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Theorem distrlem4prl 7356
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 7173 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )  ->  (
w  <Q  v  <->  ( u  .Q  w )  <Q  (
u  .Q  v ) ) )
21adantl 273 . . . . . 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 964 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  A  e.  P. )
4 simpll 501 . . . . . . 7  |-  ( ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) )  ->  x  e.  ( 1st `  A
) )
5 prop 7247 . . . . . . . 8  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
6 elprnql 7253 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
75, 6sylan 279 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  x  e.  Q. )
83, 4, 7syl2an 285 . . . . . 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 503 . . . . . . 7  |-  ( ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A
)  /\  z  e.  ( 1st `  C ) ) )  ->  f  e.  ( 1st `  A
) )
10 elprnql 7253 . . . . . . . 8  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
115, 10sylan 279 . . . . . . 7  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
123, 9, 11syl2an 285 . . . . . 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 968 . . . . . . 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 510 . . . . . . 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 7247 . . . . . . . 8  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
16 elprnql 7253 . . . . . . . 8  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
1715, 16sylan 279 . . . . . . 7  |-  ( ( B  e.  P.  /\  y  e.  ( 1st `  B ) )  -> 
y  e.  Q. )
1813, 14, 17syl2anc 406 . . . . . 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 7155 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q. )  ->  ( w  .Q  v
)  =  ( v  .Q  w ) )
2019adantl 273 . . . . . 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 5906 . . . . 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 7172 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q.  /\  u  e.  Q. )  ->  (
w  <Q  v  <->  ( u  +Q  w )  <Q  (
u  +Q  v ) ) )
2322adantl 273 . . . . . 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 7148 . . . . . . 7  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  e.  Q. )
258, 18, 24syl2anc 406 . . . . . 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 7148 . . . . . . 7  |-  ( ( f  e.  Q.  /\  y  e.  Q. )  ->  ( f  .Q  y
)  e.  Q. )
2712, 18, 26syl2anc 406 . . . . . 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 969 . . . . . . . 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 512 . . . . . . . 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 7247 . . . . . . . . 9  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
31 elprnql 7253 . . . . . . . . 9  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
3230, 31sylan 279 . . . . . . . 8  |-  ( ( C  e.  P.  /\  z  e.  ( 1st `  C ) )  -> 
z  e.  Q. )
3328, 29, 32syl2anc 406 . . . . . . 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 7148 . . . . . . 7  |-  ( ( f  e.  Q.  /\  z  e.  Q. )  ->  ( f  .Q  z
)  e.  Q. )
3512, 33, 34syl2anc 406 . . . . . 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 7153 . . . . . . 7  |-  ( ( w  e.  Q.  /\  v  e.  Q. )  ->  ( w  +Q  v
)  =  ( v  +Q  w ) )
3736adantl 273 . . . . . 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 5906 . . . . 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 187 . . . 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 967 . . . . . 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 7309 . . . . . . . 8  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
42413adant1 982 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C )  e. 
P. )
4342adantr 272 . . . . . 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 7344 . . . . . 6  |-  ( ( A  e.  P.  /\  ( B  +P.  C )  e.  P. )  -> 
( A  .P.  ( B  +P.  C ) )  e.  P. )
4540, 43, 44syl2anc 406 . . . . 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 7159 . . . . . . 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 1199 . . . . . 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 511 . . . . . . 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 7240 . . . . . . . . . 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 7147 . . . . . . . . . 10  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
5149, 50genpprecll 7286 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( ( y  e.  ( 1st `  B
)  /\  z  e.  ( 1st `  C ) )  ->  ( y  +Q  z )  e.  ( 1st `  ( B  +P.  C ) ) ) )
5251imp 123 . . . . . . . 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 1200 . . . . . . 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 7241 . . . . . . . . 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 7148 . . . . . . . . 9  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  .Q  h
)  e.  Q. )
5654, 55genpprecll 7286 . . . . . . . 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 123 . . . . . . 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 1200 . . . . . 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 2193 . . . . 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 7247 . . . . . 6  |-  ( ( A  .P.  ( B  +P.  C ) )  e.  P.  ->  <. ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) ,  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) >.  e.  P. )
61 prcdnql 7256 . . . . . 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 279 . . . . 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 406 . . . 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 149 . . 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 5906 . . . . 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 7148 . . . . . . 7  |-  ( ( x  e.  Q.  /\  z  e.  Q. )  ->  ( x  .Q  z
)  e.  Q. )
678, 33, 66syl2anc 406 . . . . . 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 7172 . . . . . 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 1199 . . . . 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 187 . . . 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 7159 . . . . . . 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 1199 . . . . . 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 509 . . . . . . 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 7286 . . . . . . . 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 123 . . . . . . 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 1200 . . . . . 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 2193 . . . . 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 7256 . . . . . 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 279 . . . . 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 406 . . . 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 149 . . 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 689 . 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 7170 . . . . 5  |-  <Q  Or  Q.
84 nqtri3or 7168 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  \/  x  =  f  \/  f  <Q  x ) )
8583, 84sotritrieq 4215 . . . 4  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  =  f  <->  -.  ( x  <Q  f  \/  f  <Q  x ) ) )
868, 12, 85syl2anc 406 . . 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 5747 . . . . . . 7  |-  ( x  =  f  ->  (
x  .Q  z )  =  ( f  .Q  z ) )
8887oveq2d 5756 . . . . . 6  |-  ( x  =  f  ->  (
( x  .Q  y
)  +Q  ( x  .Q  z ) )  =  ( ( x  .Q  y )  +Q  ( f  .Q  z
) ) )
8972, 88sylan9eq 2168 . . . . 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 272 . . . . 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 2193 . . . 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 114 . . 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 169 . 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 7169 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  x 
<Q  f )
95 ltdcnq 7169 . . . . . 6  |-  ( ( f  e.  Q.  /\  x  e.  Q. )  -> DECID  f 
<Q  x )
9695ancoms 266 . . . . 5  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  -> DECID  f 
<Q  x )
97 dcor 902 . . . . 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 406 . . 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 803 . . 3  |-  (DECID  ( x 
<Q  f  \/  f  <Q  x )  <->  ( (
x  <Q  f  \/  f  <Q  x )  \/  -.  ( x  <Q  f  \/  f  <Q  x )
) )
10199, 100sylib 121 . 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 690 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 103    <-> wb 104    \/ wo 680  DECID wdc 802    /\ w3a 945    = wceq 1314    e. wcel 1463   <.cop 3498   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    +Q cplq 7054    .Q cmq 7055    <Q cltq 7057   P.cnp 7063    +P. cpp 7065    .P. cmp 7066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-imp 7241
This theorem is referenced by:  distrlem5prl  7358
  Copyright terms: Public domain W3C validator