Metamath Proof Explorer

Theorem mulsrid

Description: Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025)

Ref Expression
Assertion mulsrid Could not format assertion : No typesetting found for |- ( A e. No -> ( A x.s 1s ) = A ) with typecode |-

Proof

Step Hyp Ref Expression
1 oveq1 Could not format ( x = xO -> ( x x.s 1s ) = ( xO x.s 1s ) ) : No typesetting found for |- ( x = xO -> ( x x.s 1s ) = ( xO x.s 1s ) ) with typecode |-
2 id Could not format ( x = xO -> x = xO ) : No typesetting found for |- ( x = xO -> x = xO ) with typecode |-
3 1 2 eqeq12d Could not format ( x = xO -> ( ( x x.s 1s ) = x <-> ( xO x.s 1s ) = xO ) ) : No typesetting found for |- ( x = xO -> ( ( x x.s 1s ) = x <-> ( xO x.s 1s ) = xO ) ) with typecode |-
4 oveq1 Could not format ( x = A -> ( x x.s 1s ) = ( A x.s 1s ) ) : No typesetting found for |- ( x = A -> ( x x.s 1s ) = ( A x.s 1s ) ) with typecode |-
5 id ⊢x=A→x=A
6 4 5 eqeq12d Could not format ( x = A -> ( ( x x.s 1s ) = x <-> ( A x.s 1s ) = A ) ) : No typesetting found for |- ( x = A -> ( ( x x.s 1s ) = x <-> ( A x.s 1s ) = A ) ) with typecode |-
7 1sno Could not format 1s e. No : No typesetting found for |- 1s e. No with typecode |-
8 mulsval Could not format ( ( x e. No /\ 1s e. No ) -> ( x x.s 1s ) = ( ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) |s ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) ) ) : No typesetting found for |- ( ( x e. No /\ 1s e. No ) -> ( x x.s 1s ) = ( ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) |s ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) ) ) with typecode |-
9 7 8 mpan2 Could not format ( x e. No -> ( x x.s 1s ) = ( ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) |s ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) ) ) : No typesetting found for |- ( x e. No -> ( x x.s 1s ) = ( ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) |s ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) ) ) with typecode |-
10 9 adantr Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( x x.s 1s ) = ( ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) |s ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( x x.s 1s ) = ( ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) |s ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) ) ) with typecode |-
11 elun1 Could not format ( p e. ( _Left ` x ) -> p e. ( ( _Left ` x ) u. ( _Right ` x ) ) ) : No typesetting found for |- ( p e. ( _Left ` x ) -> p e. ( ( _Left ` x ) u. ( _Right ` x ) ) ) with typecode |-
12 oveq1 Could not format ( xO = p -> ( xO x.s 1s ) = ( p x.s 1s ) ) : No typesetting found for |- ( xO = p -> ( xO x.s 1s ) = ( p x.s 1s ) ) with typecode |-
13 id Could not format ( xO = p -> xO = p ) : No typesetting found for |- ( xO = p -> xO = p ) with typecode |-
14 12 13 eqeq12d Could not format ( xO = p -> ( ( xO x.s 1s ) = xO <-> ( p x.s 1s ) = p ) ) : No typesetting found for |- ( xO = p -> ( ( xO x.s 1s ) = xO <-> ( p x.s 1s ) = p ) ) with typecode |-
15 14 rspcva Could not format ( ( p e. ( ( _Left ` x ) u. ( _Right ` x ) ) /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( p x.s 1s ) = p ) : No typesetting found for |- ( ( p e. ( ( _Left ` x ) u. ( _Right ` x ) ) /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( p x.s 1s ) = p ) with typecode |-
16 11 15 sylan Could not format ( ( p e. ( _Left ` x ) /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( p x.s 1s ) = p ) : No typesetting found for |- ( ( p e. ( _Left ` x ) /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( p x.s 1s ) = p ) with typecode |-
17 16 ancoms Could not format ( ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO /\ p e. ( _Left ` x ) ) -> ( p x.s 1s ) = p ) : No typesetting found for |- ( ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO /\ p e. ( _Left ` x ) ) -> ( p x.s 1s ) = p ) with typecode |-
18 17 adantll Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( p x.s 1s ) = p ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( p x.s 1s ) = p ) with typecode |-
19 muls01 Could not format ( x e. No -> ( x x.s 0s ) = 0s ) : No typesetting found for |- ( x e. No -> ( x x.s 0s ) = 0s ) with typecode |-
20 19 adantr Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( x x.s 0s ) = 0s ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( x x.s 0s ) = 0s ) with typecode |-
21 20 adantr Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( x x.s 0s ) = 0s ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( x x.s 0s ) = 0s ) with typecode |-
22 18 21 oveq12d Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( ( p x.s 1s ) +s ( x x.s 0s ) ) = ( p +s 0s ) ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( ( p x.s 1s ) +s ( x x.s 0s ) ) = ( p +s 0s ) ) with typecode |-
23 leftssno Could not format ( _Left ` x ) C_ No : No typesetting found for |- ( _Left ` x ) C_ No with typecode |-
24 23 sseli Could not format ( p e. ( _Left ` x ) -> p e. No ) : No typesetting found for |- ( p e. ( _Left ` x ) -> p e. No ) with typecode |-
25 24 adantl Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> p e. No ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> p e. No ) with typecode |-
26 25 addsridd Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( p +s 0s ) = p ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( p +s 0s ) = p ) with typecode |-
27 22 26 eqtrd Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( ( p x.s 1s ) +s ( x x.s 0s ) ) = p ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( ( p x.s 1s ) +s ( x x.s 0s ) ) = p ) with typecode |-
28 muls01 Could not format ( p e. No -> ( p x.s 0s ) = 0s ) : No typesetting found for |- ( p e. No -> ( p x.s 0s ) = 0s ) with typecode |-
29 25 28 syl Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( p x.s 0s ) = 0s ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( p x.s 0s ) = 0s ) with typecode |-
30 27 29 oveq12d Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) = ( p -s 0s ) ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) = ( p -s 0s ) ) with typecode |-
31 subsid1 Could not format ( p e. No -> ( p -s 0s ) = p ) : No typesetting found for |- ( p e. No -> ( p -s 0s ) = p ) with typecode |-
32 25 31 syl Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( p -s 0s ) = p ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( p -s 0s ) = p ) with typecode |-
33 30 32 eqtrd Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) = p ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) = p ) with typecode |-
34 33 eqeq2d Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) <-> a = p ) ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) <-> a = p ) ) with typecode |-
35 equcom ⊢a=p↔p=a
36 34 35 bitrdi Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) <-> p = a ) ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ p e. ( _Left ` x ) ) -> ( a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) <-> p = a ) ) with typecode |-
37 36 rexbidva Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( E. p e. ( _Left ` x ) a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) <-> E. p e. ( _Left ` x ) p = a ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( E. p e. ( _Left ` x ) a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) <-> E. p e. ( _Left ` x ) p = a ) ) with typecode |-
38 left1s Could not format ( _Left ` 1s ) = { 0s } : No typesetting found for |- ( _Left ` 1s ) = { 0s } with typecode |-
39 38 rexeqi Could not format ( E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> E. q e. { 0s } a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) ) : No typesetting found for |- ( E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> E. q e. { 0s } a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) ) with typecode |-
40 0sno Could not format 0s e. No : No typesetting found for |- 0s e. No with typecode |-
41 40 elexi Could not format 0s e. _V : No typesetting found for |- 0s e. _V with typecode |-
42 oveq2 Could not format ( q = 0s -> ( x x.s q ) = ( x x.s 0s ) ) : No typesetting found for |- ( q = 0s -> ( x x.s q ) = ( x x.s 0s ) ) with typecode |-
43 42 oveq2d Could not format ( q = 0s -> ( ( p x.s 1s ) +s ( x x.s q ) ) = ( ( p x.s 1s ) +s ( x x.s 0s ) ) ) : No typesetting found for |- ( q = 0s -> ( ( p x.s 1s ) +s ( x x.s q ) ) = ( ( p x.s 1s ) +s ( x x.s 0s ) ) ) with typecode |-
44 oveq2 Could not format ( q = 0s -> ( p x.s q ) = ( p x.s 0s ) ) : No typesetting found for |- ( q = 0s -> ( p x.s q ) = ( p x.s 0s ) ) with typecode |-
45 43 44 oveq12d Could not format ( q = 0s -> ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) : No typesetting found for |- ( q = 0s -> ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) with typecode |-
46 45 eqeq2d Could not format ( q = 0s -> ( a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) ) : No typesetting found for |- ( q = 0s -> ( a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) ) with typecode |-
47 41 46 rexsn Could not format ( E. q e. { 0s } a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) : No typesetting found for |- ( E. q e. { 0s } a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) with typecode |-
48 39 47 bitri Could not format ( E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) : No typesetting found for |- ( E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) with typecode |-
49 48 rexbii Could not format ( E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> E. p e. ( _Left ` x ) a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) : No typesetting found for |- ( E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> E. p e. ( _Left ` x ) a = ( ( ( p x.s 1s ) +s ( x x.s 0s ) ) -s ( p x.s 0s ) ) ) with typecode |-
50 risset Could not format ( a e. ( _Left ` x ) <-> E. p e. ( _Left ` x ) p = a ) : No typesetting found for |- ( a e. ( _Left ` x ) <-> E. p e. ( _Left ` x ) p = a ) with typecode |-
51 37 49 50 3bitr4g Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> a e. ( _Left ` x ) ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) <-> a e. ( _Left ` x ) ) ) with typecode |-
52 51 eqabcdv Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } = ( _Left ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } = ( _Left ` x ) ) with typecode |-
53 rex0 Could not format -. E. s e. (/) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) : No typesetting found for |- -. E. s e. (/) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) with typecode |-
54 right1s Could not format ( _Right ` 1s ) = (/) : No typesetting found for |- ( _Right ` 1s ) = (/) with typecode |-
55 54 rexeqi Could not format ( E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) <-> E. s e. (/) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) ) : No typesetting found for |- ( E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) <-> E. s e. (/) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) ) with typecode |-
56 53 55 mtbir Could not format -. E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) : No typesetting found for |- -. E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) with typecode |-
57 56 a1i Could not format ( r e. ( _Right ` x ) -> -. E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) ) : No typesetting found for |- ( r e. ( _Right ` x ) -> -. E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) ) with typecode |-
58 57 nrex Could not format -. E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) : No typesetting found for |- -. E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) with typecode |-
59 58 abf Could not format { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } = (/) : No typesetting found for |- { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } = (/) with typecode |-
60 59 a1i Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } = (/) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } = (/) ) with typecode |-
61 52 60 uneq12d Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) = ( ( _Left ` x ) u. (/) ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) = ( ( _Left ` x ) u. (/) ) ) with typecode |-
62 un0 Could not format ( ( _Left ` x ) u. (/) ) = ( _Left ` x ) : No typesetting found for |- ( ( _Left ` x ) u. (/) ) = ( _Left ` x ) with typecode |-
63 61 62 eqtrdi Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) = ( _Left ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) = ( _Left ` x ) ) with typecode |-
64 rex0 Could not format -. E. u e. (/) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) : No typesetting found for |- -. E. u e. (/) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) with typecode |-
65 54 rexeqi Could not format ( E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) <-> E. u e. (/) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) ) : No typesetting found for |- ( E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) <-> E. u e. (/) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) ) with typecode |-
66 64 65 mtbir Could not format -. E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) : No typesetting found for |- -. E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) with typecode |-
67 66 a1i Could not format ( t e. ( _Left ` x ) -> -. E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) ) : No typesetting found for |- ( t e. ( _Left ` x ) -> -. E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) ) with typecode |-
68 67 nrex Could not format -. E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) : No typesetting found for |- -. E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) with typecode |-
69 68 abf Could not format { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } = (/) : No typesetting found for |- { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } = (/) with typecode |-
70 69 a1i Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } = (/) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } = (/) ) with typecode |-
71 elun2 Could not format ( v e. ( _Right ` x ) -> v e. ( ( _Left ` x ) u. ( _Right ` x ) ) ) : No typesetting found for |- ( v e. ( _Right ` x ) -> v e. ( ( _Left ` x ) u. ( _Right ` x ) ) ) with typecode |-
72 oveq1 Could not format ( xO = v -> ( xO x.s 1s ) = ( v x.s 1s ) ) : No typesetting found for |- ( xO = v -> ( xO x.s 1s ) = ( v x.s 1s ) ) with typecode |-
73 id Could not format ( xO = v -> xO = v ) : No typesetting found for |- ( xO = v -> xO = v ) with typecode |-
74 72 73 eqeq12d Could not format ( xO = v -> ( ( xO x.s 1s ) = xO <-> ( v x.s 1s ) = v ) ) : No typesetting found for |- ( xO = v -> ( ( xO x.s 1s ) = xO <-> ( v x.s 1s ) = v ) ) with typecode |-
75 74 rspcva Could not format ( ( v e. ( ( _Left ` x ) u. ( _Right ` x ) ) /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( v x.s 1s ) = v ) : No typesetting found for |- ( ( v e. ( ( _Left ` x ) u. ( _Right ` x ) ) /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( v x.s 1s ) = v ) with typecode |-
76 71 75 sylan Could not format ( ( v e. ( _Right ` x ) /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( v x.s 1s ) = v ) : No typesetting found for |- ( ( v e. ( _Right ` x ) /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( v x.s 1s ) = v ) with typecode |-
77 76 ancoms Could not format ( ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO /\ v e. ( _Right ` x ) ) -> ( v x.s 1s ) = v ) : No typesetting found for |- ( ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO /\ v e. ( _Right ` x ) ) -> ( v x.s 1s ) = v ) with typecode |-
78 77 adantll Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( v x.s 1s ) = v ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( v x.s 1s ) = v ) with typecode |-
79 20 adantr Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( x x.s 0s ) = 0s ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( x x.s 0s ) = 0s ) with typecode |-
80 78 79 oveq12d Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( ( v x.s 1s ) +s ( x x.s 0s ) ) = ( v +s 0s ) ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( ( v x.s 1s ) +s ( x x.s 0s ) ) = ( v +s 0s ) ) with typecode |-
81 rightssno Could not format ( _Right ` x ) C_ No : No typesetting found for |- ( _Right ` x ) C_ No with typecode |-
82 81 sseli Could not format ( v e. ( _Right ` x ) -> v e. No ) : No typesetting found for |- ( v e. ( _Right ` x ) -> v e. No ) with typecode |-
83 82 adantl Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> v e. No ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> v e. No ) with typecode |-
84 83 addsridd Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( v +s 0s ) = v ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( v +s 0s ) = v ) with typecode |-
85 80 84 eqtrd Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( ( v x.s 1s ) +s ( x x.s 0s ) ) = v ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( ( v x.s 1s ) +s ( x x.s 0s ) ) = v ) with typecode |-
86 muls01 Could not format ( v e. No -> ( v x.s 0s ) = 0s ) : No typesetting found for |- ( v e. No -> ( v x.s 0s ) = 0s ) with typecode |-
87 83 86 syl Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( v x.s 0s ) = 0s ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( v x.s 0s ) = 0s ) with typecode |-
88 85 87 oveq12d Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) = ( v -s 0s ) ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) = ( v -s 0s ) ) with typecode |-
89 subsid1 Could not format ( v e. No -> ( v -s 0s ) = v ) : No typesetting found for |- ( v e. No -> ( v -s 0s ) = v ) with typecode |-
90 83 89 syl Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( v -s 0s ) = v ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( v -s 0s ) = v ) with typecode |-
91 88 90 eqtrd Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) = v ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) = v ) with typecode |-
92 91 eqeq2d Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( d = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) <-> d = v ) ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( d = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) <-> d = v ) ) with typecode |-
93 38 rexeqi Could not format ( E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> E. w e. { 0s } d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) ) : No typesetting found for |- ( E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> E. w e. { 0s } d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) ) with typecode |-
94 oveq2 Could not format ( w = 0s -> ( x x.s w ) = ( x x.s 0s ) ) : No typesetting found for |- ( w = 0s -> ( x x.s w ) = ( x x.s 0s ) ) with typecode |-
95 94 oveq2d Could not format ( w = 0s -> ( ( v x.s 1s ) +s ( x x.s w ) ) = ( ( v x.s 1s ) +s ( x x.s 0s ) ) ) : No typesetting found for |- ( w = 0s -> ( ( v x.s 1s ) +s ( x x.s w ) ) = ( ( v x.s 1s ) +s ( x x.s 0s ) ) ) with typecode |-
96 oveq2 Could not format ( w = 0s -> ( v x.s w ) = ( v x.s 0s ) ) : No typesetting found for |- ( w = 0s -> ( v x.s w ) = ( v x.s 0s ) ) with typecode |-
97 95 96 oveq12d Could not format ( w = 0s -> ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) ) : No typesetting found for |- ( w = 0s -> ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) ) with typecode |-
98 97 eqeq2d Could not format ( w = 0s -> ( d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> d = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) ) ) : No typesetting found for |- ( w = 0s -> ( d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> d = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) ) ) with typecode |-
99 41 98 rexsn Could not format ( E. w e. { 0s } d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> d = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) ) : No typesetting found for |- ( E. w e. { 0s } d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> d = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) ) with typecode |-
100 93 99 bitri Could not format ( E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> d = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) ) : No typesetting found for |- ( E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> d = ( ( ( v x.s 1s ) +s ( x x.s 0s ) ) -s ( v x.s 0s ) ) ) with typecode |-
101 equcom ⊢v=d↔d=v
102 92 100 101 3bitr4g Could not format ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> v = d ) ) : No typesetting found for |- ( ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) /\ v e. ( _Right ` x ) ) -> ( E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> v = d ) ) with typecode |-
103 102 rexbidva Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> E. v e. ( _Right ` x ) v = d ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> E. v e. ( _Right ` x ) v = d ) ) with typecode |-
104 risset Could not format ( d e. ( _Right ` x ) <-> E. v e. ( _Right ` x ) v = d ) : No typesetting found for |- ( d e. ( _Right ` x ) <-> E. v e. ( _Right ` x ) v = d ) with typecode |-
105 103 104 bitr4di Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> d e. ( _Right ` x ) ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) <-> d e. ( _Right ` x ) ) ) with typecode |-
106 105 eqabcdv Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } = ( _Right ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } = ( _Right ` x ) ) with typecode |-
107 70 106 uneq12d Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) = ( (/) u. ( _Right ` x ) ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) = ( (/) u. ( _Right ` x ) ) ) with typecode |-
108 0un Could not format ( (/) u. ( _Right ` x ) ) = ( _Right ` x ) : No typesetting found for |- ( (/) u. ( _Right ` x ) ) = ( _Right ` x ) with typecode |-
109 107 108 eqtrdi Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) = ( _Right ` x ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) = ( _Right ` x ) ) with typecode |-
110 63 109 oveq12d Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) |s ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) ) = ( ( _Left ` x ) |s ( _Right ` x ) ) ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( ( { a | E. p e. ( _Left ` x ) E. q e. ( _Left ` 1s ) a = ( ( ( p x.s 1s ) +s ( x x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` x ) E. s e. ( _Right ` 1s ) b = ( ( ( r x.s 1s ) +s ( x x.s s ) ) -s ( r x.s s ) ) } ) |s ( { c | E. t e. ( _Left ` x ) E. u e. ( _Right ` 1s ) c = ( ( ( t x.s 1s ) +s ( x x.s u ) ) -s ( t x.s u ) ) } u. { d | E. v e. ( _Right ` x ) E. w e. ( _Left ` 1s ) d = ( ( ( v x.s 1s ) +s ( x x.s w ) ) -s ( v x.s w ) ) } ) ) = ( ( _Left ` x ) |s ( _Right ` x ) ) ) with typecode |-
111 lrcut Could not format ( x e. No -> ( ( _Left ` x ) |s ( _Right ` x ) ) = x ) : No typesetting found for |- ( x e. No -> ( ( _Left ` x ) |s ( _Right ` x ) ) = x ) with typecode |-
112 111 adantr Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( ( _Left ` x ) |s ( _Right ` x ) ) = x ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( ( _Left ` x ) |s ( _Right ` x ) ) = x ) with typecode |-
113 10 110 112 3eqtrd Could not format ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( x x.s 1s ) = x ) : No typesetting found for |- ( ( x e. No /\ A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO ) -> ( x x.s 1s ) = x ) with typecode |-
114 113 ex Could not format ( x e. No -> ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO -> ( x x.s 1s ) = x ) ) : No typesetting found for |- ( x e. No -> ( A. xO e. ( ( _Left ` x ) u. ( _Right ` x ) ) ( xO x.s 1s ) = xO -> ( x x.s 1s ) = x ) ) with typecode |-
115 3 6 114 noinds Could not format ( A e. No -> ( A x.s 1s ) = A ) : No typesetting found for |- ( A e. No -> ( A x.s 1s ) = A ) with typecode |-