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Theorem mulasssrg 7401
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 7370 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 mulsrpr 7389 . 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 7389 . 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 7389 . 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 7389 . 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 7228 . . . . 5  |-  ( ( x  e.  P.  /\  z  e.  P. )  ->  ( x  .P.  z
)  e.  P. )
76ad2ant2r 494 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  .P.  z )  e.  P. )
8 mulclpr 7228 . . . . 5  |-  ( ( y  e.  P.  /\  w  e.  P. )  ->  ( y  .P.  w
)  e.  P. )
98ad2ant2l 493 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  .P.  w )  e.  P. )
10 addclpr 7193 . . . 4  |-  ( ( ( x  .P.  z
)  e.  P.  /\  ( y  .P.  w
)  e.  P. )  ->  ( ( x  .P.  z )  +P.  (
y  .P.  w )
)  e.  P. )
117, 9, 10syl2anc 404 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  z )  +P.  ( y  .P.  w
) )  e.  P. )
12 mulclpr 7228 . . . . 5  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  .P.  w
)  e.  P. )
1312ad2ant2rl 496 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  .P.  w )  e.  P. )
14 mulclpr 7228 . . . . 5  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  .P.  z
)  e.  P. )
1514ad2ant2lr 495 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  .P.  z )  e.  P. )
16 addclpr 7193 . . . 4  |-  ( ( ( x  .P.  w
)  e.  P.  /\  ( y  .P.  z
)  e.  P. )  ->  ( ( x  .P.  w )  +P.  (
y  .P.  z )
)  e.  P. )
1713, 15, 16syl2anc 404 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
x  .P.  w )  +P.  ( y  .P.  z
) )  e.  P. )
1811, 17jca 301 . 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 7228 . . . . 5  |-  ( ( z  e.  P.  /\  v  e.  P. )  ->  ( z  .P.  v
)  e.  P. )
2019ad2ant2r 494 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( z  .P.  v )  e.  P. )
21 mulclpr 7228 . . . . 5  |-  ( ( w  e.  P.  /\  u  e.  P. )  ->  ( w  .P.  u
)  e.  P. )
2221ad2ant2l 493 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  .P.  u )  e.  P. )
23 addclpr 7193 . . . 4  |-  ( ( ( z  .P.  v
)  e.  P.  /\  ( w  .P.  u )  e.  P. )  -> 
( ( z  .P.  v )  +P.  (
w  .P.  u )
)  e.  P. )
2420, 22, 23syl2anc 404 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  v )  +P.  ( w  .P.  u
) )  e.  P. )
25 mulclpr 7228 . . . . 5  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  .P.  u
)  e.  P. )
2625ad2ant2rl 496 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( z  .P.  u )  e.  P. )
27 mulclpr 7228 . . . . 5  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  .P.  v
)  e.  P. )
2827ad2ant2lr 495 . . . 4  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  .P.  v )  e.  P. )
29 addclpr 7193 . . . 4  |-  ( ( ( z  .P.  u
)  e.  P.  /\  ( w  .P.  v )  e.  P. )  -> 
( ( z  .P.  u )  +P.  (
w  .P.  v )
)  e.  P. )
3026, 28, 29syl2anc 404 . . 3  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  .P.  u )  +P.  ( w  .P.  v
) )  e.  P. )
3124, 30jca 301 . 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 7236 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  =  ( g  .P.  f ) )
3332adantl 272 . . 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 7244 . . . . . 6  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
r  .P.  ( s  +P.  t ) )  =  ( ( r  .P.  s )  +P.  (
r  .P.  t )
) )
3534adantl 272 . . . . 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 946 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  f  e.  P. )
37 simp2 947 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  g  e.  P. )
38 simp3 948 . . . . 5  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  h  e.  P. )
39 addclpr 7193 . . . . . 6  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  +P.  s
)  e.  P. )
4039adantl 272 . . . . 5  |-  ( ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  /\  ( r  e.  P.  /\  s  e.  P. )
)  ->  ( r  +P.  s )  e.  P. )
41 mulcomprg 7236 . . . . . 6  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  .P.  s
)  =  ( s  .P.  r ) )
4241adantl 272 . . . . 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 5859 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  .P.  h )  =  ( ( f  .P.  h )  +P.  ( g  .P.  h
) ) )
4443adantl 272 . . 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 7237 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  .P.  g
)  .P.  h )  =  ( f  .P.  ( g  .P.  h
) ) )
4645adantl 272 . . 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 7228 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  e.  P. )
4847adantl 272 . . 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 970 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  x  e.  P. )
50 simp1r 971 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  y  e.  P. )
51 simp2l 972 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  z  e.  P. )
52 simp2r 973 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  w  e.  P. )
53 simp3l 974 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  v  e.  P. )
54 simp3r 975 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  u  e.  P. )
55 addcomprg 7234 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
5655adantl 272 . . 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 7235 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
5857adantl 272 . . 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 7193 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
6059adantl 272 . . 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 5875 . 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 5875 . 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 6442 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 927    = wceq 1296    e. wcel 1445  (class class class)co 5690   P.cnp 6947    +P. cpp 6949    .P. cmp 6950    ~R cer 6952   R.cnr 6953    .R cmr 6958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-2o 6220  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-enq0 7080  df-nq0 7081  df-0nq0 7082  df-plq0 7083  df-mq0 7084  df-inp 7122  df-iplp 7124  df-imp 7125  df-enr 7369  df-nr 7370  df-mr 7372
This theorem is referenced by:  axmulass  7505
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