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Theorem mirfv 25596
Description: Value of the point inversion function 𝑀. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-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)
Assertion
Ref Expression
mirfv (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐺   𝑧,𝑀   𝑧,𝐼   𝑧,𝑃   𝜑,𝑧   𝑧,
Allowed substitution hints:   𝑆(𝑧)   𝐿(𝑧)

Proof of Theorem mirfv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mirfv.m . . 3 𝑀 = (𝑆𝐴)
2 mirval.p . . . 4 𝑃 = (Base‘𝐺)
3 mirval.d . . . 4 = (dist‘𝐺)
4 mirval.i . . . 4 𝐼 = (Itv‘𝐺)
5 mirval.l . . . 4 𝐿 = (LineG‘𝐺)
6 mirval.s . . . 4 𝑆 = (pInvG‘𝐺)
7 mirval.g . . . 4 (𝜑𝐺 ∈ TarskiG)
8 mirval.a . . . 4 (𝜑𝐴𝑃)
92, 3, 4, 5, 6, 7, 8mirval 25595 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
101, 9syl5eq 2697 . 2 (𝜑𝑀 = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
11 simplr 807 . . . . . 6 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → 𝑦 = 𝐵)
1211oveq2d 6706 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 𝑦) = (𝐴 𝐵))
1312eqeq2d 2661 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → ((𝐴 𝑧) = (𝐴 𝑦) ↔ (𝐴 𝑧) = (𝐴 𝐵)))
1411oveq2d 6706 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵))
1514eleq2d 2716 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵)))
1613, 15anbi12d 747 . . 3 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
1716riotabidva 6667 . 2 ((𝜑𝑦 = 𝐵) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
18 mirfv.b . 2 (𝜑𝐵𝑃)
19 riotaex 6655 . . 3 (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ V
2019a1i 11 . 2 (𝜑 → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ V)
2110, 17, 18, 20fvmptd 6327 1 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381  pInvGcmir 25592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-mir 25593
This theorem is referenced by:  mircgr  25597  mirbtwn  25598  ismir  25599  mirf  25600  mireq  25605
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