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Theorem mirval 26374
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 26372 . . . 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 6669 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 mirval.p . . . . . 6 𝑃 = (Base‘𝐺)
53, 4syl6eqr 2879 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
6 fveq2 6669 . . . . . . . . . . 11 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
7 mirval.d . . . . . . . . . . 11 = (dist‘𝐺)
86, 7syl6eqr 2879 . . . . . . . . . 10 (𝑔 = 𝐺 → (dist‘𝑔) = )
98oveqd 7167 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑧) = (𝑥 𝑧))
108oveqd 7167 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑦) = (𝑥 𝑦))
119, 10eqeq12d 2842 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ↔ (𝑥 𝑧) = (𝑥 𝑦)))
12 fveq2 6669 . . . . . . . . . . 11 (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
13 mirval.i . . . . . . . . . . 11 𝐼 = (Itv‘𝐺)
1412, 13syl6eqr 2879 . . . . . . . . . 10 (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
1514oveqd 7167 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑧(Itv‘𝑔)𝑦) = (𝑧𝐼𝑦))
1615eleq2d 2903 . . . . . . . 8 (𝑔 = 𝐺 → (𝑥 ∈ (𝑧(Itv‘𝑔)𝑦) ↔ 𝑥 ∈ (𝑧𝐼𝑦)))
1711, 16anbi12d 630 . . . . . . 7 (𝑔 = 𝐺 → (((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦)) ↔ ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))
185, 17riotaeqbidv 7111 . . . . . 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 5148 . . . . 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 5148 . . . 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 3520 . . . 4 (𝜑𝐺 ∈ V)
234fvexi 6683 . . . . . 6 𝑃 ∈ V
2423mptex 6983 . . . . 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 6789 . . 3 (𝜑 → (pInvG‘𝐺) = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
271, 26syl5eq 2873 . 2 (𝜑𝑆 = (𝑥𝑃 ↦ (𝑦𝑃 ↦ (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
28 simpll 763 . . . . . . . 8 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → 𝑥 = 𝐴)
2928oveq1d 7165 . . . . . . 7 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑧) = (𝐴 𝑧))
3028oveq1d 7165 . . . . . . 7 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 𝑦) = (𝐴 𝑦))
3129, 30eqeq12d 2842 . . . . . 6 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → ((𝑥 𝑧) = (𝑥 𝑦) ↔ (𝐴 𝑧) = (𝐴 𝑦)))
3228eleq1d 2902 . . . . . 6 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝑦)))
3331, 32anbi12d 630 . . . . 5 (((𝑥 = 𝐴𝑦𝑃) ∧ 𝑧𝑃) → (((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3433riotabidva 7127 . . . 4 ((𝑥 = 𝐴𝑦𝑃) → (𝑧𝑃 ((𝑥 𝑧) = (𝑥 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))) = (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3534mpteq2dva 5158 . . 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 6983 . . 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 6773 1 (𝜑 → (𝑆𝐴) = (𝑦𝑃 ↦ (𝑧𝑃 ((𝐴 𝑧) = (𝐴 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  cmpt 5143  cfv 6354  crio 7107  (class class class)co 7150  Basecbs 16478  distcds 16569  TarskiGcstrkg 26149  Itvcitv 26155  LineGclng 26156  pInvGcmir 26371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-mir 26372
This theorem is referenced by:  mirfv  26375  mirf  26379
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