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Theorem mirval 25296
Description: Value of the point inversion function 𝑆. 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)
Assertion
Ref Expression
mirval (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐺,𝑧   𝑦,𝐼,𝑧   𝑦,𝑃,𝑧   𝜑,𝑦,𝑧   𝑦, ,𝑧
Allowed substitution hints:   𝑆(𝑦,𝑧)   𝐿(𝑦,𝑧)

Proof of Theorem mirval
Dummy variables 𝑥 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mirval.s . . 3 𝑆 = (pInvG‘𝐺)
2 df-mir 25294 . . . . 5 pInvG = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔) ↦ (𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦))))))
32a1i 11 . . . 4 (𝜑 → pInvG = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔) ↦ (𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦)))))))
4 fveq2 6088 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5 mirval.p . . . . . . 7 𝑃 = (Base‘𝐺)
64, 5syl6eqr 2661 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
7 fveq2 6088 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
8 mirval.d . . . . . . . . . . . 12 = (dist‘𝐺)
97, 8syl6eqr 2661 . . . . . . . . . . 11 (𝑔 = 𝐺 → (dist‘𝑔) = )
109oveqd 6544 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑧) = (𝑥 𝑧))
119oveqd 6544 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑦) = (𝑥 𝑦))
1210, 11eqeq12d 2624 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ↔ (𝑥 𝑧) = (𝑥 𝑦)))
13 fveq2 6088 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
14 mirval.i . . . . . . . . . . . 12 𝐼 = (Itv‘𝐺)
1513, 14syl6eqr 2661 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
1615oveqd 6544 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑧(Itv‘𝑔)𝑦) = (𝑧𝐼𝑦))
1716eleq2d 2672 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥 ∈ (𝑧(Itv‘𝑔)𝑦) ↔ 𝑥 ∈ (𝑧𝐼𝑦)))
1812, 17anbi12d 742 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦)) ↔ ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))
196, 18riotaeqbidv 6492 . . . . . . 7 (𝑔 = 𝐺 → (𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦))) = (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))
206, 19mpteq12dv 4657 . . . . . 6 (𝑔 = 𝐺 → (𝑦 ∈ (Base‘𝑔) ↦ (𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦)))) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))))
216, 20mpteq12dv 4657 . . . . 5 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔) ↦ (𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦))))) = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
2221adantl 480 . . . 4 ((𝜑𝑔 = 𝐺) → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔) ↦ (𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦))))) = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
23 mirval.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
24 elex 3184 . . . . 5 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
2523, 24syl 17 . . . 4 (𝜑𝐺 ∈ V)
26 fvex 6098 . . . . . . 7 (Base‘𝐺) ∈ V
275, 26eqeltri 2683 . . . . . 6 𝑃 ∈ V
2827mptex 6368 . . . . 5 (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))) ∈ V
2928a1i 11 . . . 4 (𝜑 → (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))) ∈ V)
303, 22, 25, 29fvmptd 6182 . . 3 (𝜑 → (pInvG‘𝐺) = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
311, 30syl5eq 2655 . 2 (𝜑𝑆 = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
32 simpll 785 . . . . . . . 8 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → 𝑥 = 𝐴)
3332oveq1d 6542 . . . . . . 7 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑧) = (𝐴 𝑧))
3432oveq1d 6542 . . . . . . 7 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑦) = (𝐴 𝑦))
3533, 34eqeq12d 2624 . . . . . 6 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → ((𝑥 𝑧) = (𝑥 𝑦) ↔ (𝐴 𝑧) = (𝐴 𝑦)))
3632eleq1d 2671 . . . . . 6 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝑦)))
3735, 36anbi12d 742 . . . . 5 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3837riotabidva 6505 . . . 4 ((𝑥 = 𝐴𝑦𝑃) → (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3938mpteq2dva 4666 . . 3 (𝑥 = 𝐴 → (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
4039adantl 480 . 2 ((𝜑𝑥 = 𝐴) → (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
41 mirval.a . 2 (𝜑𝐴𝑃)
4227mptex 6368 . . 3 (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V
4342a1i 11 . 2 (𝜑 → (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V)
4431, 40, 41, 43fvmptd 6182 1 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
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 15644  distcds 15726  TarskiGcstrkg 25074  Itvcitv 25080  LineGclng 25081  pInvGcmir 25293
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 2033  ax-13 2233  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 25294
This theorem is referenced by:  mirfv  25297  mirf  25301
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