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Theorem mirval 27646
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 = (Baseβ€˜πΊ)
mirval.d βˆ’ = (distβ€˜πΊ)
mirval.i 𝐼 = (Itvβ€˜πΊ)
mirval.l 𝐿 = (LineGβ€˜πΊ)
mirval.s 𝑆 = (pInvGβ€˜πΊ)
mirval.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
mirval.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
Assertion
Ref Expression
mirval (πœ‘ β†’ (π‘†β€˜π΄) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐺,𝑧   𝑦,𝐼,𝑧   𝑦,𝑃,𝑧   πœ‘,𝑦,𝑧   𝑦, βˆ’ ,𝑧
Allowed substitution hints:   𝑆(𝑦,𝑧)   𝐿(𝑦,𝑧)

Proof of Theorem mirval
Dummy variables π‘₯ 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mirval.s . . 3 𝑆 = (pInvGβ€˜πΊ)
2 df-mir 27644 . . . 4 pInvG = (𝑔 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘”) ↦ (𝑦 ∈ (Baseβ€˜π‘”) ↦ (℩𝑧 ∈ (Baseβ€˜π‘”)((π‘₯(distβ€˜π‘”)𝑧) = (π‘₯(distβ€˜π‘”)𝑦) ∧ π‘₯ ∈ (𝑧(Itvβ€˜π‘”)𝑦))))))
3 fveq2 6846 . . . . . 6 (𝑔 = 𝐺 β†’ (Baseβ€˜π‘”) = (Baseβ€˜πΊ))
4 mirval.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
53, 4eqtr4di 2791 . . . . 5 (𝑔 = 𝐺 β†’ (Baseβ€˜π‘”) = 𝑃)
6 fveq2 6846 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = (distβ€˜πΊ))
7 mirval.d . . . . . . . . . . 11 βˆ’ = (distβ€˜πΊ)
86, 7eqtr4di 2791 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = βˆ’ )
98oveqd 7378 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘₯(distβ€˜π‘”)𝑧) = (π‘₯ βˆ’ 𝑧))
108oveqd 7378 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘₯(distβ€˜π‘”)𝑦) = (π‘₯ βˆ’ 𝑦))
119, 10eqeq12d 2749 . . . . . . . 8 (𝑔 = 𝐺 β†’ ((π‘₯(distβ€˜π‘”)𝑧) = (π‘₯(distβ€˜π‘”)𝑦) ↔ (π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦)))
12 fveq2 6846 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (Itvβ€˜π‘”) = (Itvβ€˜πΊ))
13 mirval.i . . . . . . . . . . 11 𝐼 = (Itvβ€˜πΊ)
1412, 13eqtr4di 2791 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (Itvβ€˜π‘”) = 𝐼)
1514oveqd 7378 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (𝑧(Itvβ€˜π‘”)𝑦) = (𝑧𝐼𝑦))
1615eleq2d 2820 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘₯ ∈ (𝑧(Itvβ€˜π‘”)𝑦) ↔ π‘₯ ∈ (𝑧𝐼𝑦)))
1711, 16anbi12d 632 . . . . . . 7 (𝑔 = 𝐺 β†’ (((π‘₯(distβ€˜π‘”)𝑧) = (π‘₯(distβ€˜π‘”)𝑦) ∧ π‘₯ ∈ (𝑧(Itvβ€˜π‘”)𝑦)) ↔ ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦))))
185, 17riotaeqbidv 7320 . . . . . 6 (𝑔 = 𝐺 β†’ (℩𝑧 ∈ (Baseβ€˜π‘”)((π‘₯(distβ€˜π‘”)𝑧) = (π‘₯(distβ€˜π‘”)𝑦) ∧ π‘₯ ∈ (𝑧(Itvβ€˜π‘”)𝑦))) = (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦))))
195, 18mpteq12dv 5200 . . . . 5 (𝑔 = 𝐺 β†’ (𝑦 ∈ (Baseβ€˜π‘”) ↦ (℩𝑧 ∈ (Baseβ€˜π‘”)((π‘₯(distβ€˜π‘”)𝑧) = (π‘₯(distβ€˜π‘”)𝑦) ∧ π‘₯ ∈ (𝑧(Itvβ€˜π‘”)𝑦)))) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦)))))
205, 19mpteq12dv 5200 . . . 4 (𝑔 = 𝐺 β†’ (π‘₯ ∈ (Baseβ€˜π‘”) ↦ (𝑦 ∈ (Baseβ€˜π‘”) ↦ (℩𝑧 ∈ (Baseβ€˜π‘”)((π‘₯(distβ€˜π‘”)𝑧) = (π‘₯(distβ€˜π‘”)𝑦) ∧ π‘₯ ∈ (𝑧(Itvβ€˜π‘”)𝑦))))) = (π‘₯ ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦))))))
21 mirval.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ TarskiG)
2221elexd 3467 . . . 4 (πœ‘ β†’ 𝐺 ∈ V)
234fvexi 6860 . . . . . 6 𝑃 ∈ V
2423mptex 7177 . . . . 5 (π‘₯ ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦))))) ∈ V
2524a1i 11 . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦))))) ∈ V)
262, 20, 22, 25fvmptd3 6975 . . 3 (πœ‘ β†’ (pInvGβ€˜πΊ) = (π‘₯ ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦))))))
271, 26eqtrid 2785 . 2 (πœ‘ β†’ 𝑆 = (π‘₯ ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦))))))
28 simpll 766 . . . . . . . 8 (((π‘₯ = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) β†’ π‘₯ = 𝐴)
2928oveq1d 7376 . . . . . . 7 (((π‘₯ = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) β†’ (π‘₯ βˆ’ 𝑧) = (𝐴 βˆ’ 𝑧))
3028oveq1d 7376 . . . . . . 7 (((π‘₯ = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) β†’ (π‘₯ βˆ’ 𝑦) = (𝐴 βˆ’ 𝑦))
3129, 30eqeq12d 2749 . . . . . 6 (((π‘₯ = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) β†’ ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ↔ (𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝑦)))
3228eleq1d 2819 . . . . . 6 (((π‘₯ = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) β†’ (π‘₯ ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝑦)))
3331, 32anbi12d 632 . . . . 5 (((π‘₯ = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) β†’ (((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3433riotabidva 7337 . . . 4 ((π‘₯ = 𝐴 ∧ 𝑦 ∈ 𝑃) β†’ (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦))) = (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
3534mpteq2dva 5209 . . 3 (π‘₯ = 𝐴 β†’ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦)))) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
3635adantl 483 . 2 ((πœ‘ ∧ π‘₯ = 𝐴) β†’ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((π‘₯ βˆ’ 𝑧) = (π‘₯ βˆ’ 𝑦) ∧ π‘₯ ∈ (𝑧𝐼𝑦)))) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
37 mirval.a . 2 (πœ‘ β†’ 𝐴 ∈ 𝑃)
3823mptex 7177 . . 3 (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V
3938a1i 11 . 2 (πœ‘ β†’ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V)
4027, 36, 37, 39fvmptd 6959 1 (πœ‘ β†’ (π‘†β€˜π΄) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 βˆ’ 𝑧) = (𝐴 βˆ’ 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447   ↦ cmpt 5192  β€˜cfv 6500  β„©crio 7316  (class class class)co 7361  Basecbs 17091  distcds 17150  TarskiGcstrkg 27418  Itvcitv 27424  LineGclng 27425  pInvGcmir 27643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-mir 27644
This theorem is referenced by:  mirfv  27647  mirf  27651
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