ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  distrlem5pru Unicode version

Theorem distrlem5pru 6713
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 6698 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 933 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 6698 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 932 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-iplp 6594 . . . . 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 6501 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelvu 6639 . . . 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 397 . . 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 6595 . . . . . . . 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 6502 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelvu 6639 . . . . . . 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 932 . . . . . 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 445 . . . . 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 6595 . . . . . . . . 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 6639 . . . . . . . 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 933 . . . . . . 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 6711 . . . . . . . . . . . . . . 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 5546 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2065 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2114 . . . . . . . . . . . . . . . . 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 156 . . . . . . . . . . . . . . . 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 119 . . . . . . . . . . . . . . 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 150 . . . . . . . . . . . . . 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 353 . . . . . . . . . . . . 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 79 . . . . . . . . . . . 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 353 . . . . . . . . . . 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 76 . . . . . . . . . 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 2453 . . . . . . . . 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 2454 . . . . . . . 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 77 . . . . . . 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 143 . . . . . 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 246 . . . . 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 143 . . . 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 2454 . . 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 143 . 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 2976 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 101    <-> wb 102    /\ w3a 894    = wceq 1257    e. wcel 1407   E.wrex 2322    C_ wss 2942   ` cfv 4927  (class class class)co 5537   2ndc2nd 5791    +Q cplq 6408    .Q cmq 6409   P.cnp 6417    +P. cpp 6419    .P. cmp 6420
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 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-po 4058  df-iso 4059  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-2o 6030  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-mpq 6471  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-mqqs 6476  df-1nqqs 6477  df-rq 6478  df-ltnqqs 6479  df-enq0 6550  df-nq0 6551  df-0nq0 6552  df-plq0 6553  df-mq0 6554  df-inp 6592  df-iplp 6594  df-imp 6595
This theorem is referenced by:  distrprg  6714
  Copyright terms: Public domain W3C validator