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Theorem signswmnd 29439
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 . . . . . 6  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
21signspval 29434 . . . . 5  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 } )  ->  ( u  .+^  v )  =  if ( v  =  0 ,  u ,  v ) )
3 ifcl 3922 . . . . 5  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 } )  ->  if (
v  =  0 ,  u ,  v )  e.  { -u 1 ,  0 ,  1 } )
42, 3eqeltrd 2528 . . . 4  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 } )  ->  ( u  .+^  v )  e.  { -u 1 ,  0 ,  1 } )
51signspval 29434 . . . . . . . . . . . . 13  |-  ( ( ( u  .+^  v )  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  ->  ( ( u 
.+^  v )  .+^  w )  =  if ( w  =  0 ,  ( u  .+^  v ) ,  w
) )
64, 5stoic3 1659 . . . . . . . . . . . 12  |-  ( ( 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 ) )
7 iftrue 3886 . . . . . . . . . . . 12  |-  ( w  =  0  ->  if ( w  =  0 ,  ( u  .+^  v ) ,  w
)  =  ( u 
.+^  v ) )
86, 7sylan9eq 2504 . . . . . . . . . . 11  |-  ( ( ( 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 )
)
98adantr 467 . . . . . . . . . 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 ) 
.+^  w )  =  ( u  .+^  v ) )
1023adant3 1027 . . . . . . . . . . 11  |-  ( ( 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 ) )
1110ad2antrr 731 . . . . . . . . . 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 ) )
12 iftrue 3886 . . . . . . . . . . 11  |-  ( v  =  0  ->  if ( v  =  0 ,  u ,  v )  =  u )
1312adantl 468 . . . . . . . . . 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 )  =  u )
149, 11, 133eqtrd 2488 . . . . . . . . 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 )
15 simp1 1007 . . . . . . . . . . . 12  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  ->  u  e.  { -u 1 ,  0 ,  1 } )
161signspval 29434 . . . . . . . . . . . . . 14  |-  ( ( v  e.  { -u
1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  ->  ( v  .+^  w )  =  if ( w  =  0 ,  v ,  w
) )
17163adant1 1025 . . . . . . . . . . . . 13  |-  ( ( 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 ) )
18 simpl2 1011 . . . . . . . . . . . . . 14  |-  ( ( ( 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 } )
19 simpl3 1012 . . . . . . . . . . . . . 14  |-  ( ( ( 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 } )
2018, 19ifclda 3912 . . . . . . . . . . . . 13  |-  ( ( 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 } )
2117, 20eqeltrd 2528 . . . . . . . . . . . 12  |-  ( ( 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 } )
221signspval 29434 . . . . . . . . . . . 12  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  (
v  .+^  w )  e. 
{ -u 1 ,  0 ,  1 } )  ->  ( u  .+^  ( v  .+^  w ) )  =  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w ) ) )
2315, 21, 22syl2anc 666 . . . . . . . . . . 11  |-  ( ( 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 ) ) )
2423ad2antrr 731 . . . . . . . . . 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  .+^  w ) )  =  if ( ( v 
.+^  w )  =  0 ,  u ,  ( v  .+^  w ) ) )
25 iftrue 3886 . . . . . . . . . . . . 13  |-  ( w  =  0  ->  if ( w  =  0 ,  v ,  w
)  =  v )
2617, 25sylan9eq 2504 . . . . . . . . . . . 12  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  w  =  0 )  ->  ( v  .+^  w )  =  v )
27 id 22 . . . . . . . . . . . 12  |-  ( v  =  0  ->  v  =  0 )
2826, 27sylan9eq 2504 . . . . . . . . . . 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 )
2928iftrued 3888 . . . . . . . . . 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  .+^  w )  =  0 ,  u ,  ( v  .+^  w )
)  =  u )
3024, 29eqtrd 2484 . . . . . . . . 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 )
3114, 30eqtr4d 2487 . . . . . . . 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  .+^  ( v 
.+^  w ) ) )
326ad2antrr 731 . . . . . . . . . 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 )  .+^  w )  =  if ( w  =  0 ,  ( u  .+^  v ) ,  w ) )
337ad2antlr 732 . . . . . . . . . 10  |-  ( ( ( ( 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 ) )
3410ad2antrr 731 . . . . . . . . . . 11  |-  ( ( ( ( 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 ) )
35 iffalse 3889 . . . . . . . . . . . 12  |-  ( -.  v  =  0  ->  if ( v  =  0 ,  u ,  v )  =  v )
3635adantl 468 . . . . . . . . . . 11  |-  ( ( ( ( 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 )
3734, 36eqtrd 2484 . . . . . . . . . 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 )  =  v )
3832, 33, 373eqtrd 2488 . . . . . . . . 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 )  =  v )
3923ad2antrr 731 . . . . . . . . . 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 
.+^  w ) )  =  if ( ( v  .+^  w )  =  0 ,  u ,  ( v  .+^  w ) ) )
40 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( ( u  e. 
{ -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e. 
{ -u 1 ,  0 ,  1 } )  /\  w  =  0 )  /\  -.  v  =  0 )  ->  -.  v  =  0
)
4117ad2antrr 731 . . . . . . . . . . . . . 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 ) )
4225ad2antlr 732 . . . . . . . . . . . . . 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 )
4341, 42eqtrd 2484 . . . . . . . . . . . . 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 )
4443eqeq1d 2452 . . . . . . . . . . . 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 ) )
4540, 44mtbird 303 . . . . . . . . . . 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 )
4645iffalsed 3891 . . . . . . . . . 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  .+^  w )  =  0 ,  u ,  ( v  .+^  w )
)  =  ( v 
.+^  w ) )
4739, 46, 433eqtrd 2488 . . . . . . . . 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 ) )  =  v )
4838, 47eqtr4d 2487 . . . . . . . 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  .+^  ( v  .+^  w ) ) )
4931, 48pm2.61dan 799 . . . . . . 7  |-  ( ( ( 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 ) ) )
50 iffalse 3889 . . . . . . . . 9  |-  ( -.  w  =  0  ->  if ( w  =  0 ,  ( u  .+^  v ) ,  w
)  =  w )
516, 50sylan9eq 2504 . . . . . . . 8  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( (
u  .+^  v )  .+^  w )  =  w )
5223adantr 467 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
53 simpr 463 . . . . . . . . . . 11  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  -.  w  =  0 )
54 iffalse 3889 . . . . . . . . . . . . 13  |-  ( -.  w  =  0  ->  if ( w  =  0 ,  v ,  w
)  =  w )
5517, 54sylan9eq 2504 . . . . . . . . . . . 12  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( v  .+^  w )  =  w )
5655eqeq1d 2452 . . . . . . . . . . 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
) )
5753, 56mtbird 303 . . . . . . . . . 10  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  -.  (
v  .+^  w )  =  0 )
5857iffalsed 3891 . . . . . . . . 9  |-  ( ( ( 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 ) )
5952, 58, 553eqtrd 2488 . . . . . . . 8  |-  ( ( ( u  e.  { -u 1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  /\  -.  w  =  0
)  ->  ( u  .+^  ( v  .+^  w ) )  =  w )
6051, 59eqtr4d 2487 . . . . . . 7  |-  ( ( ( 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 ) ) )
6149, 60pm2.61dan 799 . . . . . 6  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  -> 
( ( u  .+^  v )  .+^  w )  =  ( u  .+^  ( v  .+^  w ) ) )
62613expa 1207 . . . . 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 ) ) )
6362ralrimiva 2801 . . . 4  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 } )  ->  A. w  e.  { -u 1 ,  0 ,  1 }  ( ( u  .+^  v )  .+^  w )  =  ( u  .+^  ( v  .+^  w ) ) )
644, 63jca 535 . . 3  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 } )  ->  ( (
u  .+^  v )  e. 
{ -u 1 ,  0 ,  1 }  /\  A. w  e.  { -u
1 ,  0 ,  1 }  ( ( u  .+^  v )  .+^  w )  =  ( u  .+^  ( v  .+^  w ) ) ) )
6564rgen2a 2814 . 2  |-  A. u  e.  { -u 1 ,  0 ,  1 } A. v  e.  { -u 1 ,  0 ,  1 }  ( ( u  .+^  v )  e.  { -u 1 ,  0 ,  1 }  /\  A. w  e. 
{ -u 1 ,  0 ,  1 }  (
( u  .+^  v ) 
.+^  w )  =  ( u  .+^  ( v 
.+^  w ) ) )
66 c0ex 9634 . . . 4  |-  0  e.  _V
6766tpid2 4085 . . 3  |-  0  e.  { -u 1 ,  0 ,  1 }
681signsw0glem 29435 . . 3  |-  A. u  e.  { -u 1 ,  0 ,  1 }  ( ( 0  .+^  u )  =  u  /\  ( u  .+^  0 )  =  u )
69 oveq1 6295 . . . . . . 7  |-  ( e  =  0  ->  (
e  .+^  u )  =  ( 0  .+^  u ) )
7069eqeq1d 2452 . . . . . 6  |-  ( e  =  0  ->  (
( e  .+^  u )  =  u  <->  ( 0 
.+^  u )  =  u ) )
71 oveq2 6296 . . . . . . 7  |-  ( e  =  0  ->  (
u  .+^  e )  =  ( u  .+^  0
) )
7271eqeq1d 2452 . . . . . 6  |-  ( e  =  0  ->  (
( u  .+^  e )  =  u  <->  ( u  .+^  0 )  =  u ) )
7370, 72anbi12d 716 . . . . 5  |-  ( e  =  0  ->  (
( ( e  .+^  u )  =  u  /\  ( u  .+^  e )  =  u )  <->  ( ( 0 
.+^  u )  =  u  /\  ( u 
.+^  0 )  =  u ) ) )
7473ralbidv 2826 . . . 4  |-  ( 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 ) ) )
7574rspcev 3149 . . 3  |-  ( ( 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 ) )
7667, 68, 75mp2an 677 . 2  |-  E. e  e.  { -u 1 ,  0 ,  1 } A. u  e.  { -u 1 ,  0 ,  1 }  ( ( e  .+^  u )  =  u  /\  (
u  .+^  e )  =  u )
77 signsw.w . . . 4  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
781, 77signswbase 29436 . . 3  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
791, 77signswplusg 29437 . . 3  |-  .+^  =  ( +g  `  W )
8078, 79ismnd 16532 . 2  |-  ( W  e.  Mnd  <->  ( A. u  e.  { -u 1 ,  0 ,  1 } A. v  e. 
{ -u 1 ,  0 ,  1 }  (
( u  .+^  v )  e.  { -u 1 ,  0 ,  1 }  /\  A. w  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 ) ) )
8165, 76, 80mpbir2an 930 1  |-  W  e. 
Mnd
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736   E.wrex 2737   ifcif 3880   {cpr 3969   {ctp 3971   <.cop 3973   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290   0cc0 9536   1c1 9537   -ucneg 9858   ndxcnx 15111   Basecbs 15114   +g cplusg 15183   Mndcmnd 16528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mgm 16481  df-sgrp 16520  df-mnd 16530
This theorem is referenced by:  signstcl  29447  signstf  29448  signstf0  29450  signstfvn  29451
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