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Theorem mirfv 26594
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 26593 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
101, 9syl5eq 2785 . 2 (𝜑𝑀 = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
11 simplr 769 . . . . . 6 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → 𝑦 = 𝐵)
1211oveq2d 7180 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 𝑦) = (𝐴 𝐵))
1312eqeq2d 2749 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → ((𝐴 𝑧) = (𝐴 𝑦) ↔ (𝐴 𝑧) = (𝐴 𝐵)))
1411oveq2d 7180 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵))
1514eleq2d 2818 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵)))
1613, 15anbi12d 634 . . 3 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
1716riotabidva 7141 . 2 ((𝜑𝑦 = 𝐵) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
18 mirfv.b . 2 (𝜑𝐵𝑃)
19 riotaex 7125 . . 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 6776 1 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  Vcvv 3397  cmpt 5107  cfv 6333  crio 7120  (class class class)co 7164  Basecbs 16579  distcds 16670  TarskiGcstrkg 26368  Itvcitv 26374  LineGclng 26375  pInvGcmir 26590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-mir 26591
This theorem is referenced by:  mircgr  26595  mirbtwn  26596  ismir  26597  mirf  26598  mireq  26603
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