ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  distrlem1pru Unicode version

Theorem distrlem1pru 7582
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem1pru  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  C_  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) ) )

Proof of Theorem distrlem1pru
Dummy variables  x  y  z  w  v  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addclpr 7536 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
2 df-imp 7468 . . . . . 6  |-  .P.  =  ( y  e.  P. ,  z  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  y )  /\  h  e.  ( 1st `  z
)  /\  f  =  ( g  .Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  y )  /\  h  e.  ( 2nd `  z
)  /\  f  =  ( g  .Q  h
) ) } >. )
3 mulclnq 7375 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
42, 3genpelvu 7512 . . . . 5  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  <->  E. x  e.  ( 2nd `  A
) E. v  e.  ( 2nd `  ( B  +P.  C ) ) w  =  ( x  .Q  v ) ) )
51, 4sylan2 286 . . . 4  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  C  e.  P. )
)  ->  ( w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  <->  E. x  e.  ( 2nd `  A
) E. v  e.  ( 2nd `  ( B  +P.  C ) ) w  =  ( x  .Q  v ) ) )
653impb 1199 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  <->  E. x  e.  ( 2nd `  A ) E. v  e.  ( 2nd `  ( B  +P.  C ) ) w  =  ( x  .Q  v ) ) )
7 df-iplp 7467 . . . . . . . . . . 11  |-  +P.  =  ( w  e.  P. ,  x  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  w )  /\  h  e.  ( 1st `  x
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  w )  /\  h  e.  ( 2nd `  x
)  /\  f  =  ( g  +Q  h
) ) } >. )
8 addclnq 7374 . . . . . . . . . . 11  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
97, 8genpelvu 7512 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( v  e.  ( 2nd `  ( B  +P.  C ) )  <->  E. y  e.  ( 2nd `  B ) E. z  e.  ( 2nd `  C ) v  =  ( y  +Q  z
) ) )
1093adant1 1015 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( 2nd `  ( B  +P.  C
) )  <->  E. y  e.  ( 2nd `  B
) E. z  e.  ( 2nd `  C
) v  =  ( y  +Q  z ) ) )
1110adantr 276 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  (
v  e.  ( 2nd `  ( B  +P.  C
) )  <->  E. y  e.  ( 2nd `  B
) E. z  e.  ( 2nd `  C
) v  =  ( y  +Q  z ) ) )
12 prop 7474 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 elprnqu 7481 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1412, 13sylan 283 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
15143ad2antl1 1159 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1615adantrr 479 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  x  e.  Q. )
1716adantr 276 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  ->  x  e.  Q. )
18 prop 7474 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 elprnqu 7481 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
2018, 19sylan 283 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
21 prop 7474 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
22 elprnqu 7481 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 2nd `  C ) )  -> 
z  e.  Q. )
2321, 22sylan 283 . . . . . . . . . . . . . . . . 17  |-  ( ( C  e.  P.  /\  z  e.  ( 2nd `  C ) )  -> 
z  e.  Q. )
2420, 23anim12i 338 . . . . . . . . . . . . . . . 16  |-  ( ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  /\  ( C  e.  P.  /\  z  e.  ( 2nd `  C ) ) )  ->  ( y  e. 
Q.  /\  z  e.  Q. ) )
2524an4s 588 . . . . . . . . . . . . . . 15  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) ) )  ->  (
y  e.  Q.  /\  z  e.  Q. )
)
26253adantl1 1153 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) ) )  ->  (
y  e.  Q.  /\  z  e.  Q. )
)
2726ad2ant2r 509 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( y  e.  Q.  /\  z  e.  Q. )
)
28 3anass 982 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  <->  ( x  e.  Q.  /\  ( y  e.  Q.  /\  z  e.  Q. ) ) )
2917, 27, 28sylanbrc 417 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )
)
30 simprr 531 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  w  =  ( x  .Q  v ) )
31 simpr 110 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  v  =  ( y  +Q  z ) )
3230, 31anim12i 338 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( w  =  ( x  .Q  v )  /\  v  =  ( y  +Q  z ) ) )
33 oveq2 5883 . . . . . . . . . . . . . . 15  |-  ( v  =  ( y  +Q  z )  ->  (
x  .Q  v )  =  ( x  .Q  ( y  +Q  z
) ) )
3433eqeq2d 2189 . . . . . . . . . . . . . 14  |-  ( v  =  ( y  +Q  z )  ->  (
w  =  ( x  .Q  v )  <->  w  =  ( x  .Q  (
y  +Q  z ) ) ) )
3534biimpac 298 . . . . . . . . . . . . 13  |-  ( ( w  =  ( x  .Q  v )  /\  v  =  ( y  +Q  z ) )  ->  w  =  ( x  .Q  ( y  +Q  z
) ) )
36 distrnqg 7386 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  .Q  ( y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z
) ) )
3736eqeq2d 2189 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
w  =  ( x  .Q  ( y  +Q  z ) )  <->  w  =  ( ( x  .Q  y )  +Q  (
x  .Q  z ) ) ) )
3835, 37imbitrid 154 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( w  =  ( x  .Q  v )  /\  v  =  ( y  +Q  z ) )  ->  w  =  ( ( x  .Q  y )  +Q  (
x  .Q  z ) ) ) )
3929, 32, 38sylc 62 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  ->  w  =  ( (
x  .Q  y )  +Q  ( x  .Q  z ) ) )
40 mulclpr 7571 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
41403adant3 1017 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
4241ad2antrr 488 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( A  .P.  B
)  e.  P. )
43 mulclpr 7571 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
44433adant2 1016 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
4544ad2antrr 488 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( A  .P.  C
)  e.  P. )
46 simpll 527 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  y  e.  ( 2nd `  B
) )
472, 3genppreclu 7514 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  ->  ( x  .Q  y )  e.  ( 2nd `  ( A  .P.  B ) ) ) )
48473adant3 1017 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( x  e.  ( 2nd `  A )  /\  y  e.  ( 2nd `  B ) )  ->  ( x  .Q  y )  e.  ( 2nd `  ( A  .P.  B ) ) ) )
4948impl 380 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  x  e.  ( 2nd `  A ) )  /\  y  e.  ( 2nd `  B
) )  ->  (
x  .Q  y )  e.  ( 2nd `  ( A  .P.  B ) ) )
5049adantlrr 483 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  y  e.  ( 2nd `  B ) )  -> 
( x  .Q  y
)  e.  ( 2nd `  ( A  .P.  B
) ) )
5146, 50sylan2 286 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( x  .Q  y
)  e.  ( 2nd `  ( A  .P.  B
) ) )
52 simplr 528 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  z  e.  ( 2nd `  C
) )
532, 3genppreclu 7514 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( x  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) )  ->  ( x  .Q  z )  e.  ( 2nd `  ( A  .P.  C ) ) ) )
54533adant2 1016 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( x  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  C ) )  ->  ( x  .Q  z )  e.  ( 2nd `  ( A  .P.  C ) ) ) )
5554impl 380 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  x  e.  ( 2nd `  A ) )  /\  z  e.  ( 2nd `  C
) )  ->  (
x  .Q  z )  e.  ( 2nd `  ( A  .P.  C ) ) )
5655adantlrr 483 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  z  e.  ( 2nd `  C ) )  -> 
( x  .Q  z
)  e.  ( 2nd `  ( A  .P.  C
) ) )
5752, 56sylan2 286 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( x  .Q  z
)  e.  ( 2nd `  ( A  .P.  C
) ) )
587, 8genppreclu 7514 . . . . . . . . . . . . 13  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( ( ( x  .Q  y )  e.  ( 2nd `  ( A  .P.  B ) )  /\  ( x  .Q  z )  e.  ( 2nd `  ( A  .P.  C ) ) )  ->  ( (
x  .Q  y )  +Q  ( x  .Q  z ) )  e.  ( 2nd `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) )
5958imp 124 . . . . . . . . . . . 12  |-  ( ( ( ( A  .P.  B )  e.  P.  /\  ( A  .P.  C )  e.  P. )  /\  ( ( x  .Q  y )  e.  ( 2nd `  ( A  .P.  B ) )  /\  ( x  .Q  z )  e.  ( 2nd `  ( A  .P.  C ) ) ) )  ->  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  e.  ( 2nd `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) )
6042, 45, 51, 57, 59syl22anc 1239 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( ( x  .Q  y )  +Q  (
x  .Q  z ) )  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) )
6139, 60eqeltrd 2254 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) ) )  ->  w  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) )
6261exp32 365 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  (
( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) )  ->  ( v  =  ( y  +Q  z )  ->  w  e.  ( 2nd `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) ) )
6362rexlimdvv 2601 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  ( E. y  e.  ( 2nd `  B ) E. z  e.  ( 2nd `  C ) v  =  ( y  +Q  z
)  ->  w  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) )
6411, 63sylbid 150 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  (
v  e.  ( 2nd `  ( B  +P.  C
) )  ->  w  e.  ( 2nd `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) )
6564exp32 365 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 2nd `  A )  ->  (
w  =  ( x  .Q  v )  -> 
( v  e.  ( 2nd `  ( B  +P.  C ) )  ->  w  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) ) ) )
6665com34 83 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 2nd `  A )  ->  (
v  e.  ( 2nd `  ( B  +P.  C
) )  ->  (
w  =  ( x  .Q  v )  ->  w  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) ) ) )
6766impd 254 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( x  e.  ( 2nd `  A )  /\  v  e.  ( 2nd `  ( B  +P.  C ) ) )  ->  ( w  =  ( x  .Q  v )  ->  w  e.  ( 2nd `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) ) )
6867rexlimdvv 2601 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. x  e.  ( 2nd `  A ) E. v  e.  ( 2nd `  ( B  +P.  C
) ) w  =  ( x  .Q  v
)  ->  w  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) )
696, 68sylbid 150 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  ->  w  e.  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) )
7069ssrdv 3162 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  C_  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456    C_ wss 3130   <.cop 3596   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279    +Q cplq 7281    .Q cmq 7282   P.cnp 7290    +P. cpp 7292    .P. cmp 7293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iplp 7467  df-imp 7468
This theorem is referenced by:  distrprg  7587
  Copyright terms: Public domain W3C validator