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Theorem mirval 27597
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 27595 . . . 4 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))))))
3 fveq2 6842 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 mirval.p . . . . . 6 𝑃 = (Base‘𝐺)
53, 4eqtr4di 2794 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
6 fveq2 6842 . . . . . . . . . . 11 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
7 mirval.d . . . . . . . . . . 11 = (dist‘𝐺)
86, 7eqtr4di 2794 . . . . . . . . . 10 (𝑔 = 𝐺 → (dist‘𝑔) = )
98oveqd 7374 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑧) = (𝑥 𝑧))
108oveqd 7374 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑦) = (𝑥 𝑦))
119, 10eqeq12d 2752 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ↔ (𝑥 𝑧) = (𝑥 𝑦)))
12 fveq2 6842 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
13 mirval.i . . . . . . . . . . 11 𝐼 = (Itv‘𝐺)
1412, 13eqtr4di 2794 . . . . . . . . . 10 (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
1514oveqd 7374 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑧(Itv‘𝑔)𝑦) = (𝑧𝐼𝑦))
1615eleq2d 2823 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥 ∈ (𝑧(Itv‘𝑔)𝑦) ↔ 𝑥 ∈ (𝑧𝐼𝑦)))
1711, 16anbi12d 631 . . . . . . 7 (𝑔 = 𝐺 → (((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦)) ↔ ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))
185, 17riotaeqbidv 7316 . . . . . 6 (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))))
195, 18mpteq12dv 5196 . . . . 5 (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)))))
205, 19mpteq12dv 5196 . . . 4 (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))))))
21 mirval.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
2221elexd 3465 . . . 4 (𝜑𝐺 ∈ V)
234fvexi 6856 . . . . . 6 𝑃 ∈ V
2423mptex 7173 . . . . 5 (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))) ∈ V
2524a1i 11 . . . 4 (𝜑 → (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))) ∈ V)
262, 20, 22, 25fvmptd3 6971 . . 3 (𝜑 → (pInvG‘𝐺) = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
271, 26eqtrid 2788 . 2 (𝜑𝑆 = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
28 simpll 765 . . . . . . . 8 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → 𝑥 = 𝐴)
2928oveq1d 7372 . . . . . . 7 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑧) = (𝐴 𝑧))
3028oveq1d 7372 . . . . . . 7 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑦) = (𝐴 𝑦))
3129, 30eqeq12d 2752 . . . . . 6 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → ((𝑥 𝑧) = (𝑥 𝑦) ↔ (𝐴 𝑧) = (𝐴 𝑦)))
3228eleq1d 2822 . . . . . 6 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝑦)))
3331, 32anbi12d 631 . . . . 5 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3433riotabidva 7333 . . . 4 ((𝑥 = 𝐴𝑦𝑃) → (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3534mpteq2dva 5205 . . 3 (𝑥 = 𝐴 → (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
3635adantl 482 . 2 ((𝜑𝑥 = 𝐴) → (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
37 mirval.a . 2 (𝜑𝐴𝑃)
3823mptex 7173 . . 3 (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V
3938a1i 11 . 2 (𝜑 → (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V)
4027, 36, 37, 39fvmptd 6955 1 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cmpt 5188  cfv 6496  crio 7312  (class class class)co 7357  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  Itvcitv 27375  LineGclng 27376  pInvGcmir 27594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-mir 27595
This theorem is referenced by:  mirfv  27598  mirf  27602
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