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Theorem distrlem1prl 6708
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 6663 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
2 df-imp 6595 . . . . . 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 6502 . . . . . 6  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
42, 3genpelvl 6638 . . . . 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 274 . . . 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 1109 . . 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 6594 . . . . . . . . . . 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 6501 . . . . . . . . . . 11  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
97, 8genpelvl 6638 . . . . . . . . . 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 931 . . . . . . . . 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 265 . . . . . . . 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 6601 . . . . . . . . . . . . . . . . 17  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 elprnql 6607 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1412, 13sylan 271 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
15143ad2antl1 1075 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1615adantrr 456 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  x  e.  Q. )
1716adantr 265 . . . . . . . . . . . . 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 6601 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 elprnql 6607 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
2018, 19sylan 271 . . . . . . . . . . . . . . . . 17  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
21 prop 6601 . . . . . . . . . . . . . . . . . 18  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
22 elprnql 6607 . . . . . . . . . . . . . . . . . 18  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
2321, 22sylan 271 . . . . . . . . . . . . . . . . 17  |-  ( ( C  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
2420, 23anim12i 325 . . . . . . . . . . . . . . . 16  |-  ( ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  /\  ( C  e.  P.  /\  z  e.  ( 1st `  C ) ) )  ->  ( y  e. 
Q.  /\  z  e.  Q. ) )
2524an4s 530 . . . . . . . . . . . . . . 15  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) ) )  ->  (
y  e.  Q.  /\  z  e.  Q. )
)
26253adantl1 1069 . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . 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 898 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  <->  ( x  e.  Q.  /\  ( y  e.  Q.  /\  z  e.  Q. ) ) )
2917, 27, 28sylanbrc 402 . . . . . . . . . . . 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 492 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  w  =  ( x  .Q  v ) ) )  ->  w  =  ( x  .Q  v ) )
31 simpr 107 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  v  =  ( y  +Q  z ) )
3230, 31anim12i 325 . . . . . . . . . . . 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 5545 . . . . . . . . . . . . . . 15  |-  ( v  =  ( y  +Q  z )  ->  (
x  .Q  v )  =  ( x  .Q  ( y  +Q  z
) ) )
3433eqeq2d 2065 . . . . . . . . . . . . . 14  |-  ( v  =  ( y  +Q  z )  ->  (
w  =  ( x  .Q  v )  <->  w  =  ( x  .Q  (
y  +Q  z ) ) ) )
3534biimpac 286 . . . . . . . . . . . . 13  |-  ( ( w  =  ( x  .Q  v )  /\  v  =  ( y  +Q  z ) )  ->  w  =  ( x  .Q  ( y  +Q  z
) ) )
36 distrnqg 6513 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  .Q  ( y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z
) ) )
3736eqeq2d 2065 . . . . . . . . . . . . 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 147 . . . . . . . . . . . 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 60 . . . . . . . . . . 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 6698 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
41403adant3 933 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  B )  e. 
P. )
4241ad2antrr 465 . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C
)  e.  P. )
44433adant2 932 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  C )  e. 
P. )
4544ad2antrr 465 . . . . . . . . . . . 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 489 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  y  e.  ( 1st `  B
) )
472, 3genpprecll 6640 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  ->  ( x  .Q  y )  e.  ( 1st `  ( A  .P.  B ) ) ) )
48473adant3 933 . . . . . . . . . . . . . . 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 366 . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . 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 274 . . . . . . . . . . . 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 490 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) )  /\  v  =  ( y  +Q  z
) )  ->  z  e.  ( 1st `  C
) )
532, 3genpprecll 6640 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( ( x  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) )  ->  ( x  .Q  z )  e.  ( 1st `  ( A  .P.  C ) ) ) )
54533adant2 932 . . . . . . . . . . . . . . 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 366 . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . 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 274 . . . . . . . . . . . 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 6640 . . . . . . . . . . . . 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 119 . . . . . . . . . . . 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 1145 . . . . . . . . . . 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 2128 . . . . . . . . . 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 351 . . . . . . . . 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 2454 . . . . . . . 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 143 . . . . . . 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 351 . . . . . 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 81 . . . . 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 246 . . . 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 2454 . . 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 143 . 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 2976 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 101    <-> wb 102    /\ w3a 894    = wceq 1257    e. wcel 1407   E.wrex 2322    C_ wss 2942   <.cop 3403   ` cfv 4927  (class class class)co 5537   1stc1st 5790   2ndc2nd 5791   Q.cnq 6406    +Q cplq 6408    .Q cmq 6409   P.cnp 6417    +P. cpp 6419    .P. cmp 6420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-po 4058  df-iso 4059  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-2o 6030  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-mpq 6471  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-mqqs 6476  df-1nqqs 6477  df-rq 6478  df-ltnqqs 6479  df-enq0 6550  df-nq0 6551  df-0nq0 6552  df-plq0 6553  df-mq0 6554  df-inp 6592  df-iplp 6594  df-imp 6595
This theorem is referenced by:  distrprg  6714
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