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
9 hpgid.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG)
10 hpgid.d . . . 4 (πœ‘ β†’ 𝐷 ∈ ran 𝐿)
11 hpgid.a . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑃)
12 hpgcom.b . . . 4 (πœ‘ β†’ 𝐡 ∈ 𝑃)
135, 6, 7, 8, 9, 10, 11, 12hpgbr 27799 . . 3 (πœ‘ β†’ (𝐴((hpGβ€˜πΊ)β€˜π·)𝐡 ↔ βˆƒπ‘ ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐡𝑂𝑐)))
145, 6, 7, 8, 9, 10, 12, 11hpgbr 27799 . . 3 (πœ‘ β†’ (𝐡((hpGβ€˜πΊ)β€˜π·)𝐴 ↔ βˆƒπ‘ ∈ 𝑃 (𝐡𝑂𝑐 ∧ 𝐴𝑂𝑐)))
154, 13, 143bitr4d 310 . 2 (πœ‘ β†’ (𝐴((hpGβ€˜πΊ)β€˜π·)𝐡 ↔ 𝐡((hpGβ€˜πΊ)β€˜π·)𝐴))
161, 15mpbid 231 1 (πœ‘ β†’ 𝐡((hpGβ€˜πΊ)β€˜π·)𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3069   βˆ– cdif 3925   class class class wbr 5125  {copab 5187  ran crn 5654  β€˜cfv 6516  (class class class)co 7377  Basecbs 17109  TarskiGcstrkg 27466  Itvcitv 27472  LineGclng 27473  hpGchpg 27796
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-hpg 27797
This theorem is referenced by:  trgcopyeulem  27844  tgasa1  27897
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