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Theorem signswmnd 28154
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 28149 . . . . . . 7  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 } )  ->  ( u  .+^  v )  =  if ( v  =  0 ,  u ,  v ) )
323adant3 1016 . . . . . 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 2539 . . . . . . 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 2539 . . . . . . 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 999 . . . . . . 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 1000 . . . . . . 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 3974 . . . . . 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 2555 . . . . 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 998 . . . . . . . . . . . 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 28149 . . . . . . . . . . . 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 3945 . . . . . . . . . . 11  |-  ( w  =  0  ->  if ( w  =  0 ,  ( u  .+^  v ) ,  w
)  =  ( u 
.+^  v ) )
1412, 13sylan9eq 2528 . . . . . . . . . 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 3945 . . . . . . . . . 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 2512 . . . . . . . 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 996 . . . . . . . . . . 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 28149 . . . . . . . . . . . . 13  |-  ( ( v  e.  { -u
1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  ->  ( v  .+^  w )  =  if ( w  =  0 ,  v ,  w
) )
22213adant1 1014 . . . . . . . . . . . 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 2539 . . . . . . . . . . . . 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 2539 . . . . . . . . . . . . 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 1000 . . . . . . . . . . . . 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 3974 . . . . . . . . . . . 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 2555 . . . . . . . . . . 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 28149 . . . . . . . . . . 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 3945 . . . . . . . . . . . 12  |-  ( w  =  0  ->  if ( w  =  0 ,  v ,  w
)  =  v )
3322, 32sylan9eq 2528 . . . . . . . . . . 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 2528 . . . . . . . . . 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 3945 . . . . . . . . . 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 2508 . . . . . . . 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 2511 . . . . . . 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 3948 . . . . . . . . . . 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 2508 . . . . . . . . 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 2512 . . . . . . . 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 2508 . . . . . . . . . . . . 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 2469 . . . . . . . . . . . 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 3948 . . . . . . . . . 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 2512 . . . . . . . 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 2511 . . . . . . 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 3948 . . . . . . . 8  |-  ( -.  w  =  0  ->  if ( w  =  0 ,  ( u  .+^  v ) ,  w
)  =  w )
6112, 60sylan9eq 2528 . . . . . . 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 3948 . . . . . . . . . . . . 13  |-  ( -.  w  =  0  ->  if ( w  =  0 ,  v ,  w
)  =  w )
6522, 64sylan9eq 2528 . . . . . . . . . . . 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 2469 . . . . . . . . . . 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 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 )
7161, 70eqtr4d 2511 . . . . . 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 2890 . . 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 9586 . . . . 5  |-  0  e.  _V
7675tpid2 4141 . . . 4  |-  0  e.  { -u 1 ,  0 ,  1 }
771signsw0glem 28150 . . . 4  |-  A. u  e.  { -u 1 ,  0 ,  1 }  ( ( 0  .+^  u )  =  u  /\  ( u  .+^  0 )  =  u )
78 oveq1 6289 . . . . . . . 8  |-  ( e  =  0  ->  (
e  .+^  u )  =  ( 0  .+^  u ) )
7978eqeq1d 2469 . . . . . . 7  |-  ( e  =  0  ->  (
( e  .+^  u )  =  u  <->  ( 0 
.+^  u )  =  u ) )
80 oveq2 6290 . . . . . . . 8  |-  ( e  =  0  ->  (
u  .+^  e )  =  ( u  .+^  0
) )
8180eqeq1d 2469 . . . . . . 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 2903 . . . . 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 3214 . . . 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 28151 . . 3  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
891, 87signswplusg 28152 . . 3  |-  .+^  =  ( +g  `  W )
9088, 89ismnd 15730 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   ifcif 3939   {cpr 4029   {ctp 4031   <.cop 4033   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   0cc0 9488   1c1 9489   -ucneg 9802   ndxcnx 14483   Basecbs 14486   +g cplusg 14551   Mndcmnd 15722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-plusg 14564  df-mnd 15728
This theorem is referenced by:  signstcl  28162  signstf  28163  signstf0  28165  signstfvn  28166
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