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Theorem distrlem5pr 9441
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 9434 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 1025 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 9434 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 1024 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-plp 9397 . . . . 5  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  { f  |  E. g  e.  x  E. h  e.  y  f  =  ( g  +Q  h ) } )
6 addclnq 9359 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelv 9414 . . . 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 665 . . 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 9398 . . . . . . . 8  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. g  e.  w  E. h  e.  v  x  =  ( g  .Q  h ) } )
10 mulclnq 9361 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelv 9414 . . . . . . 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 1024 . . . . . 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 708 . . . . 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 9398 . . . . . . . . 9  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { f  |  E. g  e.  w  E. h  e.  v  f  =  ( g  .Q  h ) } )
1514, 10genpelv 9414 . . . . . . . 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 1025 . . . . . . 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 9440 . . . . . . . . . . . . . . 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 6305 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2434 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2492 . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . 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 610 . . . . . . . . . . . . 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 84 . . . . . . . . . . . 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 610 . . . . . . . . . . 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 81 . . . . . . . . . 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 2920 . . . . . . . . 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 2921 . . . . . . . 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 82 . . . . . . 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 218 . . . . . 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 432 . . . . 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 218 . . . 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 2921 . . 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 218 . 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 3467 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   E.wrex 2774    C_ wss 3433  (class class class)co 6296    +Q cplq 9269    .Q cmq 9270   P.cnp 9273    +P. cpp 9275    .P. cmp 9276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-omul 7186  df-er 7362  df-ni 9286  df-pli 9287  df-mi 9288  df-lti 9289  df-plpq 9322  df-mpq 9323  df-ltpq 9324  df-enq 9325  df-nq 9326  df-erq 9327  df-plq 9328  df-mq 9329  df-1nq 9330  df-rq 9331  df-ltnq 9332  df-np 9395  df-plp 9397  df-mp 9398
This theorem is referenced by:  distrpr  9442
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