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Theorem distrlem5prl 6741
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 6727 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 935 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 6727 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 934 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-iplp 6623 . . . . 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 6530 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelvl 6667 . . . 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 397 . . 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 6624 . . . . . . . 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 6531 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelvl 6667 . . . . . . 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 934 . . . . . 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 445 . . . . 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 6624 . . . . . . . . 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 6667 . . . . . . . 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 935 . . . . . . 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 6739 . . . . . . . . . . . . . . 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 5548 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2067 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2116 . . . . . . . . . . . . . . . . 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 156 . . . . . . . . . . . . . . . 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 119 . . . . . . . . . . . . . . 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 150 . . . . . . . . . . . . . 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 353 . . . . . . . . . . . . 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 79 . . . . . . . . . . . 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 353 . . . . . . . . . . 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 76 . . . . . . . . . 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 2455 . . . . . . . . 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 2456 . . . . . . . 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 77 . . . . . . 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 143 . . . . . 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 246 . . . . 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 143 . . . 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 2456 . . 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 143 . 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 2978 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 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324    C_ wss 2944   ` cfv 4929  (class class class)co 5539   1stc1st 5792    +Q cplq 6437    .Q cmq 6438   P.cnp 6446    +P. cpp 6448    .P. cmp 6449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iplp 6623  df-imp 6624
This theorem is referenced by:  distrprg  6743
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