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Theorem distrlem5pru 7125
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem5pru  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) )  C_  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )

Proof of Theorem distrlem5pru
Dummy variables  x  y  z  w  v  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 7110 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 963 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 7110 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 962 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-iplp 7006 . . . . 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 6913 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelvu 7051 . . . 4  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( w  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  <->  E. v  e.  ( 2nd `  ( A  .P.  B ) ) E. u  e.  ( 2nd `  ( A  .P.  C ) ) w  =  ( v  +Q  u ) ) )
82, 4, 7syl2anc 403 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  <->  E. v  e.  ( 2nd `  ( A  .P.  B ) ) E. u  e.  ( 2nd `  ( A  .P.  C ) ) w  =  ( v  +Q  u ) ) )
9 df-imp 7007 . . . . . . . 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 6914 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelvu 7051 . . . . . . 7  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( u  e.  ( 2nd `  ( A  .P.  C ) )  <->  E. f  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  C ) u  =  ( f  .Q  z
) ) )
12113adant2 962 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
u  e.  ( 2nd `  ( A  .P.  C
) )  <->  E. f  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  C
) u  =  ( f  .Q  z ) ) )
1312anbi2d 452 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( 2nd `  ( A  .P.  B ) )  /\  u  e.  ( 2nd `  ( A  .P.  C ) ) )  <->  ( v  e.  ( 2nd `  ( A  .P.  B ) )  /\  E. f  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  C
) u  =  ( f  .Q  z ) ) ) )
14 df-imp 7007 . . . . . . . . 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, 10genpelvu 7051 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( v  e.  ( 2nd `  ( A  .P.  B ) )  <->  E. x  e.  ( 2nd `  A ) E. y  e.  ( 2nd `  B ) v  =  ( x  .Q  y
) ) )
16153adant3 963 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( 2nd `  ( A  .P.  B
) )  <->  E. x  e.  ( 2nd `  A
) E. y  e.  ( 2nd `  B
) v  =  ( x  .Q  y ) ) )
17 distrlem4pru 7123 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
18 oveq12 5643 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2099 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2150 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  ->  (
w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  <->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
2119, 20syl6bi 161 . . . . . . . . . . . . . . . 16  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  ->  ( w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  <->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
2221imp 122 . . . . . . . . . . . . . . 15  |-  ( ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  /\  w  =  ( v  +Q  u
) )  ->  (
w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  <->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
2317, 22syl5ibrcom 155 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  /\  w  =  ( v  +Q  u ) )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
2423exp4b 359 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) )  ->  (
( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
2524com3l 80 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ( 2nd `  A )  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) )  ->  (
( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  ->  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
2625exp4b 359 . . . . . . . . . . 11  |-  ( ( x  e.  ( 2nd `  A )  /\  y  e.  ( 2nd `  B
) )  ->  (
( f  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  C ) )  ->  ( v  =  ( x  .Q  y )  ->  (
u  =  ( f  .Q  z )  -> 
( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) ) ) )
2726com23 77 . . . . . . . . . 10  |-  ( ( x  e.  ( 2nd `  A )  /\  y  e.  ( 2nd `  B
) )  ->  (
v  =  ( x  .Q  y )  -> 
( ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) )  ->  ( u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) ) ) )
2827rexlimivv 2494 . . . . . . . . 9  |-  ( E. x  e.  ( 2nd `  A ) E. y  e.  ( 2nd `  B
) v  =  ( x  .Q  y )  ->  ( ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) )  ->  ( u  =  ( f  .Q  z )  ->  (
( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) ) )
2928rexlimdvv 2495 . . . . . . . 8  |-  ( E. x  e.  ( 2nd `  A ) E. y  e.  ( 2nd `  B
) v  =  ( x  .Q  y )  ->  ( E. f  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  C
) u  =  ( f  .Q  z )  ->  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  ->  (
w  =  ( v  +Q  u )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
3029com3r 78 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. x  e.  ( 2nd `  A ) E. y  e.  ( 2nd `  B ) v  =  ( x  .Q  y
)  ->  ( E. f  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  C
) u  =  ( f  .Q  z )  ->  ( w  =  ( v  +Q  u
)  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
3116, 30sylbid 148 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( 2nd `  ( A  .P.  B
) )  ->  ( E. f  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  C ) u  =  ( f  .Q  z
)  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) ) )
3231impd 251 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( 2nd `  ( A  .P.  B ) )  /\  E. f  e.  ( 2nd `  A
) E. z  e.  ( 2nd `  C
) u  =  ( f  .Q  z ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3313, 32sylbid 148 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( v  e.  ( 2nd `  ( A  .P.  B ) )  /\  u  e.  ( 2nd `  ( A  .P.  C ) ) )  ->  ( w  =  ( v  +Q  u )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) ) )
3433rexlimdvv 2495 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. v  e.  ( 2nd `  ( A  .P.  B ) ) E. u  e.  ( 2nd `  ( A  .P.  C ) ) w  =  ( v  +Q  u )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
358, 34sylbid 148 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  ->  w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
3635ssrdv 3029 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) )  C_  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360    C_ wss 2997   ` cfv 5002  (class class class)co 5634   2ndc2nd 5892    +Q cplq 6820    .Q cmq 6821   P.cnp 6829    +P. cpp 6831    .P. cmp 6832
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-imp 7007
This theorem is referenced by:  distrprg  7126
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