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Theorem distrlem5prl 6890
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem5prl  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) )  C_  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )

Proof of Theorem distrlem5prl
Dummy variables  x  y  z  w  v  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 6876 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 959 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 6876 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 958 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-iplp 6772 . . . . 5  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  x )  /\  h  e.  ( 1st `  y
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  x )  /\  h  e.  ( 2nd `  y
)  /\  f  =  ( g  +Q  h
) ) } >. )
6 addclnq 6679 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelvl 6816 . . . 4  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( w  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  <->  E. v  e.  ( 1st `  ( A  .P.  B ) ) E. u  e.  ( 1st `  ( A  .P.  C ) ) w  =  ( v  +Q  u ) ) )
82, 4, 7syl2anc 403 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  <->  E. v  e.  ( 1st `  ( A  .P.  B ) ) E. u  e.  ( 1st `  ( A  .P.  C ) ) w  =  ( v  +Q  u ) ) )
9 df-imp 6773 . . . . . . . 8  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  w )  /\  h  e.  ( 1st `  v
)  /\  x  =  ( g  .Q  h
) ) } ,  { x  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  w )  /\  h  e.  ( 2nd `  v
)  /\  x  =  ( g  .Q  h
) ) } >. )
10 mulclnq 6680 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelvl 6816 . . . . . . 7  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( u  e.  ( 1st `  ( A  .P.  C ) )  <->  E. f  e.  ( 1st `  A ) E. z  e.  ( 1st `  C ) u  =  ( f  .Q  z
) ) )
12113adant2 958 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
u  e.  ( 1st `  ( A  .P.  C
) )  <->  E. f  e.  ( 1st `  A
) E. z  e.  ( 1st `  C
) u  =  ( f  .Q  z ) ) )
1312anbi2d 452 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( 1st `  ( A  .P.  B ) )  /\  u  e.  ( 1st `  ( A  .P.  C ) ) )  <->  ( v  e.  ( 1st `  ( A  .P.  B ) )  /\  E. f  e.  ( 1st `  A
) E. z  e.  ( 1st `  C
) u  =  ( f  .Q  z ) ) ) )
14 df-imp 6773 . . . . . . . . 9  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  w )  /\  h  e.  ( 1st `  v
)  /\  f  =  ( g  .Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  w )  /\  h  e.  ( 2nd `  v
)  /\  f  =  ( g  .Q  h
) ) } >. )
1514, 10genpelvl 6816 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( v  e.  ( 1st `  ( A  .P.  B ) )  <->  E. x  e.  ( 1st `  A ) E. y  e.  ( 1st `  B ) v  =  ( x  .Q  y
) ) )
16153adant3 959 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( 1st `  ( A  .P.  B
) )  <->  E. x  e.  ( 1st `  A
) E. y  e.  ( 1st `  B
) v  =  ( x  .Q  y ) ) )
17 distrlem4prl 6888 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
18 oveq12 5572 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2094 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2145 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  ->  (
w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  <->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
2119, 20syl6bi 161 . . . . . . . . . . . . . . . 16  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  ->  ( w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  <->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
2221imp 122 . . . . . . . . . . . . . . 15  |-  ( ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  /\  w  =  ( v  +Q  u
) )  ->  (
w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  <->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
2317, 22syl5ibrcom 155 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  /\  w  =  ( v  +Q  u ) )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
2423exp4b 359 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) )  ->  (
( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
2524com3l 80 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) )  ->  (
( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  ->  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
2625exp4b 359 . . . . . . . . . . 11  |-  ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B
) )  ->  (
( f  e.  ( 1st `  A )  /\  z  e.  ( 1st `  C ) )  ->  ( v  =  ( x  .Q  y )  ->  (
u  =  ( f  .Q  z )  -> 
( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) ) ) )
2726com23 77 . . . . . . . . . 10  |-  ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B
) )  ->  (
v  =  ( x  .Q  y )  -> 
( ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) )  ->  ( u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) ) ) )
2827rexlimivv 2487 . . . . . . . . 9  |-  ( E. x  e.  ( 1st `  A ) E. y  e.  ( 1st `  B
) v  =  ( x  .Q  y )  ->  ( ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) )  ->  ( u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) ) )
2928rexlimdvv 2488 . . . . . . . 8  |-  ( E. x  e.  ( 1st `  A ) E. y  e.  ( 1st `  B
) v  =  ( x  .Q  y )  ->  ( E. f  e.  ( 1st `  A
) E. z  e.  ( 1st `  C
) u  =  ( f  .Q  z )  ->  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
3029com3r 78 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. x  e.  ( 1st `  A ) E. y  e.  ( 1st `  B ) v  =  ( x  .Q  y
)  ->  ( E. f  e.  ( 1st `  A ) E. z  e.  ( 1st `  C
) u  =  ( f  .Q  z )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
3116, 30sylbid 148 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( 1st `  ( A  .P.  B
) )  ->  ( E. f  e.  ( 1st `  A ) E. z  e.  ( 1st `  C ) u  =  ( f  .Q  z
)  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
3231impd 251 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( 1st `  ( A  .P.  B ) )  /\  E. f  e.  ( 1st `  A
) E. z  e.  ( 1st `  C
) u  =  ( f  .Q  z ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3313, 32sylbid 148 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( 1st `  ( A  .P.  B ) )  /\  u  e.  ( 1st `  ( A  .P.  C ) ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3433rexlimdvv 2488 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. v  e.  ( 1st `  ( A  .P.  B ) ) E. u  e.  ( 1st `  ( A  .P.  C ) ) w  =  ( v  +Q  u )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
358, 34sylbid 148 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  ->  w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
3635ssrdv 3014 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) )  C_  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   E.wrex 2354    C_ wss 2982   ` cfv 4952  (class class class)co 5563   1stc1st 5816    +Q cplq 6586    .Q cmq 6587   P.cnp 6595    +P. cpp 6597    .P. cmp 6598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-2o 6086  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-mpq 6649  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-mqqs 6654  df-1nqqs 6655  df-rq 6656  df-ltnqqs 6657  df-enq0 6728  df-nq0 6729  df-0nq0 6730  df-plq0 6731  df-mq0 6732  df-inp 6770  df-iplp 6772  df-imp 6773
This theorem is referenced by:  distrprg  6892
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