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Theorem signswmnd 29020
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 29015 . . . . 5  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 } )  ->  ( u  .+^  v )  =  if ( v  =  0 ,  u ,  v ) )
3 ifcl 3927 . . . . 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 2490 . . . 4  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  v  e.  { -u 1 ,  0 ,  1 } )  ->  ( u  .+^  v )  e.  { -u 1 ,  0 ,  1 } )
51signspval 29015 . . . . . . . . . . . . 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 1630 . . . . . . . . . . . 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 3891 . . . . . . . . . . . 12  |-  ( w  =  0  ->  if ( w  =  0 ,  ( u  .+^  v ) ,  w
)  =  ( u 
.+^  v ) )
86, 7sylan9eq 2463 . . . . . . . . . . 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 463 . . . . . . . . . 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 1017 . . . . . . . . . . 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 724 . . . . . . . . . 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 3891 . . . . . . . . . . 11  |-  ( v  =  0  ->  if ( v  =  0 ,  u ,  v )  =  u )
1312adantl 464 . . . . . . . . . 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 2447 . . . . . . . . 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 997 . . . . . . . . . . . 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 29015 . . . . . . . . . . . . . 14  |-  ( ( v  e.  { -u
1 ,  0 ,  1 }  /\  w  e.  { -u 1 ,  0 ,  1 } )  ->  ( v  .+^  w )  =  if ( w  =  0 ,  v ,  w
) )
17163adant1 1015 . . . . . . . . . . . . 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 1001 . . . . . . . . . . . . . 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 1002 . . . . . . . . . . . . . 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 3917 . . . . . . . . . . . . 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 2490 . . . . . . . . . . . 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 29015 . . . . . . . . . . . 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 659 . . . . . . . . . . 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 724 . . . . . . . . . 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 3891 . . . . . . . . . . . . 13  |-  ( w  =  0  ->  if ( w  =  0 ,  v ,  w
)  =  v )
2617, 25sylan9eq 2463 . . . . . . . . . . . 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 2463 . . . . . . . . . . 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 3893 . . . . . . . . . 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 2443 . . . . . . . . 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 2446 . . . . . . . 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 724 . . . . . . . . . 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 725 . . . . . . . . . 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 724 . . . . . . . . . . 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 3894 . . . . . . . . . . . 12  |-  ( -.  v  =  0  ->  if ( v  =  0 ,  u ,  v )  =  v )
3635adantl 464 . . . . . . . . . . 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 2443 . . . . . . . . . 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 2447 . . . . . . . . 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 724 . . . . . . . . . 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 459 . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . 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 2404 . . . . . . . . . . . 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 299 . . . . . . . . . . 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 3896 . . . . . . . . . 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 2447 . . . . . . . . 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 2446 . . . . . . . 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 792 . . . . . . 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 3894 . . . . . . . . 9  |-  ( -.  w  =  0  ->  if ( w  =  0 ,  ( u  .+^  v ) ,  w
)  =  w )
516, 50sylan9eq 2463 . . . . . . . 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 463 . . . . . . . . 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 459 . . . . . . . . . . 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 3894 . . . . . . . . . . . . 13  |-  ( -.  w  =  0  ->  if ( w  =  0 ,  v ,  w
)  =  w )
5517, 54sylan9eq 2463 . . . . . . . . . . . 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 2404 . . . . . . . . . . 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 299 . . . . . . . . . 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 3896 . . . . . . . . 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 2447 . . . . . . . 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 2446 . . . . . . 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 792 . . . . . 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 1197 . . . . 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 2818 . . . 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 530 . . 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 2831 . 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 9620 . . . 4  |-  0  e.  _V
6766tpid2 4086 . . 3  |-  0  e.  { -u 1 ,  0 ,  1 }
681signsw0glem 29016 . . 3  |-  A. u  e.  { -u 1 ,  0 ,  1 }  ( ( 0  .+^  u )  =  u  /\  ( u  .+^  0 )  =  u )
69 oveq1 6285 . . . . . . 7  |-  ( e  =  0  ->  (
e  .+^  u )  =  ( 0  .+^  u ) )
7069eqeq1d 2404 . . . . . 6  |-  ( e  =  0  ->  (
( e  .+^  u )  =  u  <->  ( 0 
.+^  u )  =  u ) )
71 oveq2 6286 . . . . . . 7  |-  ( e  =  0  ->  (
u  .+^  e )  =  ( u  .+^  0
) )
7271eqeq1d 2404 . . . . . 6  |-  ( e  =  0  ->  (
( u  .+^  e )  =  u  <->  ( u  .+^  0 )  =  u ) )
7370, 72anbi12d 709 . . . . 5  |-  ( e  =  0  ->  (
( ( e  .+^  u )  =  u  /\  ( u  .+^  e )  =  u )  <->  ( ( 0 
.+^  u )  =  u  /\  ( u 
.+^  0 )  =  u ) ) )
7473ralbidv 2843 . . . 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 3160 . . 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 670 . 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 29017 . . 3  |-  { -u
1 ,  0 ,  1 }  =  (
Base `  W )
791, 77signswplusg 29018 . . 3  |-  .+^  =  ( +g  `  W )
8078, 79ismnd 16247 . 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 921 1  |-  W  e. 
Mnd
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   ifcif 3885   {cpr 3974   {ctp 3976   <.cop 3978   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   0cc0 9522   1c1 9523   -ucneg 9842   ndxcnx 14838   Basecbs 14841   +g cplusg 14909   Mndcmnd 16243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-plusg 14922  df-mgm 16196  df-sgrp 16235  df-mnd 16245
This theorem is referenced by:  signstcl  29028  signstf  29029  signstf0  29031  signstfvn  29032
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