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Theorem mulasssrg 7786
Description: Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
Assertion
Ref Expression
mulasssrg  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  .R  B
)  .R  C )  =  ( A  .R  ( B  .R  C ) ) )

Proof of Theorem mulasssrg
Dummy variables  f  g  h  r  s  t  u  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7755 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 mulsrpr 7774 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
( x  .P.  z
)  +P.  ( y  .P.  w ) ) ,  ( ( x  .P.  w )  +P.  (
y  .P.  z )
) >. ]  ~R  )
3 mulsrpr 7774 . 2  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  .R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
( z  .P.  v
)  +P.  ( w  .P.  u ) ) ,  ( ( z  .P.  u )  +P.  (
w  .P.  v )
) >. ]  ~R  )
4 mulsrpr 7774 . 2  |-  ( ( ( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P.  /\  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. ( ( x  .P.  z )  +P.  (
y  .P.  w )
) ,  ( ( x  .P.  w )  +P.  ( y  .P.  z ) ) >. ]  ~R  .R  [ <. v ,  u >. ]  ~R  )  =  [ <. (
( ( ( x  .P.  z )  +P.  ( y  .P.  w
) )  .P.  v
)  +P.  ( (
( x  .P.  w
)  +P.  ( y  .P.  z ) )  .P.  u ) ) ,  ( ( ( ( x  .P.  z )  +P.  ( y  .P.  w ) )  .P.  u )  +P.  (
( ( x  .P.  w )  +P.  (
y  .P.  z )
)  .P.  v )
) >. ]  ~R  )
5 mulsrpr 7774 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( ( ( z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P.  /\  ( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. ( ( z  .P.  v
)  +P.  ( w  .P.  u ) ) ,  ( ( z  .P.  u )  +P.  (
w  .P.  v )
) >. ]  ~R  )  =  [ <. ( ( x  .P.  ( ( z  .P.  v )  +P.  ( w  .P.  u
) ) )  +P.  ( y  .P.  (
( z  .P.  u
)  +P.  ( w  .P.  v ) ) ) ) ,  ( ( x  .P.  ( ( z  .P.  u )  +P.  ( w  .P.  v ) ) )  +P.  ( y  .P.  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) ) ) >. ]  ~R  )
6 mulclpr 7600 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
76ad2ant2r 509 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  .P.  z )  e.  P. )
8 mulclpr 7600 . . . . 5  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
98ad2ant2l 508 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  .P.  w )  e.  P. )
10 addclpr 7565 . . . 4  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
117, 9, 10syl2anc 411 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
12 mulclpr 7600 . . . . 5  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
1312ad2ant2rl 511 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  .P.  w )  e.  P. )
14 mulclpr 7600 . . . . 5  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
1514ad2ant2lr 510 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  .P.  z )  e.  P. )
16 addclpr 7565 . . . 4  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1713, 15, 16syl2anc 411 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1811, 17jca 306 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
( x  .P.  z
)  +P.  ( y  .P.  w ) )  e. 
P.  /\  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. ) )
19 mulclpr 7600 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
2019ad2ant2r 509 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( z  .P.  v )  e.  P. )
21 mulclpr 7600 . . . . 5  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  .P.  u
)  e.  P. )
2221ad2ant2l 508 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  .P.  u )  e.  P. )
23 addclpr 7565 . . . 4  |-  ( ( ( z  .P.  v
)  e.  P.  /\  ( w  .P.  u )  e.  P. )  -> 
( ( z  .P.  v )  +P.  (
w  .P.  u )
)  e.  P. )
2420, 22, 23syl2anc 411 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P. )
25 mulclpr 7600 . . . . 5  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  .P.  u
)  e.  P. )
2625ad2ant2rl 511 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( z  .P.  u )  e.  P. )
27 mulclpr 7600 . . . . 5  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  .P.  v
)  e.  P. )
2827ad2ant2lr 510 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  .P.  v )  e.  P. )
29 addclpr 7565 . . . 4  |-  ( ( ( z  .P.  u
)  e.  P.  /\  ( w  .P.  v )  e.  P. )  -> 
( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
3026, 28, 29syl2anc 411 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. )
3124, 30jca 306 . 2  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( z  .P.  v
)  +P.  ( w  .P.  u ) )  e. 
P.  /\  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. ) )
32 mulcomprg 7608 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  =  ( g  .P.  f ) )
3332adantl 277 . . 3  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  .P.  g
)  =  ( g  .P.  f ) )
34 distrprg 7616 . . . . . 6  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
r  .P.  ( s  +P.  t ) )  =  ( ( r  .P.  s )  +P.  (
r  .P.  t )
) )
3534adantl 277 . . . . 5  |-  ( ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  /\  ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )
)  ->  ( r  .P.  ( s  +P.  t
) )  =  ( ( r  .P.  s
)  +P.  ( r  .P.  t ) ) )
36 simp1 999 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  f  e.  P. )
37 simp2 1000 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  g  e.  P. )
38 simp3 1001 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  h  e.  P. )
39 addclpr 7565 . . . . . 6  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  +P.  s
)  e.  P. )
4039adantl 277 . . . . 5  |-  ( ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  /\  ( r  e.  P.  /\  s  e.  P. )
)  ->  ( r  +P.  s )  e.  P. )
41 mulcomprg 7608 . . . . . 6  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  .P.  s
)  =  ( s  .P.  r ) )
4241adantl 277 . . . . 5  |-  ( ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  /\  ( r  e.  P.  /\  s  e.  P. )
)  ->  ( r  .P.  s )  =  ( s  .P.  r ) )
4335, 36, 37, 38, 40, 42caovdir2d 6072 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  .P.  h )  =  ( ( f  .P.  h )  +P.  ( g  .P.  h
) ) )
4443adantl 277 . . 3  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P.  /\  h  e.  P. ) )  ->  (
( f  +P.  g
)  .P.  h )  =  ( ( f  .P.  h )  +P.  ( g  .P.  h
) ) )
45 mulassprg 7609 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  .P.  g
)  .P.  h )  =  ( f  .P.  ( g  .P.  h
) ) )
4645adantl 277 . . 3  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P.  /\  h  e.  P. ) )  ->  (
( f  .P.  g
)  .P.  h )  =  ( f  .P.  ( g  .P.  h
) ) )
47 mulclpr 7600 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  e.  P. )
4847adantl 277 . . 3  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  .P.  g
)  e.  P. )
49 simp1l 1023 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  x  e.  P. )
50 simp1r 1024 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  y  e.  P. )
51 simp2l 1025 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  z  e.  P. )
52 simp2r 1026 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  w  e.  P. )
53 simp3l 1027 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  v  e.  P. )
54 simp3r 1028 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  u  e.  P. )
55 addcomprg 7606 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
5655adantl 277 . . 3  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
57 addassprg 7607 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
5857adantl 277 . . 3  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P.  /\  h  e.  P. ) )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
59 addclpr 7565 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
6059adantl 277 . . 3  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  e.  P. )
6133, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 58, 60caovlem2d 6088 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( ( x  .P.  z )  +P.  (
y  .P.  w )
)  .P.  v )  +P.  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  .P.  u
) )  =  ( ( x  .P.  (
( z  .P.  v
)  +P.  ( w  .P.  u ) ) )  +P.  ( y  .P.  ( ( z  .P.  u )  +P.  (
w  .P.  v )
) ) ) )
6233, 44, 46, 48, 49, 50, 51, 52, 54, 53, 56, 58, 60caovlem2d 6088 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( ( x  .P.  z )  +P.  (
y  .P.  w )
)  .P.  u )  +P.  ( ( ( x  .P.  w )  +P.  ( y  .P.  z
) )  .P.  v
) )  =  ( ( x  .P.  (
( z  .P.  u
)  +P.  ( w  .P.  v ) ) )  +P.  ( y  .P.  ( ( z  .P.  v )  +P.  (
w  .P.  u )
) ) ) )
631, 2, 3, 4, 5, 18, 31, 61, 62ecoviass 6670 1  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  (
( A  .R  B
)  .R  C )  =  ( A  .R  ( B  .R  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160  (class class class)co 5895   P.cnp 7319    +P. cpp 7321    .P. cmp 7322    ~R cer 7324   R.cnr 7325    .R cmr 7330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-1o 6440  df-2o 6441  df-oadd 6444  df-omul 6445  df-er 6558  df-ec 6560  df-qs 6564  df-ni 7332  df-pli 7333  df-mi 7334  df-lti 7335  df-plpq 7372  df-mpq 7373  df-enq 7375  df-nqqs 7376  df-plqqs 7377  df-mqqs 7378  df-1nqqs 7379  df-rq 7380  df-ltnqqs 7381  df-enq0 7452  df-nq0 7453  df-0nq0 7454  df-plq0 7455  df-mq0 7456  df-inp 7494  df-iplp 7496  df-imp 7497  df-enr 7754  df-nr 7755  df-mr 7757
This theorem is referenced by:  axmulass  7901
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