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Theorem distrlem1prl 7413
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem1prl  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  C_  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) ) )

Proof of Theorem distrlem1prl
Dummy variables  x  y  z  w  v  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addclpr 7368 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
2 df-imp 7300 . . . . . 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 7207 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
42, 3genpelvl 7343 . . . . 5  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  <->  E. x  e.  ( 1st `  A
) E. v  e.  ( 1st `  ( B  +P.  C ) ) w  =  ( x  .Q  v ) ) )
51, 4sylan2 284 . . . 4  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  C  e.  P. )
)  ->  ( w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  <->  E. x  e.  ( 1st `  A
) E. v  e.  ( 1st `  ( B  +P.  C ) ) w  =  ( x  .Q  v ) ) )
653impb 1178 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  <->  E. x  e.  ( 1st `  A ) E. v  e.  ( 1st `  ( B  +P.  C ) ) w  =  ( x  .Q  v ) ) )
7 df-iplp 7299 . . . . . . . . . . 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 7206 . . . . . . . . . . 11  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
97, 8genpelvl 7343 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( v  e.  ( 1st `  ( B  +P.  C ) )  <->  E. y  e.  ( 1st `  B ) E. z  e.  ( 1st `  C ) v  =  ( y  +Q  z
) ) )
1093adant1 1000 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
v  e.  ( 1st `  ( B  +P.  C
) )  <->  E. y  e.  ( 1st `  B
) E. z  e.  ( 1st `  C
) v  =  ( y  +Q  z ) ) )
1110adantr 274 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  (
v  e.  ( 1st `  ( B  +P.  C
) )  <->  E. y  e.  ( 1st `  B
) E. z  e.  ( 1st `  C
) v  =  ( y  +Q  z ) ) )
12 prop 7306 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 elprnql 7312 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1412, 13sylan 281 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
15143ad2antl1 1144 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1615adantrr 471 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  x  e.  Q. )
1716adantr 274 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  ->  x  e.  Q. )
18 prop 7306 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 elprnql 7312 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
2018, 19sylan 281 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
21 prop 7306 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
22 elprnql 7312 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
2321, 22sylan 281 . . . . . . . . . . . . . . . . 17  |-  ( ( C  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
2420, 23anim12i 336 . . . . . . . . . . . . . . . 16  |-  ( ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  /\  ( C  e.  P.  /\  z  e.  ( 1st `  C ) ) )  ->  ( y  e. 
Q.  /\  z  e.  Q. ) )
2524an4s 578 . . . . . . . . . . . . . . 15  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) ) )  ->  (
y  e.  Q.  /\  z  e.  Q. )
)
26253adantl1 1138 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) ) )  ->  (
y  e.  Q.  /\  z  e.  Q. )
)
2726ad2ant2r 501 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( y  e.  Q.  /\  z  e.  Q. )
)
28 3anass 967 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  <->  ( x  e.  Q.  /\  ( y  e.  Q.  /\  z  e.  Q. ) ) )
2917, 27, 28sylanbrc 414 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )
)
30 simprr 522 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  w  =  ( x  .Q  v ) )
31 simpr 109 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  v  =  ( y  +Q  z ) )
3230, 31anim12i 336 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( w  =  ( x  .Q  v )  /\  v  =  ( y  +Q  z ) ) )
33 oveq2 5789 . . . . . . . . . . . . . . 15  |-  ( v  =  ( y  +Q  z )  ->  (
x  .Q  v )  =  ( x  .Q  ( y  +Q  z
) ) )
3433eqeq2d 2152 . . . . . . . . . . . . . 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 7218 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  .Q  ( y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z
) ) )
3736eqeq2d 2152 . . . . . . . . . . . . 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.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  ->  w  =  ( (
x  .Q  y )  +Q  ( x  .Q  z ) ) )
40 mulclpr 7403 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
41403adant3 1002 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
4241ad2antrr 480 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( A  .P.  B
)  e.  P. )
43 mulclpr 7403 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
44433adant2 1001 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
4544ad2antrr 480 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( A  .P.  C
)  e.  P. )
46 simpll 519 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  y  e.  ( 1st `  B
) )
472, 3genpprecll 7345 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  ->  ( x  .Q  y )  e.  ( 1st `  ( A  .P.  B ) ) ) )
48473adant3 1002 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B ) )  ->  ( x  .Q  y )  e.  ( 1st `  ( A  .P.  B ) ) ) )
4948impl 378 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  x  e.  ( 1st `  A ) )  /\  y  e.  ( 1st `  B
) )  ->  (
x  .Q  y )  e.  ( 1st `  ( A  .P.  B ) ) )
5049adantlrr 475 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  y  e.  ( 1st `  B ) )  -> 
( x  .Q  y
)  e.  ( 1st `  ( A  .P.  B
) ) )
5146, 50sylan2 284 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( x  .Q  y
)  e.  ( 1st `  ( A  .P.  B
) ) )
52 simplr 520 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  z  e.  ( 1st `  C
) )
532, 3genpprecll 7345 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( x  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) )  ->  ( x  .Q  z )  e.  ( 1st `  ( A  .P.  C ) ) ) )
54533adant2 1001 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( x  e.  ( 1st `  A )  /\  z  e.  ( 1st `  C ) )  ->  ( x  .Q  z )  e.  ( 1st `  ( A  .P.  C ) ) ) )
5554impl 378 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  x  e.  ( 1st `  A ) )  /\  z  e.  ( 1st `  C
) )  ->  (
x  .Q  z )  e.  ( 1st `  ( A  .P.  C ) ) )
5655adantlrr 475 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  z  e.  ( 1st `  C ) )  -> 
( x  .Q  z
)  e.  ( 1st `  ( A  .P.  C
) ) )
5752, 56sylan2 284 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( x  .Q  z
)  e.  ( 1st `  ( A  .P.  C
) ) )
587, 8genpprecll 7345 . . . . . . . . . . . . 13  |-  ( ( ( A  .P.  B
)  e.  P.  /\  ( A  .P.  C )  e.  P. )  -> 
( ( ( x  .Q  y )  e.  ( 1st `  ( A  .P.  B ) )  /\  ( x  .Q  z )  e.  ( 1st `  ( A  .P.  C ) ) )  ->  ( (
x  .Q  y )  +Q  ( x  .Q  z ) )  e.  ( 1st `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) )
5958imp 123 . . . . . . . . . . . 12  |-  ( ( ( ( A  .P.  B )  e.  P.  /\  ( A  .P.  C )  e.  P. )  /\  ( ( x  .Q  y )  e.  ( 1st `  ( A  .P.  B ) )  /\  ( x  .Q  z )  e.  ( 1st `  ( A  .P.  C ) ) ) )  ->  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  e.  ( 1st `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) )
6042, 45, 51, 57, 59syl22anc 1218 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  -> 
( ( x  .Q  y )  +Q  (
x  .Q  z ) )  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) )
6139, 60eqeltrd 2217 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A
)  /\  w  =  ( x  .Q  v
) ) )  /\  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) ) )  ->  w  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) )
6261exp32 363 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  (
( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  ->  ( v  =  ( y  +Q  z )  ->  w  e.  ( 1st `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) ) )
6362rexlimdvv 2559 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  ( E. y  e.  ( 1st `  B ) E. z  e.  ( 1st `  C ) v  =  ( y  +Q  z
)  ->  w  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) )
6411, 63sylbid 149 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  (
v  e.  ( 1st `  ( B  +P.  C
) )  ->  w  e.  ( 1st `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) )
6564exp32 363 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 1st `  A )  ->  (
w  =  ( x  .Q  v )  -> 
( v  e.  ( 1st `  ( B  +P.  C ) )  ->  w  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) ) ) )
6665com34 83 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 1st `  A )  ->  (
v  e.  ( 1st `  ( B  +P.  C
) )  ->  (
w  =  ( x  .Q  v )  ->  w  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) ) ) )
6766impd 252 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( x  e.  ( 1st `  A )  /\  v  e.  ( 1st `  ( B  +P.  C ) ) )  ->  ( w  =  ( x  .Q  v )  ->  w  e.  ( 1st `  (
( A  .P.  B
)  +P.  ( A  .P.  C ) ) ) ) ) )
6867rexlimdvv 2559 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. x  e.  ( 1st `  A ) E. v  e.  ( 1st `  ( B  +P.  C
) ) w  =  ( x  .Q  v
)  ->  w  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) )
696, 68sylbid 149 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
w  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  ->  w  e.  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) ) )
7069ssrdv 3107 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  C_  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   E.wrex 2418    C_ wss 3075   <.cop 3534   ` cfv 5130  (class class class)co 5781   1stc1st 6043   2ndc2nd 6044   Q.cnq 7111    +Q cplq 7113    .Q cmq 7114   P.cnp 7122    +P. cpp 7124    .P. cmp 7125
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-2o 6321  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-enq0 7255  df-nq0 7256  df-0nq0 7257  df-plq0 7258  df-mq0 7259  df-inp 7297  df-iplp 7299  df-imp 7300
This theorem is referenced by:  distrprg  7419
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