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

Theorem istrkgc 27972
Description: Property of being a Tarski geometry - congruence part. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
istrkg.p 𝑃 = (Baseβ€˜πΊ)
istrkg.d βˆ’ = (distβ€˜πΊ)
istrkg.i 𝐼 = (Itvβ€˜πΊ)
Assertion
Ref Expression
istrkgc (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 (π‘₯ βˆ’ 𝑦) = (𝑦 βˆ’ π‘₯) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 ((π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧) β†’ π‘₯ = 𝑦))))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝐼   π‘₯,𝑃,𝑦,𝑧   π‘₯, βˆ’ ,𝑦,𝑧
Allowed substitution hints:   𝐺(π‘₯,𝑦,𝑧)

Proof of Theorem istrkgc
Dummy variables 𝑓 𝑑 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istrkg.p . . 3 𝑃 = (Baseβ€˜πΊ)
2 istrkg.d . . 3 βˆ’ = (distβ€˜πΊ)
3 simpl 481 . . . . 5 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ 𝑝 = 𝑃)
4 simpr 483 . . . . . . . 8 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ 𝑑 = βˆ’ )
54oveqd 7428 . . . . . . 7 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ (π‘₯𝑑𝑦) = (π‘₯ βˆ’ 𝑦))
64oveqd 7428 . . . . . . 7 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ (𝑦𝑑π‘₯) = (𝑦 βˆ’ π‘₯))
75, 6eqeq12d 2746 . . . . . 6 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ ((π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ↔ (π‘₯ βˆ’ 𝑦) = (𝑦 βˆ’ π‘₯)))
83, 7raleqbidv 3340 . . . . 5 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ (βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ↔ βˆ€π‘¦ ∈ 𝑃 (π‘₯ βˆ’ 𝑦) = (𝑦 βˆ’ π‘₯)))
93, 8raleqbidv 3340 . . . 4 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ (βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ↔ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 (π‘₯ βˆ’ 𝑦) = (𝑦 βˆ’ π‘₯)))
104oveqd 7428 . . . . . . . . 9 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ (𝑧𝑑𝑧) = (𝑧 βˆ’ 𝑧))
115, 10eqeq12d 2746 . . . . . . . 8 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) ↔ (π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧)))
1211imbi1d 340 . . . . . . 7 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ (((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦) ↔ ((π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧) β†’ π‘₯ = 𝑦)))
133, 12raleqbidv 3340 . . . . . 6 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ (βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦) ↔ βˆ€π‘§ ∈ 𝑃 ((π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧) β†’ π‘₯ = 𝑦)))
143, 13raleqbidv 3340 . . . . 5 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ (βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦) ↔ βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 ((π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧) β†’ π‘₯ = 𝑦)))
153, 14raleqbidv 3340 . . . 4 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ (βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦) ↔ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 ((π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧) β†’ π‘₯ = 𝑦)))
169, 15anbi12d 629 . . 3 ((𝑝 = 𝑃 ∧ 𝑑 = βˆ’ ) β†’ ((βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦)) ↔ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 (π‘₯ βˆ’ 𝑦) = (𝑦 βˆ’ π‘₯) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 ((π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧) β†’ π‘₯ = 𝑦))))
171, 2, 16sbcie2s 17098 . 2 (𝑓 = 𝐺 β†’ ([(Baseβ€˜π‘“) / 𝑝][(distβ€˜π‘“) / 𝑑](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦)) ↔ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 (π‘₯ βˆ’ 𝑦) = (𝑦 βˆ’ π‘₯) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 ((π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧) β†’ π‘₯ = 𝑦))))
18 df-trkgc 27966 . 2 TarskiGC = {𝑓 ∣ [(Baseβ€˜π‘“) / 𝑝][(distβ€˜π‘“) / 𝑑](βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 (π‘₯𝑑𝑦) = (𝑦𝑑π‘₯) ∧ βˆ€π‘₯ ∈ 𝑝 βˆ€π‘¦ ∈ 𝑝 βˆ€π‘§ ∈ 𝑝 ((π‘₯𝑑𝑦) = (𝑧𝑑𝑧) β†’ π‘₯ = 𝑦))}
1917, 18elab4g 3672 1 (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 (π‘₯ βˆ’ 𝑦) = (𝑦 βˆ’ π‘₯) ∧ βˆ€π‘₯ ∈ 𝑃 βˆ€π‘¦ ∈ 𝑃 βˆ€π‘§ ∈ 𝑃 ((π‘₯ βˆ’ 𝑦) = (𝑧 βˆ’ 𝑧) β†’ π‘₯ = 𝑦))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  Vcvv 3472  [wsbc 3776  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  distcds 17210  TarskiGCcstrkgc 27946  Itvcitv 27951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-trkgc 27966
This theorem is referenced by:  axtgcgrrflx  27980  axtgcgrid  27981  f1otrg  28389  xmstrkgc  28410  eengtrkg  28511
  Copyright terms: Public domain W3C validator