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

Theorem distrlem1pru 7398
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 7352 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
2 df-imp 7284 . . . . . 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 7191 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
42, 3genpelvu 7328 . . . . 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 284 . . . 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 1177 . . 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 7283 . . . . . . . . . . 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 7190 . . . . . . . . . . 11  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
97, 8genpelvu 7328 . . . . . . . . . 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 999 . . . . . . . . 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 274 . . . . . . . 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 7290 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 elprnqu 7297 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1412, 13sylan 281 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
15143ad2antl1 1143 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
1615adantrr 470 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  x  e.  Q. )
1716adantr 274 . . . . . . . . . . . . 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 7290 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 elprnqu 7297 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
2018, 19sylan 281 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
21 prop 7290 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
22 elprnqu 7297 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 2nd `  C ) )  -> 
z  e.  Q. )
2321, 22sylan 281 . . . . . . . . . . . . . . . . 17  |-  ( ( C  e.  P.  /\  z  e.  ( 2nd `  C ) )  -> 
z  e.  Q. )
2420, 23anim12i 336 . . . . . . . . . . . . . . . 16  |-  ( ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  /\  ( C  e.  P.  /\  z  e.  ( 2nd `  C ) ) )  ->  ( y  e. 
Q.  /\  z  e.  Q. ) )
2524an4s 577 . . . . . . . . . . . . . . 15  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) ) )  ->  (
y  e.  Q.  /\  z  e.  Q. )
)
26253adantl1 1137 . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . 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 966 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  <->  ( x  e.  Q.  /\  ( y  e.  Q.  /\  z  e.  Q. ) ) )
2917, 27, 28sylanbrc 413 . . . . . . . . . . . 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 521 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 2nd `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  w  =  ( x  .Q  v ) )
31 simpr 109 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  v  =  ( y  +Q  z ) )
3230, 31anim12i 336 . . . . . . . . . . . 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 5782 . . . . . . . . . . . . . . 15  |-  ( v  =  ( y  +Q  z )  ->  (
x  .Q  v )  =  ( x  .Q  ( y  +Q  z
) ) )
3433eqeq2d 2151 . . . . . . . . . . . . . 14  |-  ( v  =  ( y  +Q  z )  ->  (
w  =  ( x  .Q  v )  <->  w  =  ( x  .Q  (
y  +Q  z ) ) ) )
3534biimpac 296 . . . . . . . . . . . . 13  |-  ( ( w  =  ( x  .Q  v )  /\  v  =  ( y  +Q  z ) )  ->  w  =  ( x  .Q  ( y  +Q  z
) ) )
36 distrnqg 7202 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  .Q  ( y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z
) ) )
3736eqeq2d 2151 . . . . . . . . . . . . 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, 37syl5ib 153 . . . . . . . . . . . 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 7387 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
41403adant3 1001 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
4241ad2antrr 479 . . . . . . . . . . . 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 7387 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
44433adant2 1000 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
4544ad2antrr 479 . . . . . . . . . . . 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 518 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  y  e.  ( 2nd `  B
) )
472, 3genppreclu 7330 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  ->  ( x  .Q  y )  e.  ( 2nd `  ( A  .P.  B ) ) ) )
48473adant3 1001 . . . . . . . . . . . . . . 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 377 . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 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 284 . . . . . . . . . . . 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 519 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  z  e.  ( 2nd `  C
) )
532, 3genppreclu 7330 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( x  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) )  ->  ( x  .Q  z )  e.  ( 2nd `  ( A  .P.  C ) ) ) )
54533adant2 1000 . . . . . . . . . . . . . . 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 377 . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 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 284 . . . . . . . . . . . 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 7330 . . . . . . . . . . . . 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 123 . . . . . . . . . . . 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 1217 . . . . . . . . . . 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 2216 . . . . . . . . . 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 362 . . . . . . . . 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 2556 . . . . . . . 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 149 . . . . . . 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 362 . . . . . 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 252 . . . 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 2556 . . 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 149 . 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 3103 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 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417    C_ wss 3071   <.cop 3530   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095    +Q cplq 7097    .Q cmq 7098   P.cnp 7106    +P. cpp 7108    .P. cmp 7109
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283  df-imp 7284
This theorem is referenced by:  distrprg  7403
  Copyright terms: Public domain W3C validator