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Theorem distrlem1prl 7085
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 7040 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
2 df-imp 6972 . . . . . 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 6879 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
42, 3genpelvl 7015 . . . . 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 280 . . . 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 1137 . . 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 6971 . . . . . . . . . . 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 6878 . . . . . . . . . . 11  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
97, 8genpelvl 7015 . . . . . . . . . 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 959 . . . . . . . . 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 270 . . . . . . . 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 6978 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 elprnql 6984 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1412, 13sylan 277 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
15143ad2antl1 1103 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1615adantrr 463 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  x  e.  Q. )
1716adantr 270 . . . . . . . . . . . . 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 6978 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 elprnql 6984 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
2018, 19sylan 277 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
21 prop 6978 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
22 elprnql 6984 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
2321, 22sylan 277 . . . . . . . . . . . . . . . . 17  |-  ( ( C  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
2420, 23anim12i 331 . . . . . . . . . . . . . . . 16  |-  ( ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  /\  ( C  e.  P.  /\  z  e.  ( 1st `  C ) ) )  ->  ( y  e. 
Q.  /\  z  e.  Q. ) )
2524an4s 553 . . . . . . . . . . . . . . 15  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) ) )  ->  (
y  e.  Q.  /\  z  e.  Q. )
)
26253adantl1 1097 . . . . . . . . . . . . . 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 493 . . . . . . . . . . . . 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 926 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  <->  ( x  e.  Q.  /\  ( y  e.  Q.  /\  z  e.  Q. ) ) )
2917, 27, 28sylanbrc 408 . . . . . . . . . . . 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 499 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  w  =  ( x  .Q  v ) )
31 simpr 108 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  v  =  ( y  +Q  z ) )
3230, 31anim12i 331 . . . . . . . . . . . 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 5621 . . . . . . . . . . . . . . 15  |-  ( v  =  ( y  +Q  z )  ->  (
x  .Q  v )  =  ( x  .Q  ( y  +Q  z
) ) )
3433eqeq2d 2096 . . . . . . . . . . . . . 14  |-  ( v  =  ( y  +Q  z )  ->  (
w  =  ( x  .Q  v )  <->  w  =  ( x  .Q  (
y  +Q  z ) ) ) )
3534biimpac 292 . . . . . . . . . . . . 13  |-  ( ( w  =  ( x  .Q  v )  /\  v  =  ( y  +Q  z ) )  ->  w  =  ( x  .Q  ( y  +Q  z
) ) )
36 distrnqg 6890 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  .Q  ( y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z
) ) )
3736eqeq2d 2096 . . . . . . . . . . . . 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 152 . . . . . . . . . . . 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 61 . . . . . . . . . . 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 7075 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
41403adant3 961 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
4241ad2antrr 472 . . . . . . . . . . . 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 7075 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
44433adant2 960 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
4544ad2antrr 472 . . . . . . . . . . . 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 496 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  y  e.  ( 1st `  B
) )
472, 3genpprecll 7017 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  ->  ( x  .Q  y )  e.  ( 1st `  ( A  .P.  B ) ) ) )
48473adant3 961 . . . . . . . . . . . . . . 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 372 . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . 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 280 . . . . . . . . . . . 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 497 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  z  e.  ( 1st `  C
) )
532, 3genpprecll 7017 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( x  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) )  ->  ( x  .Q  z )  e.  ( 1st `  ( A  .P.  C ) ) ) )
54533adant2 960 . . . . . . . . . . . . . . 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 372 . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . 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 280 . . . . . . . . . . . 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 7017 . . . . . . . . . . . . 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 122 . . . . . . . . . . . 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 1173 . . . . . . . . . . 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 2161 . . . . . . . . . 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 357 . . . . . . . . 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 2491 . . . . . . . 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 148 . . . . . . 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 357 . . . . . 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 82 . . . . 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 251 . . . 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 2491 . . 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 148 . 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 3020 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 102    <-> wb 103    /\ w3a 922    = wceq 1287    e. wcel 1436   E.wrex 2356    C_ wss 2988   <.cop 3434   ` cfv 4981  (class class class)co 5613   1stc1st 5866   2ndc2nd 5867   Q.cnq 6783    +Q cplq 6785    .Q cmq 6786   P.cnp 6794    +P. cpp 6796    .P. cmp 6797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-2o 6136  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856  df-enq0 6927  df-nq0 6928  df-0nq0 6929  df-plq0 6930  df-mq0 6931  df-inp 6969  df-iplp 6971  df-imp 6972
This theorem is referenced by:  distrprg  7091
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