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

Theorem distrlem5pru 7359
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 7344 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
213adant3 984 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
3 mulclpr 7344 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
433adant2 983 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
5 df-iplp 7240 . . . . 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 7147 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
75, 6genpelvu 7285 . . . 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 406 . . 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 7241 . . . . . . . 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 7148 . . . . . . . 8  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
119, 10genpelvu 7285 . . . . . . 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 983 . . . . . 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 457 . . . . 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 7241 . . . . . . . . 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 7285 . . . . . . . 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 984 . . . . . . 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 7357 . . . . . . . . . . . . . . 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 5749 . . . . . . . . . . . . . . . . . 18  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( v  +Q  u
)  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
1918eqeq2d 2127 . . . . . . . . . . . . . . . . 17  |-  ( ( v  =  ( x  .Q  y )  /\  u  =  ( f  .Q  z ) )  -> 
( w  =  ( v  +Q  u )  <-> 
w  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
20 eleq1 2178 . . . . . . . . . . . . . . . . 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 162 . . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . . . 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 156 . . . . . . . . . . . . . 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 362 . . . . . . . . . . . . 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 81 . . . . . . . . . . . 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 362 . . . . . . . . . . 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 78 . . . . . . . . . 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 2530 . . . . . . . . 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 2531 . . . . . . . 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 79 . . . . . . 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 149 . . . . . 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 252 . . . . 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 149 . . . 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 2531 . . 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 149 . 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 3071 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 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2392    C_ wss 3039   ` cfv 5091  (class class class)co 5740   2ndc2nd 6003    +Q cplq 7054    .Q cmq 7055   P.cnp 7063    +P. cpp 7065    .P. cmp 7066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-imp 7241
This theorem is referenced by:  distrprg  7360
  Copyright terms: Public domain W3C validator