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Theorem mirfv 25269
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 25268 . . 3 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
101, 9syl5eq 2655 . 2 (𝜑𝑀 = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
11 simplr 787 . . . . . 6 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → 𝑦 = 𝐵)
1211oveq2d 6543 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 𝑦) = (𝐴 𝐵))
1312eqeq2d 2619 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → ((𝐴 𝑧) = (𝐴 𝑦) ↔ (𝐴 𝑧) = (𝐴 𝐵)))
1411oveq2d 6543 . . . . 5 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵))
1514eleq2d 2672 . . . 4 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵)))
1613, 15anbi12d 742 . . 3 (((𝜑𝑦 = 𝐵) ∧ 𝑧𝑃) → (((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
1716riotabidva 6505 . 2 ((𝜑𝑦 = 𝐵) → (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
18 mirfv.b . 2 (𝜑𝐵𝑃)
19 riotaex 6493 . . 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 6182 1 (𝜑 → (𝑀𝐵) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cmpt 4637  cfv 5790  crio 6488  (class class class)co 6527  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053  pInvGcmir 25265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-mir 25266
This theorem is referenced by:  mircgr  25270  mirbtwn  25271  ismir  25272  mirf  25273  mireq  25278
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