MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-rag Structured version   Visualization version   GIF version

Definition df-rag 28378
Description: Define the class of right angles. Definition 8.1 of [Schwabhauser] p. 57. See israg 28381. (Contributed by Thierry Arnoux, 25-Aug-2019.)
Assertion
Ref Expression
df-rag ∟G = (𝑔 ∈ V ↦ {𝑀 ∈ Word (Baseβ€˜π‘”) ∣ ((β™―β€˜π‘€) = 3 ∧ ((π‘€β€˜0)(distβ€˜π‘”)(π‘€β€˜2)) = ((π‘€β€˜0)(distβ€˜π‘”)(((pInvGβ€˜π‘”)β€˜(π‘€β€˜1))β€˜(π‘€β€˜2))))})
Distinct variable group:   𝑀,𝑔

Detailed syntax breakdown of Definition df-rag
StepHypRef Expression
1 crag 28377 . 2 class ∟G
2 vg . . 3 setvar 𝑔
3 cvv 3473 . . 3 class V
4 vw . . . . . . . 8 setvar 𝑀
54cv 1539 . . . . . . 7 class 𝑀
6 chash 14297 . . . . . . 7 class β™―
75, 6cfv 6543 . . . . . 6 class (β™―β€˜π‘€)
8 c3 12275 . . . . . 6 class 3
97, 8wceq 1540 . . . . 5 wff (β™―β€˜π‘€) = 3
10 cc0 11116 . . . . . . . 8 class 0
1110, 5cfv 6543 . . . . . . 7 class (π‘€β€˜0)
12 c2 12274 . . . . . . . 8 class 2
1312, 5cfv 6543 . . . . . . 7 class (π‘€β€˜2)
142cv 1539 . . . . . . . 8 class 𝑔
15 cds 17213 . . . . . . . 8 class dist
1614, 15cfv 6543 . . . . . . 7 class (distβ€˜π‘”)
1711, 13, 16co 7412 . . . . . 6 class ((π‘€β€˜0)(distβ€˜π‘”)(π‘€β€˜2))
18 c1 11117 . . . . . . . . . 10 class 1
1918, 5cfv 6543 . . . . . . . . 9 class (π‘€β€˜1)
20 cmir 28336 . . . . . . . . . 10 class pInvG
2114, 20cfv 6543 . . . . . . . . 9 class (pInvGβ€˜π‘”)
2219, 21cfv 6543 . . . . . . . 8 class ((pInvGβ€˜π‘”)β€˜(π‘€β€˜1))
2313, 22cfv 6543 . . . . . . 7 class (((pInvGβ€˜π‘”)β€˜(π‘€β€˜1))β€˜(π‘€β€˜2))
2411, 23, 16co 7412 . . . . . 6 class ((π‘€β€˜0)(distβ€˜π‘”)(((pInvGβ€˜π‘”)β€˜(π‘€β€˜1))β€˜(π‘€β€˜2)))
2517, 24wceq 1540 . . . . 5 wff ((π‘€β€˜0)(distβ€˜π‘”)(π‘€β€˜2)) = ((π‘€β€˜0)(distβ€˜π‘”)(((pInvGβ€˜π‘”)β€˜(π‘€β€˜1))β€˜(π‘€β€˜2)))
269, 25wa 395 . . . 4 wff ((β™―β€˜π‘€) = 3 ∧ ((π‘€β€˜0)(distβ€˜π‘”)(π‘€β€˜2)) = ((π‘€β€˜0)(distβ€˜π‘”)(((pInvGβ€˜π‘”)β€˜(π‘€β€˜1))β€˜(π‘€β€˜2))))
27 cbs 17151 . . . . . 6 class Base
2814, 27cfv 6543 . . . . 5 class (Baseβ€˜π‘”)
2928cword 14471 . . . 4 class Word (Baseβ€˜π‘”)
3026, 4, 29crab 3431 . . 3 class {𝑀 ∈ Word (Baseβ€˜π‘”) ∣ ((β™―β€˜π‘€) = 3 ∧ ((π‘€β€˜0)(distβ€˜π‘”)(π‘€β€˜2)) = ((π‘€β€˜0)(distβ€˜π‘”)(((pInvGβ€˜π‘”)β€˜(π‘€β€˜1))β€˜(π‘€β€˜2))))}
312, 3, 30cmpt 5231 . 2 class (𝑔 ∈ V ↦ {𝑀 ∈ Word (Baseβ€˜π‘”) ∣ ((β™―β€˜π‘€) = 3 ∧ ((π‘€β€˜0)(distβ€˜π‘”)(π‘€β€˜2)) = ((π‘€β€˜0)(distβ€˜π‘”)(((pInvGβ€˜π‘”)β€˜(π‘€β€˜1))β€˜(π‘€β€˜2))))})
321, 31wceq 1540 1 wff ∟G = (𝑔 ∈ V ↦ {𝑀 ∈ Word (Baseβ€˜π‘”) ∣ ((β™―β€˜π‘€) = 3 ∧ ((π‘€β€˜0)(distβ€˜π‘”)(π‘€β€˜2)) = ((π‘€β€˜0)(distβ€˜π‘”)(((pInvGβ€˜π‘”)β€˜(π‘€β€˜1))β€˜(π‘€β€˜2))))})
Colors of variables: wff setvar class
This definition is referenced by:  israg  28381
  Copyright terms: Public domain W3C validator