Metamath Proof Explorer
Description: Variation on mirmir . (Contributed by Thierry Arnoux, 10-Nov-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 |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
|
|
mirval.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
|
|
mirfv.m |
⊢ 𝑀 = ( 𝑆 ‘ 𝐴 ) |
|
|
mirmir.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
|
|
mircom.1 |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) = 𝐶 ) |
|
Assertion |
mircom |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐶 ) = 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 = ( LineG ‘ 𝐺 ) |
5 |
|
mirval.s |
⊢ 𝑆 = ( pInvG ‘ 𝐺 ) |
6 |
|
mirval.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
7 |
|
mirval.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
8 |
|
mirfv.m |
⊢ 𝑀 = ( 𝑆 ‘ 𝐴 ) |
9 |
|
mirmir.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
10 |
|
mircom.1 |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) = 𝐶 ) |
11 |
10
|
fveq2d |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑀 ‘ 𝐵 ) ) = ( 𝑀 ‘ 𝐶 ) ) |
12 |
1 2 3 4 5 6 7 8 9
|
mirmir |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑀 ‘ 𝐵 ) ) = 𝐵 ) |
13 |
11 12
|
eqtr3d |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐶 ) = 𝐵 ) |