Metamath Proof Explorer


Theorem negsbdaylem

Description: Lemma for negsbday . Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025)

Ref Expression
Assertion negsbdaylem A No bday + s A bday A

Proof

Step Hyp Ref Expression
1 2fveq3 Could not format ( x = xO -> ( bday ` ( -us ` x ) ) = ( bday ` ( -us ` xO ) ) ) : No typesetting found for |- ( x = xO -> ( bday ` ( -us ` x ) ) = ( bday ` ( -us ` xO ) ) ) with typecode |-
2 fveq2 Could not format ( x = xO -> ( bday ` x ) = ( bday ` xO ) ) : No typesetting found for |- ( x = xO -> ( bday ` x ) = ( bday ` xO ) ) with typecode |-
3 1 2 sseq12d Could not format ( x = xO -> ( ( bday ` ( -us ` x ) ) C_ ( bday ` x ) <-> ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) ) : No typesetting found for |- ( x = xO -> ( ( bday ` ( -us ` x ) ) C_ ( bday ` x ) <-> ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) ) with typecode |-
4 2fveq3 x = A bday + s x = bday + s A
5 fveq2 x = A bday x = bday A
6 4 5 sseq12d x = A bday + s x bday x bday + s A bday A
7 negsval x No + s x = + s R x | s + s L x
8 7 fveq2d x No bday + s x = bday + s R x | s + s L x
9 8 adantr Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday ` ( -us ` x ) ) = ( bday ` ( ( -us " ( _Right ` x ) ) |s ( -us " ( _Left ` x ) ) ) ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday ` ( -us ` x ) ) = ( bday ` ( ( -us " ( _Right ` x ) ) |s ( -us " ( _Left ` x ) ) ) ) ) with typecode |-
10 negscut2 x No + s R x s + s L x
11 lrold L x R x = Old bday x
12 uncom L x R x = R x L x
13 11 12 eqtr3i Old bday x = R x L x
14 13 imaeq2i + s Old bday x = + s R x L x
15 imaundi + s R x L x = + s R x + s L x
16 14 15 eqtri + s Old bday x = + s R x + s L x
17 16 imaeq2i bday + s Old bday x = bday + s R x + s L x
18 11 raleqi Could not format ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) <-> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) : No typesetting found for |- ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) <-> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) with typecode |-
19 oldbdayim Could not format ( xO e. ( _Old ` ( bday ` x ) ) -> ( bday ` xO ) e. ( bday ` x ) ) : No typesetting found for |- ( xO e. ( _Old ` ( bday ` x ) ) -> ( bday ` xO ) e. ( bday ` x ) ) with typecode |-
20 19 adantl Could not format ( ( x e. No /\ xO e. ( _Old ` ( bday ` x ) ) ) -> ( bday ` xO ) e. ( bday ` x ) ) : No typesetting found for |- ( ( x e. No /\ xO e. ( _Old ` ( bday ` x ) ) ) -> ( bday ` xO ) e. ( bday ` x ) ) with typecode |-
21 bdayelon Could not format ( bday ` ( -us ` xO ) ) e. On : No typesetting found for |- ( bday ` ( -us ` xO ) ) e. On with typecode |-
22 bdayelon bday x On
23 ontr2 Could not format ( ( ( bday ` ( -us ` xO ) ) e. On /\ ( bday ` x ) e. On ) -> ( ( ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) /\ ( bday ` xO ) e. ( bday ` x ) ) -> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) : No typesetting found for |- ( ( ( bday ` ( -us ` xO ) ) e. On /\ ( bday ` x ) e. On ) -> ( ( ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) /\ ( bday ` xO ) e. ( bday ` x ) ) -> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) with typecode |-
24 21 22 23 mp2an Could not format ( ( ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) /\ ( bday ` xO ) e. ( bday ` x ) ) -> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) : No typesetting found for |- ( ( ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) /\ ( bday ` xO ) e. ( bday ` x ) ) -> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) with typecode |-
25 24 a1i Could not format ( ( x e. No /\ xO e. ( _Old ` ( bday ` x ) ) ) -> ( ( ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) /\ ( bday ` xO ) e. ( bday ` x ) ) -> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) : No typesetting found for |- ( ( x e. No /\ xO e. ( _Old ` ( bday ` x ) ) ) -> ( ( ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) /\ ( bday ` xO ) e. ( bday ` x ) ) -> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) with typecode |-
26 20 25 mpan2d Could not format ( ( x e. No /\ xO e. ( _Old ` ( bday ` x ) ) ) -> ( ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) -> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) : No typesetting found for |- ( ( x e. No /\ xO e. ( _Old ` ( bday ` x ) ) ) -> ( ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) -> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) with typecode |-
27 26 ralimdva Could not format ( x e. No -> ( A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) -> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) : No typesetting found for |- ( x e. No -> ( A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) -> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) with typecode |-
28 27 imp Could not format ( ( x e. No /\ A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) with typecode |-
29 bdayfun Fun bday
30 imassrn + s Old bday x ran + s
31 bdaydm dom bday = No
32 negsfo + s : No onto No
33 forn + s : No onto No ran + s = No
34 32 33 ax-mp ran + s = No
35 31 34 eqtr4i dom bday = ran + s
36 30 35 sseqtrri + s Old bday x dom bday
37 funimass4 Fun bday + s Old bday x dom bday bday + s Old bday x bday x y + s Old bday x bday y bday x
38 29 36 37 mp2an bday + s Old bday x bday x y + s Old bday x bday y bday x
39 negsfn + s Fn No
40 oldssno Old bday x No
41 fveq2 Could not format ( y = ( -us ` xO ) -> ( bday ` y ) = ( bday ` ( -us ` xO ) ) ) : No typesetting found for |- ( y = ( -us ` xO ) -> ( bday ` y ) = ( bday ` ( -us ` xO ) ) ) with typecode |-
42 41 eleq1d Could not format ( y = ( -us ` xO ) -> ( ( bday ` y ) e. ( bday ` x ) <-> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) : No typesetting found for |- ( y = ( -us ` xO ) -> ( ( bday ` y ) e. ( bday ` x ) <-> ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) with typecode |-
43 42 imaeqsalv Could not format ( ( -us Fn No /\ ( _Old ` ( bday ` x ) ) C_ No ) -> ( A. y e. ( -us " ( _Old ` ( bday ` x ) ) ) ( bday ` y ) e. ( bday ` x ) <-> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) : No typesetting found for |- ( ( -us Fn No /\ ( _Old ` ( bday ` x ) ) C_ No ) -> ( A. y e. ( -us " ( _Old ` ( bday ` x ) ) ) ( bday ` y ) e. ( bday ` x ) <-> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) ) with typecode |-
44 39 40 43 mp2an Could not format ( A. y e. ( -us " ( _Old ` ( bday ` x ) ) ) ( bday ` y ) e. ( bday ` x ) <-> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) : No typesetting found for |- ( A. y e. ( -us " ( _Old ` ( bday ` x ) ) ) ( bday ` y ) e. ( bday ` x ) <-> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) with typecode |-
45 38 44 bitri Could not format ( ( bday " ( -us " ( _Old ` ( bday ` x ) ) ) ) C_ ( bday ` x ) <-> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) : No typesetting found for |- ( ( bday " ( -us " ( _Old ` ( bday ` x ) ) ) ) C_ ( bday ` x ) <-> A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) e. ( bday ` x ) ) with typecode |-
46 28 45 sylibr Could not format ( ( x e. No /\ A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday " ( -us " ( _Old ` ( bday ` x ) ) ) ) C_ ( bday ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( _Old ` ( bday ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday " ( -us " ( _Old ` ( bday ` x ) ) ) ) C_ ( bday ` x ) ) with typecode |-
47 18 46 sylan2b Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday " ( -us " ( _Old ` ( bday ` x ) ) ) ) C_ ( bday ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday " ( -us " ( _Old ` ( bday ` x ) ) ) ) C_ ( bday ` x ) ) with typecode |-
48 17 47 eqsstrrid Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday " ( ( -us " ( _Right ` x ) ) u. ( -us " ( _Left ` x ) ) ) ) C_ ( bday ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday " ( ( -us " ( _Right ` x ) ) u. ( -us " ( _Left ` x ) ) ) ) C_ ( bday ` x ) ) with typecode |-
49 scutbdaybnd + s R x s + s L x bday x On bday + s R x + s L x bday x bday + s R x | s + s L x bday x
50 22 49 mp3an2 + s R x s + s L x bday + s R x + s L x bday x bday + s R x | s + s L x bday x
51 10 48 50 syl2an2r Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday ` ( ( -us " ( _Right ` x ) ) |s ( -us " ( _Left ` x ) ) ) ) C_ ( bday ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday ` ( ( -us " ( _Right ` x ) ) |s ( -us " ( _Left ` x ) ) ) ) C_ ( bday ` x ) ) with typecode |-
52 9 51 eqsstrd Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday ` ( -us ` x ) ) C_ ( bday ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) ) -> ( bday ` ( -us ` x ) ) C_ ( bday ` x ) ) with typecode |-
53 52 ex Could not format ( x e. No -> ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) -> ( bday ` ( -us ` x ) ) C_ ( bday ` x ) ) ) : No typesetting found for |- ( x e. No -> ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( bday ` ( -us ` xO ) ) C_ ( bday ` xO ) -> ( bday ` ( -us ` x ) ) C_ ( bday ` x ) ) ) with typecode |-
54 3 6 53 noinds A No bday + s A bday A