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Theorem distrlem5pr 9417
Description: Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
distrlem5pr  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  C_  ( A  .P.  ( B  +P.  C ) ) )

Proof of Theorem distrlem5pr
Dummy variables  x  y  z  w  v  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 9410 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 1016 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 9410 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 1015 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-plp 9373 . . . . 5  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  { f  |  E. g  e.  x  E. h  e.  y  f  =  ( g  +Q  h ) } )
6 addclnq 9335 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelv 9390 . . . 4  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( w  e.  ( ( A  .P.  B
)  +P.  ( A  .P.  C ) )  <->  E. v  e.  ( A  .P.  B
) E. u  e.  ( A  .P.  C
) w  =  ( v  +Q  u ) ) )
82, 4, 7syl2anc 661 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( ( A  .P.  B )  +P.  ( A  .P.  C ) )  <->  E. v  e.  ( A  .P.  B
) E. u  e.  ( A  .P.  C
) w  =  ( v  +Q  u ) ) )
9 df-mp 9374 . . . . . . . 8  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. g  e.  w  E. h  e.  v  x  =  ( g  .Q  h ) } )
10 mulclnq 9337 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelv 9390 . . . . . . 7  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( u  e.  ( A  .P.  C )  <->  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z ) ) )
12113adant2 1015 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
u  e.  ( A  .P.  C )  <->  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z
) ) )
1312anbi2d 703 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( A  .P.  B )  /\  u  e.  ( A  .P.  C ) )  <->  ( v  e.  ( A  .P.  B
)  /\  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z
) ) ) )
14 df-mp 9374 . . . . . . . . 9  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { f  |  E. g  e.  w  E. h  e.  v  f  =  ( g  .Q  h ) } )
1514, 10genpelv 9390 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( v  e.  ( A  .P.  B )  <->  E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y ) ) )
16153adant3 1016 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( A  .P.  B )  <->  E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y
) ) )
17 distrlem4pr 9416 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  A  /\  z  e.  C )
) )  ->  (
( x  .Q  y
)  +Q  ( f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) )
18 oveq12 6304 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2481 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2539 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  ->  (
w  e.  ( A  .P.  ( B  +P.  C ) )  <->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) ) )
2119, 20syl6bi 228 . . . . . . . . . . . . . . . 16  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  ->  ( w  e.  ( A  .P.  ( B  +P.  C ) )  <-> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) ) ) )
2221imp 429 . . . . . . . . . . . . . . 15  |-  ( ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  /\  w  =  ( v  +Q  u
) )  ->  (
w  e.  ( A  .P.  ( B  +P.  C ) )  <->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( A  .P.  ( B  +P.  C ) ) ) )
2317, 22syl5ibrcom 222 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  A  /\  z  e.  C )
) )  ->  (
( ( v  =  ( x  .Q  y
)  /\  u  =  ( f  .Q  z
) )  /\  w  =  ( v  +Q  u ) )  ->  w  e.  ( A  .P.  ( B  +P.  C
) ) ) )
2423exp4b 607 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( ( x  e.  A  /\  y  e.  B )  /\  (
f  e.  A  /\  z  e.  C )
)  ->  ( (
v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
2524com3l 81 . . . . . . . . . . . 12  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( f  e.  A  /\  z  e.  C ) )  -> 
( ( v  =  ( x  .Q  y
)  /\  u  =  ( f  .Q  z
) )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
2625exp4b 607 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ( f  e.  A  /\  z  e.  C )  ->  (
v  =  ( x  .Q  y )  -> 
( u  =  ( f  .Q  z )  ->  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C
) ) ) ) ) ) ) )
2726com23 78 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( v  =  ( x  .Q  y )  ->  ( ( f  e.  A  /\  z  e.  C )  ->  (
u  =  ( f  .Q  z )  -> 
( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) ) )
2827rexlimivv 2964 . . . . . . . . 9  |-  ( E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y )  ->  (
( f  e.  A  /\  z  e.  C
)  ->  ( u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
2928rexlimdvv 2965 . . . . . . . 8  |-  ( E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y )  ->  ( E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3029com3r 79 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. x  e.  A  E. y  e.  B  v  =  ( x  .Q  y )  ->  ( E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C
) ) ) ) ) )
3116, 30sylbid 215 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( A  .P.  B )  -> 
( E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3231impd 431 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( A  .P.  B )  /\  E. f  e.  A  E. z  e.  C  u  =  ( f  .Q  z ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) )
3313, 32sylbid 215 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( A  .P.  B )  /\  u  e.  ( A  .P.  C ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) ) )
3433rexlimdvv 2965 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. v  e.  ( A  .P.  B ) E. u  e.  ( A  .P.  C ) w  =  ( v  +Q  u )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) )
358, 34sylbid 215 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( ( A  .P.  B )  +P.  ( A  .P.  C ) )  ->  w  e.  ( A  .P.  ( B  +P.  C ) ) ) )
3635ssrdv 3515 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( A  .P.  B
)  +P.  ( A  .P.  C ) )  C_  ( A  .P.  ( B  +P.  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818    C_ wss 3481  (class class class)co 6295    +Q cplq 9245    .Q cmq 9246   P.cnp 9249    +P. cpp 9251    .P. cmp 9252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-omul 7147  df-er 7323  df-ni 9262  df-pli 9263  df-mi 9264  df-lti 9265  df-plpq 9298  df-mpq 9299  df-ltpq 9300  df-enq 9301  df-nq 9302  df-erq 9303  df-plq 9304  df-mq 9305  df-1nq 9306  df-rq 9307  df-ltnq 9308  df-np 9371  df-plp 9373  df-mp 9374
This theorem is referenced by:  distrpr  9418
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