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Theorem distrlem4prl 6739
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem4prl  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 1st `  ( 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 distrlem4prl
Dummy variables  w  v  u  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 6556 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )  ->  (
w  <Q  v  <->  ( u  .Q  w )  <Q  (
u  .Q  v ) ) )
21adantl 266 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  /\  ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )
)  ->  ( w  <Q  v  <->  ( u  .Q  w )  <Q  (
u  .Q  v ) ) )
3 simp1 915 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  A  e.  P. )
4 simpll 489 . . . . . . 7  |-  ( ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) )  ->  x  e.  ( 1st `  A
) )
5 prop 6630 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
6 elprnql 6636 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
75, 6sylan 271 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
83, 4, 7syl2an 277 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  ->  x  e.  Q. )
9 simprl 491 . . . . . . 7  |-  ( ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) )  ->  f  e.  ( 1st `  A
) )
10 elprnql 6636 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
115, 10sylan 271 . . . . . . 7  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
123, 9, 11syl2an 277 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
f  e.  Q. )
13 simpl2 919 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  ->  B  e.  P. )
14 simprlr 498 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
y  e.  ( 1st `  B ) )
15 prop 6630 . . . . . . . 8  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
16 elprnql 6636 . . . . . . . 8  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
1715, 16sylan 271 . . . . . . 7  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
1813, 14, 17syl2anc 397 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
y  e.  Q. )
19 mulcomnqg 6538 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q. )  ->  ( w  .Q  v
)  =  ( v  .Q  w ) )
2019adantl 266 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  /\  ( w  e.  Q.  /\  v  e.  Q. )
)  ->  ( w  .Q  v )  =  ( v  .Q  w ) )
212, 8, 12, 18, 20caovord2d 5697 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  <Q  f  <->  ( x  .Q  y ) 
<Q  ( f  .Q  y
) ) )
22 ltanqg 6555 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )  ->  (
w  <Q  v  <->  ( u  +Q  w )  <Q  (
u  +Q  v ) ) )
2322adantl 266 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  /\  ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )
)  ->  ( w  <Q  v  <->  ( u  +Q  w )  <Q  (
u  +Q  v ) ) )
24 mulclnq 6531 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  e.  Q. )
258, 18, 24syl2anc 397 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  .Q  y
)  e.  Q. )
26 mulclnq 6531 . . . . . . 7  |-  ( ( f  e.  Q.  /\  y  e.  Q. )  ->  ( f  .Q  y
)  e.  Q. )
2712, 18, 26syl2anc 397 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( f  .Q  y
)  e.  Q. )
28 simpl3 920 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  ->  C  e.  P. )
29 simprrr 500 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
z  e.  ( 1st `  C ) )
30 prop 6630 . . . . . . . . 9  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
31 elprnql 6636 . . . . . . . . 9  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
3230, 31sylan 271 . . . . . . . 8  |-  ( ( C  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
3328, 29, 32syl2anc 397 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
z  e.  Q. )
34 mulclnq 6531 . . . . . . 7  |-  ( ( f  e.  Q.  /\  z  e.  Q. )  ->  ( f  .Q  z
)  e.  Q. )
3512, 33, 34syl2anc 397 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( f  .Q  z
)  e.  Q. )
36 addcomnqg 6536 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q. )  ->  ( w  +Q  v
)  =  ( v  +Q  w ) )
3736adantl 266 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  /\  ( w  e.  Q.  /\  v  e.  Q. )
)  ->  ( w  +Q  v )  =  ( v  +Q  w ) )
3823, 25, 27, 35, 37caovord2d 5697 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( x  .Q  y )  <Q  (
f  .Q  y )  <-> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  <Q  ( (
f  .Q  y )  +Q  ( f  .Q  z ) ) ) )
3921, 38bitrd 181 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  <Q  f  <->  ( ( x  .Q  y
)  +Q  ( f  .Q  z ) ) 
<Q  ( ( f  .Q  y )  +Q  (
f  .Q  z ) ) ) )
40 simpl1 918 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  ->  A  e.  P. )
41 addclpr 6692 . . . . . . . 8  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
42413adant1 933 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C )  e. 
P. )
4342adantr 265 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( B  +P.  C
)  e.  P. )
44 mulclpr 6727 . . . . . 6  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
4540, 43, 44syl2anc 397 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
46 distrnqg 6542 . . . . . . 7  |-  ( ( f  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
f  .Q  ( y  +Q  z ) )  =  ( ( f  .Q  y )  +Q  ( f  .Q  z
) ) )
4712, 18, 33, 46syl3anc 1146 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( f  .Q  (
y  +Q  z ) )  =  ( ( f  .Q  y )  +Q  ( f  .Q  z ) ) )
48 simprrl 499 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
f  e.  ( 1st `  A ) )
49 df-iplp 6623 . . . . . . . . . 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
) ) } >. )
50 addclnq 6530 . . . . . . . . . 10  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
5149, 50genpprecll 6669 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  ->  ( y  +Q  z )  e.  ( 1st `  ( B  +P.  C ) ) ) )
5251imp 119 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( y  e.  ( 1st `  B )  /\  z  e.  ( 1st `  C ) ) )  ->  (
y  +Q  z )  e.  ( 1st `  ( B  +P.  C ) ) )
5313, 28, 14, 29, 52syl22anc 1147 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( y  +Q  z
)  e.  ( 1st `  ( B  +P.  C
) ) )
54 df-imp 6624 . . . . . . . . 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
) ) } >. )
55 mulclnq 6531 . . . . . . . . 9  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
5654, 55genpprecll 6669 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( ( f  e.  ( 1st `  A
)  /\  ( y  +Q  z )  e.  ( 1st `  ( B  +P.  C ) ) )  ->  ( f  .Q  ( y  +Q  z
) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
5756imp 119 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  ( B  +P.  C
)  e.  P. )  /\  ( f  e.  ( 1st `  A )  /\  ( y  +Q  z )  e.  ( 1st `  ( B  +P.  C ) ) ) )  ->  (
f  .Q  ( y  +Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
5840, 43, 48, 53, 57syl22anc 1147 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( f  .Q  (
y  +Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
5947, 58eqeltrrd 2131 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( f  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
60 prop 6630 . . . . . 6  |-  ( ( A  .P.  ( B  +P.  C ) )  e.  P.  ->  <. ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ,  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) >.  e.  P. )
61 prcdnql 6639 . . . . . 6  |-  ( (
<. ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ,  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) >.  e.  P.  /\  ( ( f  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )  ->  ( ( ( x  .Q  y )  +Q  ( f  .Q  z ) )  <Q 
( ( f  .Q  y )  +Q  (
f  .Q  z ) )  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
6260, 61sylan 271 . . . . 5  |-  ( ( ( A  .P.  ( B  +P.  C ) )  e.  P.  /\  (
( f  .Q  y
)  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )  ->  ( ( ( x  .Q  y )  +Q  ( f  .Q  z ) )  <Q 
( ( f  .Q  y )  +Q  (
f  .Q  z ) )  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
6345, 59, 62syl2anc 397 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  <Q  (
( f  .Q  y
)  +Q  ( f  .Q  z ) )  ->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
6439, 63sylbid 143 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  <Q  f  ->  ( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
652, 12, 8, 33, 20caovord2d 5697 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( f  <Q  x  <->  ( f  .Q  z ) 
<Q  ( x  .Q  z
) ) )
66 mulclnq 6531 . . . . . . 7  |-  ( ( x  e.  Q.  /\  z  e.  Q. )  ->  ( x  .Q  z
)  e.  Q. )
678, 33, 66syl2anc 397 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  .Q  z
)  e.  Q. )
68 ltanqg 6555 . . . . . 6  |-  ( ( ( f  .Q  z
)  e.  Q.  /\  ( x  .Q  z
)  e.  Q.  /\  ( x  .Q  y
)  e.  Q. )  ->  ( ( f  .Q  z )  <Q  (
x  .Q  z )  <-> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  <Q  ( (
x  .Q  y )  +Q  ( x  .Q  z ) ) ) )
6935, 67, 25, 68syl3anc 1146 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( f  .Q  z )  <Q  (
x  .Q  z )  <-> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  <Q  ( (
x  .Q  y )  +Q  ( x  .Q  z ) ) ) )
7065, 69bitrd 181 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( f  <Q  x  <->  ( ( x  .Q  y
)  +Q  ( f  .Q  z ) ) 
<Q  ( ( x  .Q  y )  +Q  (
x  .Q  z ) ) ) )
71 distrnqg 6542 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  .Q  ( y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z
) ) )
728, 18, 33, 71syl3anc 1146 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  .Q  (
y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( x  .Q  z ) ) )
73 simprll 497 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  ->  x  e.  ( 1st `  A ) )
7454, 55genpprecll 6669 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( ( x  e.  ( 1st `  A
)  /\  ( y  +Q  z )  e.  ( 1st `  ( B  +P.  C ) ) )  ->  ( x  .Q  ( y  +Q  z
) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
7574imp 119 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  ( B  +P.  C
)  e.  P. )  /\  ( x  e.  ( 1st `  A )  /\  ( y  +Q  z )  e.  ( 1st `  ( B  +P.  C ) ) ) )  ->  (
x  .Q  ( y  +Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
7640, 43, 73, 53, 75syl22anc 1147 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  .Q  (
y  +Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
7772, 76eqeltrrd 2131 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( x  .Q  y )  +Q  (
x  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
78 prcdnql 6639 . . . . . 6  |-  ( (
<. ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ,  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) >.  e.  P.  /\  ( ( x  .Q  y )  +Q  ( x  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )  ->  ( ( ( x  .Q  y )  +Q  ( f  .Q  z ) )  <Q 
( ( x  .Q  y )  +Q  (
x  .Q  z ) )  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
7960, 78sylan 271 . . . . 5  |-  ( ( ( A  .P.  ( B  +P.  C ) )  e.  P.  /\  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )  ->  ( ( ( x  .Q  y )  +Q  ( f  .Q  z ) )  <Q 
( ( x  .Q  y )  +Q  (
x  .Q  z ) )  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
8045, 77, 79syl2anc 397 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  <Q  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  ->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
8170, 80sylbid 143 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( f  <Q  x  ->  ( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
8264, 81jaod 647 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( x  <Q  f  \/  f  <Q  x
)  ->  ( (
x  .Q  y )  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
83 ltsonq 6553 . . . . 5  |-  <Q  Or  Q.
84 nqtri3or 6551 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  \/  x  =  f  \/  f  <Q  x ) )
8583, 84sotritrieq 4089 . . . 4  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  =  f  <->  -.  ( x  <Q  f  \/  f  <Q  x ) ) )
868, 12, 85syl2anc 397 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  =  f  <->  -.  ( x  <Q  f  \/  f  <Q  x ) ) )
87 oveq1 5546 . . . . . . 7  |-  ( x  =  f  ->  (
x  .Q  z )  =  ( f  .Q  z ) )
8887oveq2d 5555 . . . . . 6  |-  ( x  =  f  ->  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) ) )
8972, 88sylan9eq 2108 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  /\  x  =  f )  ->  ( x  .Q  (
y  +Q  z ) )  =  ( ( x  .Q  y )  +Q  ( f  .Q  z ) ) )
9076adantr 265 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  /\  x  =  f )  ->  ( x  .Q  (
y  +Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
9189, 90eqeltrrd 2131 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  /\  x  =  f )  ->  ( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
9291ex 112 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  =  f  ->  ( ( x  .Q  y )  +Q  ( f  .Q  z
) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
9386, 92sylbird 163 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( -.  ( x 
<Q  f  \/  f  <Q  x )  ->  (
( x  .Q  y
)  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ) )
94 ltdcnq 6552 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  x 
<Q  f )
95 ltdcnq 6552 . . . . . 6  |-  ( ( f  e.  Q.  /\  x  e.  Q. )  -> DECID  f 
<Q  x )
9695ancoms 259 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  f 
<Q  x )
97 dcor 854 . . . . 5  |-  (DECID  x  <Q  f  ->  (DECID  f  <Q  x  -> DECID  ( x 
<Q  f  \/  f  <Q  x ) ) )
9894, 96, 97sylc 60 . . . 4  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  ( x  <Q  f  \/  f  <Q  x ) )
998, 12, 98syl2anc 397 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> DECID  (
x  <Q  f  \/  f  <Q  x ) )
100 df-dc 754 . . 3  |-  (DECID  ( x 
<Q  f  \/  f  <Q  x )  <->  ( (
x  <Q  f  \/  f  <Q  x )  \/  -.  ( x  <Q  f  \/  f  <Q  x )
) )
10199, 100sylib 131 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( x  <Q  f  \/  f  <Q  x
)  \/  -.  (
x  <Q  f  \/  f  <Q  x ) ) )
10282, 93, 101mpjaod 648 1  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( ( x  .Q  y )  +Q  (
f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639  DECID wdc 753    /\ w3a 896    = wceq 1259    e. wcel 1409   <.cop 3405   class class class wbr 3791   ` cfv 4929  (class class class)co 5539   1stc1st 5792   2ndc2nd 5793   Q.cnq 6435    +Q cplq 6437    .Q cmq 6438    <Q cltq 6440   P.cnp 6446    +P. cpp 6448    .P. cmp 6449
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 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iplp 6623  df-imp 6624
This theorem is referenced by:  distrlem5prl  6741
  Copyright terms: Public domain W3C validator