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Theorem distrlem1pru 7359
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 7313 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
2 df-imp 7245 . . . . . 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 7152 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
42, 3genpelvu 7289 . . . . 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 1162 . . 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 7244 . . . . . . . . . . 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 7151 . . . . . . . . . . 11  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
97, 8genpelvu 7289 . . . . . . . . . 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 984 . . . . . . . . 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 7251 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 elprnqu 7258 . . . . . . . . . . . . . . . . 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 1128 . . . . . . . . . . . . . . 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 7251 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 elprnqu 7258 . . . . . . . . . . . . . . . . . 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 7251 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
22 elprnqu 7258 . . . . . . . . . . . . . . . . . 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 562 . . . . . . . . . . . . . . 15  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) ) )  ->  (
y  e.  Q.  /\  z  e.  Q. )
)
26253adantl1 1122 . . . . . . . . . . . . . 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 951 . . . . . . . . . . . . 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 506 . . . . . . . . . . . . 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 5750 . . . . . . . . . . . . . . 15  |-  ( v  =  ( y  +Q  z )  ->  (
x  .Q  v )  =  ( x  .Q  ( y  +Q  z
) ) )
3433eqeq2d 2129 . . . . . . . . . . . . . 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 7163 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  .Q  ( y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z
) ) )
3736eqeq2d 2129 . . . . . . . . . . . . 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 7348 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
41403adant3 986 . . . . . . . . . . . . 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 7348 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
44433adant2 985 . . . . . . . . . . . . 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 503 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  y  e.  ( 2nd `  B
) )
472, 3genppreclu 7291 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  ->  ( x  .Q  y )  e.  ( 2nd `  ( A  .P.  B ) ) ) )
48473adant3 986 . . . . . . . . . . . . . . 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 504 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  z  e.  ( 2nd `  C
) )
532, 3genppreclu 7291 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( x  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) )  ->  ( x  .Q  z )  e.  ( 2nd `  ( A  .P.  C ) ) ) )
54533adant2 985 . . . . . . . . . . . . . . 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 7291 . . . . . . . . . . . . 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 1202 . . . . . . . . . . 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 2194 . . . . . . . . . 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 2533 . . . . . . . 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 2533 . . 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 3073 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 947    = wceq 1316    e. wcel 1465   E.wrex 2394    C_ wss 3041   <.cop 3500   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    +Q cplq 7058    .Q cmq 7059   P.cnp 7067    +P. cpp 7069    .P. cmp 7070
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-imp 7245
This theorem is referenced by:  distrprg  7364
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