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Theorem distrlem4pru 7575
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 7391 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )  ->  (
w  <Q  v  <->  ( u  .Q  w )  <Q  (
u  .Q  v ) ) )
21adantl 277 . . . . . 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 997 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  A  e.  P. )
4 simpll 527 . . . . . . 7  |-  ( ( ( x  e.  ( 2nd `  A )  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) )  ->  x  e.  ( 2nd `  A
) )
5 prop 7465 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
6 elprnqu 7472 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
75, 6sylan 283 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
83, 4, 7syl2an 289 . . . . . 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 529 . . . . . . 7  |-  ( ( ( x  e.  ( 2nd `  A )  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A
)  /\  z  e.  ( 2nd `  C ) ) )  ->  f  e.  ( 2nd `  A
) )
10 elprnqu 7472 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
115, 10sylan 283 . . . . . . 7  |-  ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
123, 9, 11syl2an 289 . . . . . 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 1002 . . . . . . 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 540 . . . . . . 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 7465 . . . . . . . 8  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
16 elprnqu 7472 . . . . . . . 8  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 2nd `  C ) )  -> 
z  e.  Q. )
1715, 16sylan 283 . . . . . . 7  |-  ( ( C  e.  P.  /\  z  e.  ( 2nd `  C ) )  -> 
z  e.  Q. )
1813, 14, 17syl2anc 411 . . . . . 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 7373 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q. )  ->  ( w  .Q  v
)  =  ( v  .Q  w ) )
2019adantl 277 . . . . . 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 6038 . . . . 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 7366 . . . . . . 7  |-  ( ( x  e.  Q.  /\  z  e.  Q. )  ->  ( x  .Q  z
)  e.  Q. )
238, 18, 22syl2anc 411 . . . . . 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 7366 . . . . . . 7  |-  ( ( f  e.  Q.  /\  z  e.  Q. )  ->  ( f  .Q  z
)  e.  Q. )
2512, 18, 24syl2anc 411 . . . . . 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 1001 . . . . . . . 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 538 . . . . . . . 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 7465 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
29 elprnqu 7472 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
3028, 29sylan 283 . . . . . . . 8  |-  ( ( B  e.  P.  /\  y  e.  ( 2nd `  B ) )  -> 
y  e.  Q. )
3126, 27, 30syl2anc 411 . . . . . . 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 7366 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  e.  Q. )
338, 31, 32syl2anc 411 . . . . . 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 7390 . . . . . 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 1238 . . . . 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 188 . . . 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 1000 . . . . . 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 7527 . . . . . . . 8  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
39383adant1 1015 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C )  e. 
P. )
4039adantr 276 . . . . . 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 7562 . . . . . 6  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
4237, 40, 41syl2anc 411 . . . . 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 7377 . . . . . . 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 1238 . . . . . 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 537 . . . . . . 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 7458 . . . . . . . . . 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 7365 . . . . . . . . . 10  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
4846, 47genppreclu 7505 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( ( y  e.  ( 2nd `  B
)  /\  z  e.  ( 2nd `  C ) )  ->  ( y  +Q  z )  e.  ( 2nd `  ( B  +P.  C ) ) ) )
4948imp 124 . . . . . . . 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 1239 . . . . . . 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 7459 . . . . . . . . 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 7366 . . . . . . . . 9  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
5351, 52genppreclu 7505 . . . . . . . 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 124 . . . . . . 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 1239 . . . . . 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 2255 . . . . 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 7465 . . . . . 6  |-  ( ( A  .P.  ( B  +P.  C ) )  e.  P.  ->  <. ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ,  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) >.  e.  P. )
58 prcunqu 7475 . . . . . 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 283 . . . . 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 411 . . . 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 150 . . 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 6038 . . . . 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 7390 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )  ->  (
w  <Q  v  <->  ( u  +Q  w )  <Q  (
u  +Q  v ) ) )
6463adantl 277 . . . . . 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 7366 . . . . . . 7  |-  ( ( f  e.  Q.  /\  y  e.  Q. )  ->  ( f  .Q  y
)  e.  Q. )
6612, 31, 65syl2anc 411 . . . . . 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 7371 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q. )  ->  ( w  +Q  v
)  =  ( v  +Q  w ) )
6867adantl 277 . . . . . 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 6038 . . . . 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 188 . . . 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 7377 . . . . . . 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 1238 . . . . . 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 539 . . . . . . 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 7505 . . . . . . . 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 124 . . . . . . 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 1239 . . . . . 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 2255 . . . . 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 7475 . . . . . 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 283 . . . . 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 411 . . . 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 150 . . 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 717 . 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 7388 . . . . 5  |-  <Q  Or  Q.
84 nqtri3or 7386 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  \/  x  =  f  \/  f  <Q  x ) )
8583, 84sotritrieq 4322 . . . 4  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  =  f  <->  -.  ( x  <Q  f  \/  f  <Q  x ) ) )
868, 12, 85syl2anc 411 . . 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 5876 . . . . . . 7  |-  ( x  =  f  ->  (
x  .Q  z )  =  ( f  .Q  z ) )
8887oveq2d 5885 . . . . . 6  |-  ( x  =  f  ->  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) ) )
8944, 88sylan9eq 2230 . . . . 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 276 . . . . 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 2255 . . . 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 115 . . 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 170 . 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 7387 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  x 
<Q  f )
95 ltdcnq 7387 . . . . . 6  |-  ( ( f  e.  Q.  /\  x  e.  Q. )  -> DECID  f 
<Q  x )
9695ancoms 268 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  f 
<Q  x )
97 dcor 935 . . . . 5  |-  (DECID  x  <Q  f  ->  (DECID  f  <Q  x  -> DECID  ( x 
<Q  f  \/  f  <Q  x ) ) )
9894, 96, 97sylc 62 . . . 4  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  ( x  <Q  f  \/  f  <Q  x ) )
998, 12, 98syl2anc 411 . . 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 835 . . 3  |-  (DECID  ( x 
<Q  f  \/  f  <Q  x )  <->  ( (
x  <Q  f  \/  f  <Q  x )  \/  -.  ( x  <Q  f  \/  f  <Q  x )
) )
10199, 100sylib 122 . 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 718 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 104    <-> wb 105    \/ wo 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270    +Q cplq 7272    .Q cmq 7273    <Q cltq 7275   P.cnp 7281    +P. cpp 7283    .P. cmp 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iplp 7458  df-imp 7459
This theorem is referenced by:  distrlem5pru  7577
  Copyright terms: Public domain W3C validator