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Theorem mirval 25841
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 25839 . . . . 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 6375 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5 mirval.p . . . . . . 7 𝑃 = (Base‘𝐺)
64, 5syl6eqr 2817 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
7 fveq2 6375 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
8 mirval.d . . . . . . . . . . . 12 = (dist‘𝐺)
97, 8syl6eqr 2817 . . . . . . . . . . 11 (𝑔 = 𝐺 → (dist‘𝑔) = )
109oveqd 6859 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑧) = (𝑥 𝑧))
119oveqd 6859 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑦) = (𝑥 𝑦))
1210, 11eqeq12d 2780 . . . . . . . . 9 (𝑔 = 𝐺 → ((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ↔ (𝑥 𝑧) = (𝑥 𝑦)))
13 fveq2 6375 . . . . . . . . . . . 12 (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
14 mirval.i . . . . . . . . . . . 12 𝐼 = (Itv‘𝐺)
1513, 14syl6eqr 2817 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
1615oveqd 6859 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑧(Itv‘𝑔)𝑦) = (𝑧𝐼𝑦))
1716eleq2d 2830 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥 ∈ (𝑧(Itv‘𝑔)𝑦) ↔ 𝑥 ∈ (𝑧𝐼𝑦)))
1812, 17anbi12d 624 . . . . . . . 8 (𝑔 = 𝐺 → (((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦)) ↔ ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))
196, 18riotaeqbidv 6806 . . . . . . 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 4892 . . . . . 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 4892 . . . . 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 473 . . . 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 3365 . . . . 5 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
2523, 24syl 17 . . . 4 (𝜑𝐺 ∈ V)
265fvexi 6389 . . . . . 6 𝑃 ∈ V
2726mptex 6679 . . . . 5 (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))) ∈ V
2827a1i 11 . . . 4 (𝜑 → (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))) ∈ V)
293, 22, 25, 28fvmptd 6477 . . 3 (𝜑 → (pInvG‘𝐺) = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
301, 29syl5eq 2811 . 2 (𝜑𝑆 = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
31 simpll 783 . . . . . . . 8 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → 𝑥 = 𝐴)
3231oveq1d 6857 . . . . . . 7 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑧) = (𝐴 𝑧))
3331oveq1d 6857 . . . . . . 7 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑦) = (𝐴 𝑦))
3432, 33eqeq12d 2780 . . . . . 6 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → ((𝑥 𝑧) = (𝑥 𝑦) ↔ (𝐴 𝑧) = (𝐴 𝑦)))
3531eleq1d 2829 . . . . . 6 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝑦)))
3634, 35anbi12d 624 . . . . 5 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3736riotabidva 6819 . . . 4 ((𝑥 = 𝐴𝑦𝑃) → (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3837mpteq2dva 4903 . . 3 (𝑥 = 𝐴 → (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
3938adantl 473 . 2 ((𝜑𝑥 = 𝐴) → (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
40 mirval.a . 2 (𝜑𝐴𝑃)
4126mptex 6679 . . 3 (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V
4241a1i 11 . 2 (𝜑 → (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V)
4330, 39, 40, 42fvmptd 6477 1 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  cmpt 4888  cfv 6068  crio 6802  (class class class)co 6842  Basecbs 16130  distcds 16223  TarskiGcstrkg 25620  Itvcitv 25626  LineGclng 25627  pInvGcmir 25838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-mir 25839
This theorem is referenced by:  mirfv  25842  mirf  25846
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