Metamath Proof Explorer


Theorem mircl

Description: Closure of the point inversion function. (Contributed by Thierry Arnoux, 20-Oct-2019)

Ref Expression
Hypotheses 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 )
mirfv.m 𝑀 = ( 𝑆𝐴 )
mircl.x ( 𝜑𝑋𝑃 )
Assertion mircl ( 𝜑 → ( 𝑀𝑋 ) ∈ 𝑃 )

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 = ( LineG ‘ 𝐺 )
5 mirval.s 𝑆 = ( pInvG ‘ 𝐺 )
6 mirval.g ( 𝜑𝐺 ∈ TarskiG )
7 mirval.a ( 𝜑𝐴𝑃 )
8 mirfv.m 𝑀 = ( 𝑆𝐴 )
9 mircl.x ( 𝜑𝑋𝑃 )
10 1 2 3 4 5 6 7 8 mirf ( 𝜑𝑀 : 𝑃𝑃 )
11 10 9 ffvelrnd ( 𝜑 → ( 𝑀𝑋 ) ∈ 𝑃 )