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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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