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Theorem signswmnd 27094
Description:  W is a monoid structure on  { -u
1 ,  0 ,  1 } which operation retains the right side, but skips zeroes. This will be used for skipping zeroes when counting sign changes. (Contributed by Thierry Arnoux, 9-Sep-2018.)
Hypotheses
Ref Expression
signsw.p  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
signsw.w  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
Assertion
Ref Expression
signswmnd  |-  W  e. 
Mnd
Distinct variable group:    a, b,  .+^
Allowed substitution hints:    W( a, b)

Proof of Theorem signswmnd
Dummy variables  u  e  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 signsw.p . . . . . . . 8  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
21signspval 27089 . . . . . . 7  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 } )  ->  ( u  .+^  v )  =  if ( v  =  0 ,  u ,  v ) )
323adant3 1008 . . . . . 6  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  -> 
( u  .+^  v )  =  if ( v  =  0 ,  u ,  v ) )
4 eleq1 2523 . . . . . . 7  |-  ( u  =  if ( v  =  0 ,  u ,  v )  -> 
( u  e.  { -u 1 ,  0 ,  1 }  <->  if (
v  =  0 ,  u ,  v )  e.  { -u 1 ,  0 ,  1 } ) )
5 eleq1 2523 . . . . . . 7  |-  ( v  =  if ( v  =  0 ,  u ,  v )  -> 
( v  e.  { -u 1 ,  0 ,  1 }  <->  if (
v  =  0 ,  u ,  v )  e.  { -u 1 ,  0 ,  1 } ) )
6 simpl1 991 . . . . . . 7  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  v  =  0 )  ->  u  e.  { -u 1 ,  0 ,  1 } )
7 simpl2 992 . . . . . . 7  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  v  =  0
)  ->  v  e.  {
-u 1 ,  0 ,  1 } )
84, 5, 6, 7ifbothda 3924 . . . . . 6  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  ->  if ( v  =  0 ,  u ,  v )  e.  { -u
1 ,  0 ,  1 } )
93, 8eqeltrd 2539 . . . . 5  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  -> 
( u  .+^  v )  e.  { -u 1 ,  0 ,  1 } )
10 simp3 990 . . . . . . . . . . . 12  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  ->  w  e.  { -u 1 ,  0 ,  1 } )
111signspval 27089 . . . . . . . . . . . 12  |-  ( ( ( u  .+^  v )  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  ->  ( ( u 
.+^  v )  .+^  w )  =  if ( w  =  0 ,  ( u  .+^  v ) ,  w
) )
129, 10, 11syl2anc 661 . . . . . . . . . . 11  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( u  .+^  v )  .+^  w )  =  if ( w  =  0 ,  ( u  .+^  v ) ,  w ) )
13 iftrue 3897 . . . . . . . . . . 11  |-  ( w  =  0  ->  if ( w  =  0 ,  ( u  .+^  v ) ,  w
)  =  ( u 
.+^  v ) )
1412, 13sylan9eq 2512 . . . . . . . . . 10  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  w  =  0 )  ->  ( ( u 
.+^  v )  .+^  w )  =  ( u  .+^  v )
)
1514adantr 465 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  v  =  0 )  ->  (
( u  .+^  v ) 
.+^  w )  =  ( u  .+^  v ) )
163ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  v  =  0 )  ->  (
u  .+^  v )  =  if ( v  =  0 ,  u ,  v ) )
17 iftrue 3897 . . . . . . . . . 10  |-  ( v  =  0  ->  if ( v  =  0 ,  u ,  v )  =  u )
1817adantl 466 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  v  =  0 )  ->  if ( v  =  0 ,  u ,  v )  =  u )
1915, 16, 183eqtrd 2496 . . . . . . . 8  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  v  =  0 )  ->  (
( u  .+^  v ) 
.+^  w )  =  u )
20 simp1 988 . . . . . . . . . . 11  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  ->  u  e.  { -u 1 ,  0 ,  1 } )
211signspval 27089 . . . . . . . . . . . . 13  |-  ( ( v  e.  { -u
1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  ->  ( v  .+^  w )  =  if ( w  =  0 ,  v ,  w
) )
22213adant1 1006 . . . . . . . . . . . 12  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  -> 
( v  .+^  w )  =  if ( w  =  0 ,  v ,  w ) )
23 eleq1 2523 . . . . . . . . . . . . 13  |-  ( v  =  if ( w  =  0 ,  v ,  w )  -> 
( v  e.  { -u 1 ,  0 ,  1 }  <->  if (
w  =  0 ,  v ,  w )  e.  { -u 1 ,  0 ,  1 } ) )
24 eleq1 2523 . . . . . . . . . . . . 13  |-  ( w  =  if ( w  =  0 ,  v ,  w )  -> 
( w  e.  { -u 1 ,  0 ,  1 }  <->  if (
w  =  0 ,  v ,  w )  e.  { -u 1 ,  0 ,  1 } ) )
25 simpl2 992 . . . . . . . . . . . . 13  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  w  =  0 )  ->  v  e.  { -u 1 ,  0 ,  1 } )
2610adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  w  e.  {
-u 1 ,  0 ,  1 } )
2723, 24, 25, 26ifbothda 3924 . . . . . . . . . . . 12  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  ->  if ( w  =  0 ,  v ,  w
)  e.  { -u
1 ,  0 ,  1 } )
2822, 27eqeltrd 2539 . . . . . . . . . . 11  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  -> 
( v  .+^  w )  e.  { -u 1 ,  0 ,  1 } )
291signspval 27089 . . . . . . . . . . 11  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  (
v  .+^  w )  e. 
{ -u 1 ,  0 ,  1 } )  ->  ( u  .+^  ( v  .+^  w ) )  =  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w ) ) )
3020, 28, 29syl2anc 661 . . . . . . . . . 10  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  -> 
( u  .+^  ( v 
.+^  w ) )  =  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w ) ) )
3130ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  v  =  0 )  ->  (
u  .+^  ( v  .+^  w ) )  =  if ( ( v 
.+^  w )  =  0 ,  u ,  ( v  .+^  w ) ) )
32 iftrue 3897 . . . . . . . . . . . 12  |-  ( w  =  0  ->  if ( w  =  0 ,  v ,  w
)  =  v )
3322, 32sylan9eq 2512 . . . . . . . . . . 11  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  w  =  0 )  ->  ( v  .+^  w )  =  v )
34 id 22 . . . . . . . . . . 11  |-  ( v  =  0  ->  v  =  0 )
3533, 34sylan9eq 2512 . . . . . . . . . 10  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  v  =  0 )  ->  (
v  .+^  w )  =  0 )
36 iftrue 3897 . . . . . . . . . 10  |-  ( ( v  .+^  w )  =  0  ->  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w )
)  =  u )
3735, 36syl 16 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  v  =  0 )  ->  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w )
)  =  u )
3831, 37eqtrd 2492 . . . . . . . 8  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  v  =  0 )  ->  (
u  .+^  ( v  .+^  w ) )  =  u )
3919, 38eqtr4d 2495 . . . . . . 7  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  v  =  0 )  ->  (
( u  .+^  v ) 
.+^  w )  =  ( u  .+^  ( v 
.+^  w ) ) )
4012ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( ( u  .+^  v )  .+^  w )  =  if ( w  =  0 ,  ( u  .+^  v ) ,  w ) )
4113ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  ->  if ( w  =  0 ,  ( u  .+^  v ) ,  w
)  =  ( u 
.+^  v ) )
423ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( u  .+^  v )  =  if ( v  =  0 ,  u ,  v ) )
43 iffalse 3899 . . . . . . . . . . 11  |-  ( -.  v  =  0  ->  if ( v  =  0 ,  u ,  v )  =  v )
4443adantl 466 . . . . . . . . . 10  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  ->  if ( v  =  0 ,  u ,  v )  =  v )
4542, 44eqtrd 2492 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( u  .+^  v )  =  v )
4640, 41, 453eqtrd 2496 . . . . . . . 8  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( ( u  .+^  v )  .+^  w )  =  v )
4730ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( u  .+^  ( v 
.+^  w ) )  =  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w ) ) )
48 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  ->  -.  v  =  0
)
4922ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( v  .+^  w )  =  if ( w  =  0 ,  v ,  w ) )
5032ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  ->  if ( w  =  0 ,  v ,  w
)  =  v )
5149, 50eqtrd 2492 . . . . . . . . . . . . 13  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( v  .+^  w )  =  v )
5251eqeq1d 2453 . . . . . . . . . . . 12  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( ( v  .+^  w )  =  0  <-> 
v  =  0 ) )
5352notbid 294 . . . . . . . . . . 11  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( -.  ( v 
.+^  w )  =  0  <->  -.  v  = 
0 ) )
5448, 53mpbird 232 . . . . . . . . . 10  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  ->  -.  ( v  .+^  w )  =  0 )
55 iffalse 3899 . . . . . . . . . 10  |-  ( -.  ( v  .+^  w )  =  0  ->  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w )
)  =  ( v 
.+^  w ) )
5654, 55syl 16 . . . . . . . . 9  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  ->  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w )
)  =  ( v 
.+^  w ) )
5747, 56, 513eqtrd 2496 . . . . . . . 8  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( u  .+^  ( v 
.+^  w ) )  =  v )
5846, 57eqtr4d 2495 . . . . . . 7  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  -> 
( ( u  .+^  v )  .+^  w )  =  ( u  .+^  ( v  .+^  w ) ) )
5939, 58pm2.61dan 789 . . . . . 6  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  w  =  0 )  ->  ( ( u 
.+^  v )  .+^  w )  =  ( u  .+^  ( v  .+^  w ) ) )
60 iffalse 3899 . . . . . . . 8  |-  ( -.  w  =  0  ->  if ( w  =  0 ,  ( u  .+^  v ) ,  w
)  =  w )
6112, 60sylan9eq 2512 . . . . . . 7  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( (
u  .+^  v )  .+^  w )  =  w )
6230adantr 465 . . . . . . . 8  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( u  .+^  ( v  .+^  w ) )  =  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w ) ) )
63 simpr 461 . . . . . . . . . 10  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  -.  w  =  0 )
64 iffalse 3899 . . . . . . . . . . . . 13  |-  ( -.  w  =  0  ->  if ( w  =  0 ,  v ,  w
)  =  w )
6522, 64sylan9eq 2512 . . . . . . . . . . . 12  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( v  .+^  w )  =  w )
6665eqeq1d 2453 . . . . . . . . . . 11  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( (
v  .+^  w )  =  0  <->  w  =  0
) )
6766notbid 294 . . . . . . . . . 10  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( -.  ( v  .+^  w )  =  0  <->  -.  w  =  0 ) )
6863, 67mpbird 232 . . . . . . . . 9  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  -.  (
v  .+^  w )  =  0 )
6968, 55syl 16 . . . . . . . 8  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  if (
( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w ) )  =  ( v  .+^  w ) )
7062, 69, 653eqtrd 2496 . . . . . . 7  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( u  .+^  ( v  .+^  w ) )  =  w )
7161, 70eqtr4d 2495 . . . . . 6  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( (
u  .+^  v )  .+^  w )  =  ( u  .+^  ( v  .+^  w ) ) )
7259, 71pm2.61dan 789 . . . . 5  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( u  .+^  v )  .+^  w )  =  ( u  .+^  ( v  .+^  w ) ) )
739, 72jca 532 . . . 4  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( u  .+^  v )  e.  { -u 1 ,  0 ,  1 }  /\  (
( u  .+^  v ) 
.+^  w )  =  ( u  .+^  ( v 
.+^  w ) ) ) )
7473rgen3 2911 . . 3  |-  A. u  e.  { -u 1 ,  0 ,  1 } A. v  e.  { -u 1 ,  0 ,  1 } A. w  e.  { -u 1 ,  0 ,  1 }  ( ( u  .+^  v )  e.  { -u 1 ,  0 ,  1 }  /\  (
( u  .+^  v ) 
.+^  w )  =  ( u  .+^  ( v 
.+^  w ) ) )
75 c0ex 9483 . . . . 5  |-  0  e.  _V
7675tpid2 4089 . . . 4  |-  0  e.  { -u 1 ,  0 ,  1 }
771signsw0glem 27090 . . . 4  |-  A. u  e.  { -u 1 ,  0 ,  1 }  ( ( 0  .+^  u )  =  u  /\  ( u  .+^  0 )  =  u )
78 oveq1 6199 . . . . . . . 8  |-  ( e  =  0  ->  (
e  .+^  u )  =  ( 0  .+^  u ) )
7978eqeq1d 2453 . . . . . . 7  |-  ( e  =  0  ->  (
( e  .+^  u )  =  u  <->  ( 0 
.+^  u )  =  u ) )
80 oveq2 6200 . . . . . . . 8  |-  ( e  =  0  ->  (
u  .+^  e )  =  ( u  .+^  0
) )
8180eqeq1d 2453 . . . . . . 7  |-  ( e  =  0  ->  (
( u  .+^  e )  =  u  <->  ( u  .+^  0 )  =  u ) )
8279, 81anbi12d 710 . . . . . 6  |-  ( e  =  0  ->  (
( ( e  .+^  u )  =  u  /\  ( u  .+^  e )  =  u )  <->  ( ( 0 
.+^  u )  =  u  /\  ( u 
.+^  0 )  =  u ) ) )
8382ralbidv 2838 . . . . 5  |-  ( e  =  0  ->  ( A. u  e.  { -u
1 ,  0 ,  1 }  ( ( e  .+^  u )  =  u  /\  (
u  .+^  e )  =  u )  <->  A. u  e.  { -u 1 ,  0 ,  1 }  ( ( 0  .+^  u )  =  u  /\  ( u  .+^  0 )  =  u ) ) )
8483rspcev 3171 . . . 4  |-  ( ( 0  e.  { -u
1 ,  0 ,  1 }  /\  A. u  e.  { -u 1 ,  0 ,  1 }  ( ( 0 
.+^  u )  =  u  /\  ( u 
.+^  0 )  =  u ) )  ->  E. e  e.  { -u
1 ,  0 ,  1 } A. u  e.  { -u 1 ,  0 ,  1 }  ( ( e  .+^  u )  =  u  /\  ( u  .+^  e )  =  u ) )
8576, 77, 84mp2an 672 . . 3  |-  E. e  e.  { -u 1 ,  0 ,  1 } A. u  e.  { -u 1 ,  0 ,  1 }  ( ( e  .+^  u )  =  u  /\  (
u  .+^  e )  =  u )
8674, 85pm3.2i 455 . 2  |-  ( A. u  e.  { -u 1 ,  0 ,  1 } A. v  e. 
{ -u 1 ,  0 ,  1 } A. w  e.  { -u 1 ,  0 ,  1 }  ( ( u 
.+^  v )  e. 
{ -u 1 ,  0 ,  1 }  /\  ( ( u  .+^  v )  .+^  w )  =  ( u  .+^  ( v  .+^  w ) ) )  /\  E. e  e.  { -u 1 ,  0 ,  1 } A. u  e. 
{ -u 1 ,  0 ,  1 }  (
( e  .+^  u )  =  u  /\  (
u  .+^  e )  =  u ) )
87 signsw.w . . . 4  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
881, 87signswbase 27091 . . 3  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
891, 87signswplusg 27092 . . 3  |-  .+^  =  ( +g  `  W )
9088, 89ismnd 15521 . 2  |-  ( W  e.  Mnd  <->  ( A. u  e.  { -u 1 ,  0 ,  1 } A. v  e. 
{ -u 1 ,  0 ,  1 } A. w  e.  { -u 1 ,  0 ,  1 }  ( ( u 
.+^  v )  e. 
{ -u 1 ,  0 ,  1 }  /\  ( ( u  .+^  v )  .+^  w )  =  ( u  .+^  ( v  .+^  w ) ) )  /\  E. e  e.  { -u 1 ,  0 ,  1 } A. u  e. 
{ -u 1 ,  0 ,  1 }  (
( e  .+^  u )  =  u  /\  (
u  .+^  e )  =  u ) ) )
9186, 90mpbir 209 1  |-  W  e. 
Mnd
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   ifcif 3891   {cpr 3979   {ctp 3981   <.cop 3983   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   0cc0 9385   1c1 9386   -ucneg 9699   ndxcnx 14275   Basecbs 14278   +g cplusg 14342   Mndcmnd 15513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-plusg 14355  df-mnd 15519
This theorem is referenced by:  signstcl  27102  signstf  27103  signstf0  27105  signstfvn  27106
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