Metamath Proof Explorer


Theorem mirfv

Description: Value of the point inversion function M . Definition 7.5 of Schwabhauser p. 49. (Contributed by Thierry Arnoux, 30-May-2019)

Ref Expression
Hypotheses mirval.p P = Base G
mirval.d - ˙ = dist G
mirval.i I = Itv G
mirval.l L = Line 𝒢 G
mirval.s S = pInv 𝒢 G
mirval.g φ G 𝒢 Tarski
mirval.a φ A P
mirfv.m M = S A
mirfv.b φ B P
Assertion mirfv φ M B = ι z P | A - ˙ z = A - ˙ B A z I B

Proof

Step Hyp Ref Expression
1 mirval.p P = Base G
2 mirval.d - ˙ = dist G
3 mirval.i I = Itv G
4 mirval.l L = Line 𝒢 G
5 mirval.s S = pInv 𝒢 G
6 mirval.g φ G 𝒢 Tarski
7 mirval.a φ A P
8 mirfv.m M = S A
9 mirfv.b φ B P
10 1 2 3 4 5 6 7 mirval φ S A = y P ι z P | A - ˙ z = A - ˙ y A z I y
11 8 10 syl5eq φ M = y P ι z P | A - ˙ z = A - ˙ y A z I y
12 simplr φ y = B z P y = B
13 12 oveq2d φ y = B z P A - ˙ y = A - ˙ B
14 13 eqeq2d φ y = B z P A - ˙ z = A - ˙ y A - ˙ z = A - ˙ B
15 12 oveq2d φ y = B z P z I y = z I B
16 15 eleq2d φ y = B z P A z I y A z I B
17 14 16 anbi12d φ y = B z P A - ˙ z = A - ˙ y A z I y A - ˙ z = A - ˙ B A z I B
18 17 riotabidva φ y = B ι z P | A - ˙ z = A - ˙ y A z I y = ι z P | A - ˙ z = A - ˙ B A z I B
19 riotaex ι z P | A - ˙ z = A - ˙ B A z I B V
20 19 a1i φ ι z P | A - ˙ z = A - ˙ B A z I B V
21 11 18 9 20 fvmptd φ M B = ι z P | A - ˙ z = A - ˙ B A z I B