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Theorem distrlem4pru 6907
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem4pru  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
Distinct variable groups:    x, y, z, f, A    x, B, y, z, f    x, C, y, z, f

Proof of Theorem distrlem4pru
Dummy variables  w  v  u  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 6723 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )  ->  (
w  <Q  v  <->  ( u  .Q  w )  <Q  (
u  .Q  v ) ) )
21adantl 271 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  /\  ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )
)  ->  ( w  <Q  v  <->  ( u  .Q  w )  <Q  (
u  .Q  v ) ) )
3 simp1 939 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  A  e.  P. )
4 simpll 496 . . . . . . 7  |-  ( ( ( x  e.  ( 2nd `  A )  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) )  ->  x  e.  ( 2nd `  A
) )
5 prop 6797 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
6 elprnqu 6804 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
75, 6sylan 277 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
83, 4, 7syl2an 283 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  ->  x  e.  Q. )
9 simprl 498 . . . . . . 7  |-  ( ( ( x  e.  ( 2nd `  A )  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) )  ->  f  e.  ( 2nd `  A
) )
10 elprnqu 6804 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
115, 10sylan 277 . . . . . . 7  |-  ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
123, 9, 11syl2an 283 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
f  e.  Q. )
13 simpl3 944 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  ->  C  e.  P. )
14 simprrr 507 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
z  e.  ( 2nd `  C ) )
15 prop 6797 . . . . . . . 8  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
16 elprnqu 6804 . . . . . . . 8  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 2nd `  C ) )  -> 
z  e.  Q. )
1715, 16sylan 277 . . . . . . 7  |-  ( ( C  e.  P.  /\  z  e.  ( 2nd `  C ) )  -> 
z  e.  Q. )
1813, 14, 17syl2anc 403 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
z  e.  Q. )
19 mulcomnqg 6705 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q. )  ->  ( w  .Q  v
)  =  ( v  .Q  w ) )
2019adantl 271 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  /\  ( w  e.  Q.  /\  v  e.  Q. )
)  ->  ( w  .Q  v )  =  ( v  .Q  w ) )
212, 8, 12, 18, 20caovord2d 5722 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( x  <Q  f  <->  ( x  .Q  z ) 
<Q  ( f  .Q  z
) ) )
22 mulclnq 6698 . . . . . . 7  |-  ( ( x  e.  Q.  /\  z  e.  Q. )  ->  ( x  .Q  z
)  e.  Q. )
238, 18, 22syl2anc 403 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( x  .Q  z
)  e.  Q. )
24 mulclnq 6698 . . . . . . 7  |-  ( ( f  e.  Q.  /\  z  e.  Q. )  ->  ( f  .Q  z
)  e.  Q. )
2512, 18, 24syl2anc 403 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( f  .Q  z
)  e.  Q. )
26 simpl2 943 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  ->  B  e.  P. )
27 simprlr 505 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
y  e.  ( 2nd `  B ) )
28 prop 6797 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
29 elprnqu 6804 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
3028, 29sylan 277 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
3126, 27, 30syl2anc 403 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
y  e.  Q. )
32 mulclnq 6698 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  e.  Q. )
338, 31, 32syl2anc 403 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( x  .Q  y
)  e.  Q. )
34 ltanqg 6722 . . . . . 6  |-  ( ( ( x  .Q  z
)  e.  Q.  /\  ( f  .Q  z
)  e.  Q.  /\  ( x  .Q  y
)  e.  Q. )  ->  ( ( x  .Q  z )  <Q  (
f  .Q  z )  <-> 
( ( x  .Q  y )  +Q  (
x  .Q  z ) )  <Q  ( (
x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
3523, 25, 33, 34syl3anc 1170 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( x  .Q  z )  <Q  (
f  .Q  z )  <-> 
( ( x  .Q  y )  +Q  (
x  .Q  z ) )  <Q  ( (
x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
3621, 35bitrd 186 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( x  <Q  f  <->  ( ( x  .Q  y
)  +Q  ( x  .Q  z ) ) 
<Q  ( ( x  .Q  y )  +Q  (
f  .Q  z ) ) ) )
37 simpl1 942 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  ->  A  e.  P. )
38 addclpr 6859 . . . . . . . 8  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
39383adant1 957 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C )  e. 
P. )
4039adantr 270 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( B  +P.  C
)  e.  P. )
41 mulclpr 6894 . . . . . 6  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
4237, 40, 41syl2anc 403 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
43 distrnqg 6709 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  .Q  ( y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z
) ) )
448, 31, 18, 43syl3anc 1170 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( x  .Q  (
y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z ) ) )
45 simprll 504 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  ->  x  e.  ( 2nd `  A ) )
46 df-iplp 6790 . . . . . . . . . 10  |-  +P.  =  ( u  e.  P. ,  v  e.  P.  |->  <. { w  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  u )  /\  h  e.  ( 1st `  v
)  /\  w  =  ( g  +Q  h
) ) } ,  { w  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  u )  /\  h  e.  ( 2nd `  v
)  /\  w  =  ( g  +Q  h
) ) } >. )
47 addclnq 6697 . . . . . . . . . 10  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
4846, 47genppreclu 6837 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  ->  ( y  +Q  z )  e.  ( 2nd `  ( B  +P.  C ) ) ) )
4948imp 122 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 2nd `  B )  /\  z  e.  ( 2nd `  C ) ) )  ->  (
y  +Q  z )  e.  ( 2nd `  ( B  +P.  C ) ) )
5026, 13, 27, 14, 49syl22anc 1171 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( y  +Q  z
)  e.  ( 2nd `  ( B  +P.  C
) ) )
51 df-imp 6791 . . . . . . . . 9  |-  .P.  =  ( u  e.  P. ,  v  e.  P.  |->  <. { w  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  u )  /\  h  e.  ( 1st `  v
)  /\  w  =  ( g  .Q  h
) ) } ,  { w  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  u )  /\  h  e.  ( 2nd `  v
)  /\  w  =  ( g  .Q  h
) ) } >. )
52 mulclnq 6698 . . . . . . . . 9  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
5351, 52genppreclu 6837 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( ( x  e.  ( 2nd `  A
)  /\  ( y  +Q  z )  e.  ( 2nd `  ( B  +P.  C ) ) )  ->  ( x  .Q  ( y  +Q  z
) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
5453imp 122 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  ( B  +P.  C
)  e.  P. )  /\  ( x  e.  ( 2nd `  A )  /\  ( y  +Q  z )  e.  ( 2nd `  ( B  +P.  C ) ) ) )  ->  (
x  .Q  ( y  +Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
5537, 40, 45, 50, 54syl22anc 1171 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( x  .Q  (
y  +Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
5644, 55eqeltrrd 2160 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( x  .Q  y )  +Q  (
x  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
57 prop 6797 . . . . . 6  |-  ( ( A  .P.  ( B  +P.  C ) )  e.  P.  ->  <. ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ,  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) >.  e.  P. )
58 prcunqu 6807 . . . . . 6  |-  ( (
<. ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ,  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) >.  e.  P.  /\  ( ( x  .Q  y )  +Q  ( x  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )  ->  ( ( ( x  .Q  y )  +Q  ( x  .Q  z ) )  <Q 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
5957, 58sylan 277 . . . . 5  |-  ( ( ( A  .P.  ( B  +P.  C ) )  e.  P.  /\  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )  ->  ( ( ( x  .Q  y )  +Q  ( x  .Q  z ) )  <Q 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
6042, 56, 59syl2anc 403 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( ( x  .Q  y )  +Q  ( x  .Q  z
) )  <Q  (
( x  .Q  y
)  +Q  ( f  .Q  z ) )  ->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
6136, 60sylbid 148 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( x  <Q  f  ->  ( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
622, 12, 8, 31, 20caovord2d 5722 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( f  <Q  x  <->  ( f  .Q  y ) 
<Q  ( x  .Q  y
) ) )
63 ltanqg 6722 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )  ->  (
w  <Q  v  <->  ( u  +Q  w )  <Q  (
u  +Q  v ) ) )
6463adantl 271 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  /\  ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )
)  ->  ( w  <Q  v  <->  ( u  +Q  w )  <Q  (
u  +Q  v ) ) )
65 mulclnq 6698 . . . . . . 7  |-  ( ( f  e.  Q.  /\  y  e.  Q. )  ->  ( f  .Q  y
)  e.  Q. )
6612, 31, 65syl2anc 403 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( f  .Q  y
)  e.  Q. )
67 addcomnqg 6703 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q. )  ->  ( w  +Q  v
)  =  ( v  +Q  w ) )
6867adantl 271 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  /\  ( w  e.  Q.  /\  v  e.  Q. )
)  ->  ( w  +Q  v )  =  ( v  +Q  w ) )
6964, 66, 33, 25, 68caovord2d 5722 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( f  .Q  y )  <Q  (
x  .Q  y )  <-> 
( ( f  .Q  y )  +Q  (
f  .Q  z ) )  <Q  ( (
x  .Q  y )  +Q  ( f  .Q  z ) ) ) )
7062, 69bitrd 186 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( f  <Q  x  <->  ( ( f  .Q  y
)  +Q  ( f  .Q  z ) ) 
<Q  ( ( x  .Q  y )  +Q  (
f  .Q  z ) ) ) )
71 distrnqg 6709 . . . . . . 7  |-  ( ( f  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
f  .Q  ( y  +Q  z ) )  =  ( ( f  .Q  y )  +Q  ( f  .Q  z
) ) )
7212, 31, 18, 71syl3anc 1170 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( f  .Q  (
y  +Q  z ) )  =  ( ( f  .Q  y )  +Q  ( f  .Q  z ) ) )
73 simprrl 506 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
f  e.  ( 2nd `  A ) )
7451, 52genppreclu 6837 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( ( f  e.  ( 2nd `  A
)  /\  ( y  +Q  z )  e.  ( 2nd `  ( B  +P.  C ) ) )  ->  ( f  .Q  ( y  +Q  z
) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
7574imp 122 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  ( B  +P.  C
)  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  ( y  +Q  z )  e.  ( 2nd `  ( B  +P.  C ) ) ) )  ->  (
f  .Q  ( y  +Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
7637, 40, 73, 50, 75syl22anc 1171 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( f  .Q  (
y  +Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
7772, 76eqeltrrd 2160 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( f  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
78 prcunqu 6807 . . . . . 6  |-  ( (
<. ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ,  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) >.  e.  P.  /\  ( ( f  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )  ->  ( ( ( f  .Q  y )  +Q  ( f  .Q  z ) )  <Q 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
7957, 78sylan 277 . . . . 5  |-  ( ( ( A  .P.  ( B  +P.  C ) )  e.  P.  /\  (
( f  .Q  y
)  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )  ->  ( ( ( f  .Q  y )  +Q  ( f  .Q  z ) )  <Q 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
8042, 77, 79syl2anc 403 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( ( f  .Q  y )  +Q  ( f  .Q  z
) )  <Q  (
( x  .Q  y
)  +Q  ( f  .Q  z ) )  ->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
8170, 80sylbid 148 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( f  <Q  x  ->  ( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
8261, 81jaod 670 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( x  <Q  f  \/  f  <Q  x
)  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
83 ltsonq 6720 . . . . 5  |-  <Q  Or  Q.
84 nqtri3or 6718 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  \/  x  =  f  \/  f  <Q  x ) )
8583, 84sotritrieq 4108 . . . 4  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  =  f  <->  -.  ( x  <Q  f  \/  f  <Q  x ) ) )
868, 12, 85syl2anc 403 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( x  =  f  <->  -.  ( x  <Q  f  \/  f  <Q  x ) ) )
87 oveq1 5571 . . . . . . 7  |-  ( x  =  f  ->  (
x  .Q  z )  =  ( f  .Q  z ) )
8887oveq2d 5580 . . . . . 6  |-  ( x  =  f  ->  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) ) )
8944, 88sylan9eq 2135 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  /\  x  =  f )  ->  ( x  .Q  (
y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
9055adantr 270 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  /\  x  =  f )  ->  ( x  .Q  (
y  +Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
9189, 90eqeltrrd 2160 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  /\  x  =  f )  ->  ( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
9291ex 113 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( x  =  f  ->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
9386, 92sylbird 168 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( -.  ( x 
<Q  f  \/  f  <Q  x )  ->  (
( x  .Q  y
)  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) ) )
94 ltdcnq 6719 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  x 
<Q  f )
95 ltdcnq 6719 . . . . . 6  |-  ( ( f  e.  Q.  /\  x  e.  Q. )  -> DECID  f 
<Q  x )
9695ancoms 264 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  f 
<Q  x )
97 dcor 877 . . . . 5  |-  (DECID  x  <Q  f  ->  (DECID  f  <Q  x  -> DECID  ( x 
<Q  f  \/  f  <Q  x ) ) )
9894, 96, 97sylc 61 . . . 4  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  ( x  <Q  f  \/  f  <Q  x ) )
998, 12, 98syl2anc 403 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> DECID  (
x  <Q  f  \/  f  <Q  x ) )
100 df-dc 777 . . 3  |-  (DECID  ( x 
<Q  f  \/  f  <Q  x )  <->  ( (
x  <Q  f  \/  f  <Q  x )  \/  -.  ( x  <Q  f  \/  f  <Q  x )
) )
10199, 100sylib 120 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( x  <Q  f  \/  f  <Q  x
)  \/  -.  (
x  <Q  f  \/  f  <Q  x ) ) )
10282, 93, 101mpjaod 671 1  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A
)  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) ) )  -> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662  DECID wdc 776    /\ w3a 920    = wceq 1285    e. wcel 1434   <.cop 3419   class class class wbr 3805   ` cfv 4952  (class class class)co 5564   1stc1st 5817   2ndc2nd 5818   Q.cnq 6602    +Q cplq 6604    .Q cmq 6605    <Q cltq 6607   P.cnp 6613    +P. cpp 6615    .P. cmp 6616
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-2o 6087  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6626  df-pli 6627  df-mi 6628  df-lti 6629  df-plpq 6666  df-mpq 6667  df-enq 6669  df-nqqs 6670  df-plqqs 6671  df-mqqs 6672  df-1nqqs 6673  df-rq 6674  df-ltnqqs 6675  df-enq0 6746  df-nq0 6747  df-0nq0 6748  df-plq0 6749  df-mq0 6750  df-inp 6788  df-iplp 6790  df-imp 6791
This theorem is referenced by:  distrlem5pru  6909
  Copyright terms: Public domain W3C validator