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Theorem mulasssrg 7573
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 7542 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 mulsrpr 7561 . 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 7561 . 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 7561 . 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 7561 . 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 7387 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
76ad2ant2r 500 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  .P.  z )  e.  P. )
8 mulclpr 7387 . . . . 5  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
98ad2ant2l 499 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  .P.  w )  e.  P. )
10 addclpr 7352 . . . 4  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
117, 9, 10syl2anc 408 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
12 mulclpr 7387 . . . . 5  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
1312ad2ant2rl 502 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  .P.  w )  e.  P. )
14 mulclpr 7387 . . . . 5  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
1514ad2ant2lr 501 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  .P.  z )  e.  P. )
16 addclpr 7352 . . . 4  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1713, 15, 16syl2anc 408 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1811, 17jca 304 . 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 7387 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
2019ad2ant2r 500 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( z  .P.  v )  e.  P. )
21 mulclpr 7387 . . . . 5  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  .P.  u
)  e.  P. )
2221ad2ant2l 499 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  .P.  u )  e.  P. )
23 addclpr 7352 . . . 4  |-  ( ( ( z  .P.  v
)  e.  P.  /\  ( w  .P.  u )  e.  P. )  -> 
( ( z  .P.  v )  +P.  (
w  .P.  u )
)  e.  P. )
2420, 22, 23syl2anc 408 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P. )
25 mulclpr 7387 . . . . 5  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  .P.  u
)  e.  P. )
2625ad2ant2rl 502 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( z  .P.  u )  e.  P. )
27 mulclpr 7387 . . . . 5  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  .P.  v
)  e.  P. )
2827ad2ant2lr 501 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  .P.  v )  e.  P. )
29 addclpr 7352 . . . 4  |-  ( ( ( z  .P.  u
)  e.  P.  /\  ( w  .P.  v )  e.  P. )  -> 
( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
3026, 28, 29syl2anc 408 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. )
3124, 30jca 304 . 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 7395 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  =  ( g  .P.  f ) )
3332adantl 275 . . 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 7403 . . . . . 6  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
r  .P.  ( s  +P.  t ) )  =  ( ( r  .P.  s )  +P.  (
r  .P.  t )
) )
3534adantl 275 . . . . 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 981 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  f  e.  P. )
37 simp2 982 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  g  e.  P. )
38 simp3 983 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  h  e.  P. )
39 addclpr 7352 . . . . . 6  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  +P.  s
)  e.  P. )
4039adantl 275 . . . . 5  |-  ( ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  /\  ( r  e.  P.  /\  s  e.  P. )
)  ->  ( r  +P.  s )  e.  P. )
41 mulcomprg 7395 . . . . . 6  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  .P.  s
)  =  ( s  .P.  r ) )
4241adantl 275 . . . . 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 5947 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  .P.  h )  =  ( ( f  .P.  h )  +P.  ( g  .P.  h
) ) )
4443adantl 275 . . 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 7396 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  .P.  g
)  .P.  h )  =  ( f  .P.  ( g  .P.  h
) ) )
4645adantl 275 . . 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 7387 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  e.  P. )
4847adantl 275 . . 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 1005 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  x  e.  P. )
50 simp1r 1006 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  y  e.  P. )
51 simp2l 1007 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  z  e.  P. )
52 simp2r 1008 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  w  e.  P. )
53 simp3l 1009 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  v  e.  P. )
54 simp3r 1010 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  u  e.  P. )
55 addcomprg 7393 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
5655adantl 275 . . 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 7394 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
5857adantl 275 . . 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 7352 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
6059adantl 275 . . 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 5963 . 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 5963 . 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 6539 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480  (class class class)co 5774   P.cnp 7106    +P. cpp 7108    .P. cmp 7109    ~R cer 7111   R.cnr 7112    .R cmr 7117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283  df-imp 7284  df-enr 7541  df-nr 7542  df-mr 7544
This theorem is referenced by:  axmulass  7688
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