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

Theorem ismir 28476
Description: Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Base‘𝐺)
mirval.d = (dist‘𝐺)
mirval.i 𝐼 = (Itv‘𝐺)
mirval.l 𝐿 = (LineG‘𝐺)
mirval.s 𝑆 = (pInvG‘𝐺)
mirval.g (𝜑𝐺 ∈ TarskiG)
mirval.a (𝜑𝐴𝑃)
mirfv.m 𝑀 = (𝑆𝐴)
mirfv.b (𝜑𝐵𝑃)
ismir.1 (𝜑𝐶𝑃)
ismir.2 (𝜑 → (𝐴 𝐶) = (𝐴 𝐵))
ismir.3 (𝜑𝐴 ∈ (𝐶𝐼𝐵))
Assertion
Ref Expression
ismir (𝜑𝐶 = (𝑀𝐵))

Proof of Theorem ismir
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mirval.p . . 3 𝑃 = (Base‘𝐺)
2 mirval.d . . 3 = (dist‘𝐺)
3 mirval.i . . 3 𝐼 = (Itv‘𝐺)
4 mirval.l . . 3 𝐿 = (LineG‘𝐺)
5 mirval.s . . 3 𝑆 = (pInvG‘𝐺)
6 mirval.g . . 3 (𝜑𝐺 ∈ TarskiG)
7 mirval.a . . 3 (𝜑𝐴𝑃)
8 mirfv.m . . 3 𝑀 = (𝑆𝐴)
9 mirfv.b . . 3 (𝜑𝐵𝑃)
101, 2, 3, 4, 5, 6, 7, 8, 9mirfv 28473 . 2 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
11 ismir.2 . . 3 (𝜑 → (𝐴 𝐶) = (𝐴 𝐵))
12 ismir.3 . . 3 (𝜑𝐴 ∈ (𝐶𝐼𝐵))
13 ismir.1 . . . 4 (𝜑𝐶𝑃)
141, 2, 3, 6, 9, 7mirreu3 28471 . . . 4 (𝜑 → ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
15 oveq2 7428 . . . . . . 7 (𝑧 = 𝐶 → (𝐴 𝑧) = (𝐴 𝐶))
1615eqeq1d 2730 . . . . . 6 (𝑧 = 𝐶 → ((𝐴 𝑧) = (𝐴 𝐵) ↔ (𝐴 𝐶) = (𝐴 𝐵)))
17 oveq1 7427 . . . . . . 7 (𝑧 = 𝐶 → (𝑧𝐼𝐵) = (𝐶𝐼𝐵))
1817eleq2d 2815 . . . . . 6 (𝑧 = 𝐶 → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ (𝐶𝐼𝐵)))
1916, 18anbi12d 631 . . . . 5 (𝑧 = 𝐶 → (((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵))))
2019riota2 7402 . . . 4 ((𝐶𝑃 ∧ ∃!𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) → (((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶))
2113, 14, 20syl2anc 583 . . 3 (𝜑 → (((𝐴 𝐶) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶))
2211, 12, 21mpbi2and 711 . 2 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶)
2310, 22eqtr2d 2769 1 (𝜑𝐶 = (𝑀𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  ∃!wreu 3371  cfv 6548  crio 7375  (class class class)co 7420  Basecbs 17180  distcds 17242  TarskiGcstrkg 28244  Itvcitv 28250  LineGclng 28251  pInvGcmir 28469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-trkgc 28265  df-trkgb 28266  df-trkgcb 28267  df-trkg 28270  df-mir 28470
This theorem is referenced by:  mirmir  28479  mireq  28482  mirinv  28483  miriso  28487  mirmir2  28491  mirauto  28501  colmid  28505  krippenlem  28507  midexlem  28509  mideulem2  28551  opphllem  28552  midcom  28599  trgcopyeulem  28622
  Copyright terms: Public domain W3C validator