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Theorem distrlem4prl 7046
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 6863 . . . . . . 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.  ( 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 939 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  A  e.  P. )
4 simpll 496 . . . . . . 7  |-  ( ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) )  ->  x  e.  ( 1st `  A
) )
5 prop 6937 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
6 elprnql 6943 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
75, 6sylan 277 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
83, 4, 7syl2an 283 . . . . . 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 498 . . . . . . 7  |-  ( ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) )  ->  f  e.  ( 1st `  A
) )
10 elprnql 6943 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
115, 10sylan 277 . . . . . . 7  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
123, 9, 11syl2an 283 . . . . . 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 943 . . . . . . 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 505 . . . . . . 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 6937 . . . . . . . 8  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
16 elprnql 6943 . . . . . . . 8  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
1715, 16sylan 277 . . . . . . 7  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
1813, 14, 17syl2anc 403 . . . . . 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 6845 . . . . . . 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.  ( 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 5749 . . . . 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 6862 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )  ->  (
w  <Q  v  <->  ( u  +Q  w )  <Q  (
u  +Q  v ) ) )
2322adantl 271 . . . . . 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 6838 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  e.  Q. )
258, 18, 24syl2anc 403 . . . . . 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 6838 . . . . . . 7  |-  ( ( f  e.  Q.  /\  y  e.  Q. )  ->  ( f  .Q  y
)  e.  Q. )
2712, 18, 26syl2anc 403 . . . . . 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 944 . . . . . . . 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 507 . . . . . . . 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 6937 . . . . . . . . 9  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
31 elprnql 6943 . . . . . . . . 9  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
3230, 31sylan 277 . . . . . . . 8  |-  ( ( C  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
3328, 29, 32syl2anc 403 . . . . . . 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 6838 . . . . . . 7  |-  ( ( f  e.  Q.  /\  z  e.  Q. )  ->  ( f  .Q  z
)  e.  Q. )
3512, 33, 34syl2anc 403 . . . . . 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 6843 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q. )  ->  ( w  +Q  v
)  =  ( v  +Q  w ) )
3736adantl 271 . . . . . 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 5749 . . . . 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 186 . . . 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 942 . . . . . 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 6999 . . . . . . . 8  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
42413adant1 957 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C )  e. 
P. )
4342adantr 270 . . . . . 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 7034 . . . . . 6  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
4540, 43, 44syl2anc 403 . . . . 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 6849 . . . . . . 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 1170 . . . . . 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 506 . . . . . . 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 6930 . . . . . . . . . 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 6837 . . . . . . . . . 10  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
5149, 50genpprecll 6976 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  ->  ( y  +Q  z )  e.  ( 1st `  ( B  +P.  C ) ) ) )
5251imp 122 . . . . . . . 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 1171 . . . . . . 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 6931 . . . . . . . . 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 6838 . . . . . . . . 9  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
5654, 55genpprecll 6976 . . . . . . . 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 122 . . . . . . 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 1171 . . . . . 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 2160 . . . . 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 6937 . . . . . 6  |-  ( ( A  .P.  ( B  +P.  C ) )  e.  P.  ->  <. ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ,  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) >.  e.  P. )
61 prcdnql 6946 . . . . . 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 277 . . . . 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 403 . . . 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 148 . . 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 5749 . . . . 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 6838 . . . . . . 7  |-  ( ( x  e.  Q.  /\  z  e.  Q. )  ->  ( x  .Q  z
)  e.  Q. )
678, 33, 66syl2anc 403 . . . . . 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 6862 . . . . . 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 1170 . . . . 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 186 . . . 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 6849 . . . . . . 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 1170 . . . . . 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 504 . . . . . . 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 6976 . . . . . . . 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 122 . . . . . . 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 1171 . . . . . 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 2160 . . . . 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 6946 . . . . . 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 277 . . . . 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 403 . . . 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 148 . . 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 670 . 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 6860 . . . . 5  |-  <Q  Or  Q.
84 nqtri3or 6858 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  \/  x  =  f  \/  f  <Q  x ) )
8583, 84sotritrieq 4116 . . . 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.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) ) )  -> 
( x  =  f  <->  -.  ( x  <Q  f  \/  f  <Q  x ) ) )
87 oveq1 5598 . . . . . . 7  |-  ( x  =  f  ->  (
x  .Q  z )  =  ( f  .Q  z ) )
8887oveq2d 5607 . . . . . 6  |-  ( x  =  f  ->  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) ) )
8972, 88sylan9eq 2135 . . . . 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 270 . . . . 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 2160 . . . 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 113 . . 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 168 . 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 6859 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  x 
<Q  f )
95 ltdcnq 6859 . . . . . 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.  ( 1st `  A
)  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  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.  ( 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 671 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 102    <-> wb 103    \/ wo 662  DECID wdc 776    /\ w3a 920    = wceq 1285    e. wcel 1434   <.cop 3425   class class class wbr 3811   ` cfv 4969  (class class class)co 5591   1stc1st 5844   2ndc2nd 5845   Q.cnq 6742    +Q cplq 6744    .Q cmq 6745    <Q cltq 6747   P.cnp 6753    +P. cpp 6755    .P. cmp 6756
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 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
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 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-iplp 6930  df-imp 6931
This theorem is referenced by:  distrlem5prl  7048
  Copyright terms: Public domain W3C validator