Metamath Proof Explorer


Theorem scutval

Description: The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021)

Ref Expression
Assertion scutval
|- ( A < ( A |s B ) = ( iota_ x e. { y e. No | ( A <

Proof

Step Hyp Ref Expression
1 ssltex1
 |-  ( A < A e. _V )
2 ssltss1
 |-  ( A < A C_ No )
3 1 2 elpwd
 |-  ( A < A e. ~P No )
4 df-br
 |-  ( A < <. A , B >. e. <
5 4 biimpi
 |-  ( A < <. A , B >. e. <
6 ssltex2
 |-  ( A < B e. _V )
7 elimasng
 |-  ( ( A e. _V /\ B e. _V ) -> ( B e. ( < <. A , B >. e. <
8 1 6 7 syl2anc
 |-  ( A < ( B e. ( < <. A , B >. e. <
9 5 8 mpbird
 |-  ( A < B e. ( <
10 riotaex
 |-  ( iota_ x e. { y e. No | ( A <
11 breq1
 |-  ( a = A -> ( a < A <
12 breq2
 |-  ( b = B -> ( { y } < { y } <
13 11 12 bi2anan9
 |-  ( ( a = A /\ b = B ) -> ( ( a < ( A <
14 13 rabbidv
 |-  ( ( a = A /\ b = B ) -> { y e. No | ( a <
15 14 imaeq2d
 |-  ( ( a = A /\ b = B ) -> ( bday " { y e. No | ( a <
16 15 inteqd
 |-  ( ( a = A /\ b = B ) -> |^| ( bday " { y e. No | ( a <
17 16 eqeq2d
 |-  ( ( a = A /\ b = B ) -> ( ( bday ` x ) = |^| ( bday " { y e. No | ( a < ( bday ` x ) = |^| ( bday " { y e. No | ( A <
18 14 17 riotaeqbidv
 |-  ( ( a = A /\ b = B ) -> ( iota_ x e. { y e. No | ( a <
19 sneq
 |-  ( a = A -> { a } = { A } )
20 19 imaeq2d
 |-  ( a = A -> ( <
21 df-scut
 |-  |s = ( a e. ~P No , b e. ( < ( iota_ x e. { y e. No | ( a <
22 18 20 21 ovmpox
 |-  ( ( A e. ~P No /\ B e. ( < ( A |s B ) = ( iota_ x e. { y e. No | ( A <
23 10 22 mp3an3
 |-  ( ( A e. ~P No /\ B e. ( < ( A |s B ) = ( iota_ x e. { y e. No | ( A <
24 3 9 23 syl2anc
 |-  ( A < ( A |s B ) = ( iota_ x e. { y e. No | ( A <