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Theorem distrlem5pr 9217
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 9210 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 1008 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 9210 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 1007 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-plp 9173 . . . . 5  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  { f  |  E. g  e.  x  E. h  e.  y  f  =  ( g  +Q  h ) } )
6 addclnq 9135 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelv 9190 . . . 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 9174 . . . . . . . 8  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. g  e.  w  E. h  e.  v  x  =  ( g  .Q  h ) } )
10 mulclnq 9137 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelv 9190 . . . . . . 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 1007 . . . . . 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 9174 . . . . . . . . 9  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { f  |  E. g  e.  w  E. h  e.  v  f  =  ( g  .Q  h ) } )
1514, 10genpelv 9190 . . . . . . . 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 1008 . . . . . . 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 9216 . . . . . . . . . . . . . . 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 6121 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2454 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2503 . . . . . . . . . . . . . . . . 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 2867 . . . . . . . . 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 2868 . . . . . . . 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 2868 . . 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 3383 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 965    = wceq 1369    e. wcel 1756   E.wrex 2737    C_ wss 3349  (class class class)co 6112    +Q cplq 9043    .Q cmq 9044   P.cnp 9047    +P. cpp 9049    .P. cmp 9050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-omul 6946  df-er 7122  df-ni 9062  df-pli 9063  df-mi 9064  df-lti 9065  df-plpq 9098  df-mpq 9099  df-ltpq 9100  df-enq 9101  df-nq 9102  df-erq 9103  df-plq 9104  df-mq 9105  df-1nq 9106  df-rq 9107  df-ltnq 9108  df-np 9171  df-plp 9173  df-mp 9174
This theorem is referenced by:  distrpr  9218
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