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Theorem mirfv 26369
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 26368 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
101, 9syl5eq 2865 . 2 (𝜑𝑀 = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
11 simplr 765 . . . . . 6 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → 𝑦 = 𝐵)
1211oveq2d 7161 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 𝑦) = (𝐴 𝐵))
1312eqeq2d 2829 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → ((𝐴 𝑧) = (𝐴 𝑦) ↔ (𝐴 𝑧) = (𝐴 𝐵)))
1411oveq2d 7161 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵))
1514eleq2d 2895 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵)))
1613, 15anbi12d 630 . . 3 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
1716riotabidva 7122 . 2 ((𝜑𝑦 = 𝐵) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
18 mirfv.b . 2 (𝜑𝐵𝑃)
19 riotaex 7107 . . 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 6767 1 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  cmpt 5137  cfv 6348  crio 7102  (class class class)co 7145  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149  LineGclng 26150  pInvGcmir 26365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-mir 26366
This theorem is referenced by:  mircgr  26370  mirbtwn  26371  ismir  26372  mirf  26373  mireq  26378
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