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Theorem mirfv 25900
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 25899 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
101, 9syl5eq 2843 . 2 (𝜑𝑀 = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
11 simplr 786 . . . . . 6 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → 𝑦 = 𝐵)
1211oveq2d 6892 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 𝑦) = (𝐴 𝐵))
1312eqeq2d 2807 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → ((𝐴 𝑧) = (𝐴 𝑦) ↔ (𝐴 𝑧) = (𝐴 𝐵)))
1411oveq2d 6892 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵))
1514eleq2d 2862 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵)))
1613, 15anbi12d 625 . . 3 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
1716riotabidva 6853 . 2 ((𝜑𝑦 = 𝐵) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
18 mirfv.b . 2 (𝜑𝐵𝑃)
19 riotaex 6841 . . 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 6511 1 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3383  cmpt 4920  cfv 6099  crio 6836  (class class class)co 6876  Basecbs 16181  distcds 16273  TarskiGcstrkg 25678  Itvcitv 25684  LineGclng 25685  pInvGcmir 25896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-mir 25897
This theorem is referenced by:  mircgr  25901  mirbtwn  25902  ismir  25903  mirf  25904  mireq  25909
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