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Theorem mulextsr1lem 6922
Description: Lemma for mulextsr1 6923. (Contributed by Jim Kingdon, 17-Feb-2020.)
Assertion
Ref Expression
mulextsr1lem  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( ( X  .P.  U )  +P.  ( Y  .P.  V ) )  +P.  ( ( Z  .P.  V )  +P.  ( W  .P.  U
) ) )  <P 
( ( ( X  .P.  V )  +P.  ( Y  .P.  U
) )  +P.  (
( Z  .P.  U
)  +P.  ( W  .P.  V ) ) )  ->  ( ( X  +P.  W )  <P 
( Y  +P.  Z
)  \/  ( Z  +P.  Y )  <P 
( W  +P.  X
) ) ) )

Proof of Theorem mulextsr1lem
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcomprg 6734 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
21adantl 266 . . . . . 6  |-  ( ( ( ( X  e. 
P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. ) )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
3 addclpr 6693 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
43adantl 266 . . . . . . 7  |-  ( ( ( ( X  e. 
P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. ) )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  e.  P. )
5 simp2l 941 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  Z  e.  P. )
6 simp3r 944 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  V  e.  P. )
7 mulclpr 6728 . . . . . . . 8  |-  ( ( Z  e.  P.  /\  V  e.  P. )  ->  ( Z  .P.  V
)  e.  P. )
85, 6, 7syl2anc 397 . . . . . . 7  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( Z  .P.  V )  e.  P. )
9 simp1r 940 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  Y  e.  P. )
10 mulclpr 6728 . . . . . . . 8  |-  ( ( Y  e.  P.  /\  V  e.  P. )  ->  ( Y  .P.  V
)  e.  P. )
119, 6, 10syl2anc 397 . . . . . . 7  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( Y  .P.  V )  e.  P. )
124, 8, 11caovcld 5682 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( Z  .P.  V )  +P.  ( Y  .P.  V
) )  e.  P. )
13 simp1l 939 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  X  e.  P. )
14 simp3l 943 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  U  e.  P. )
15 mulclpr 6728 . . . . . . . 8  |-  ( ( X  e.  P.  /\  U  e.  P. )  ->  ( X  .P.  U
)  e.  P. )
1613, 14, 15syl2anc 397 . . . . . . 7  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( X  .P.  U )  e.  P. )
17 simp2r 942 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  W  e.  P. )
18 mulclpr 6728 . . . . . . . 8  |-  ( ( W  e.  P.  /\  U  e.  P. )  ->  ( W  .P.  U
)  e.  P. )
1917, 14, 18syl2anc 397 . . . . . . 7  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( W  .P.  U )  e.  P. )
204, 16, 19caovcld 5682 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( X  .P.  U )  +P.  ( W  .P.  U
) )  e.  P. )
212, 12, 20caovcomd 5685 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( Z  .P.  V
)  +P.  ( Y  .P.  V ) )  +P.  ( ( X  .P.  U )  +P.  ( W  .P.  U ) ) )  =  ( ( ( X  .P.  U
)  +P.  ( W  .P.  U ) )  +P.  ( ( Z  .P.  V )  +P.  ( Y  .P.  V ) ) ) )
22 addassprg 6735 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
2322adantl 266 . . . . . 6  |-  ( ( ( ( X  e. 
P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. ) )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )
)  ->  ( (
f  +P.  g )  +P.  h )  =  ( f  +P.  ( g  +P.  h ) ) )
2416, 11, 8, 2, 23, 19, 4caov411d 5714 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( X  .P.  U
)  +P.  ( Y  .P.  V ) )  +P.  ( ( Z  .P.  V )  +P.  ( W  .P.  U ) ) )  =  ( ( ( Z  .P.  V
)  +P.  ( Y  .P.  V ) )  +P.  ( ( X  .P.  U )  +P.  ( W  .P.  U ) ) ) )
25 distrprg 6744 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
) )
2625adantl 266 . . . . . . 7  |-  ( ( ( ( X  e. 
P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. ) )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )
)  ->  ( f  .P.  ( g  +P.  h
) )  =  ( ( f  .P.  g
)  +P.  ( f  .P.  h ) ) )
27 mulcomprg 6736 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  =  ( g  .P.  f ) )
2827adantl 266 . . . . . . 7  |-  ( ( ( ( X  e. 
P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. ) )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  .P.  g )  =  ( g  .P.  f ) )
2926, 13, 17, 14, 4, 28caovdir2d 5705 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( X  +P.  W )  .P. 
U )  =  ( ( X  .P.  U
)  +P.  ( W  .P.  U ) ) )
3026, 5, 9, 6, 4, 28caovdir2d 5705 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( Z  +P.  Y )  .P. 
V )  =  ( ( Z  .P.  V
)  +P.  ( Y  .P.  V ) ) )
3129, 30oveq12d 5558 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( X  +P.  W
)  .P.  U )  +P.  ( ( Z  +P.  Y )  .P.  V ) )  =  ( ( ( X  .P.  U
)  +P.  ( W  .P.  U ) )  +P.  ( ( Z  .P.  V )  +P.  ( Y  .P.  V ) ) ) )
3221, 24, 313eqtr4d 2098 . . . 4  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( X  .P.  U
)  +P.  ( Y  .P.  V ) )  +P.  ( ( Z  .P.  V )  +P.  ( W  .P.  U ) ) )  =  ( ( ( X  +P.  W
)  .P.  U )  +P.  ( ( Z  +P.  Y )  .P.  V ) ) )
33 mulclpr 6728 . . . . . . 7  |-  ( ( X  e.  P.  /\  V  e.  P. )  ->  ( X  .P.  V
)  e.  P. )
3413, 6, 33syl2anc 397 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( X  .P.  V )  e.  P. )
35 mulclpr 6728 . . . . . . 7  |-  ( ( Y  e.  P.  /\  U  e.  P. )  ->  ( Y  .P.  U
)  e.  P. )
369, 14, 35syl2anc 397 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( Y  .P.  U )  e.  P. )
37 mulclpr 6728 . . . . . . 7  |-  ( ( Z  e.  P.  /\  U  e.  P. )  ->  ( Z  .P.  U
)  e.  P. )
385, 14, 37syl2anc 397 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( Z  .P.  U )  e.  P. )
39 mulclpr 6728 . . . . . . 7  |-  ( ( W  e.  P.  /\  V  e.  P. )  ->  ( W  .P.  V
)  e.  P. )
4017, 6, 39syl2anc 397 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( W  .P.  V )  e.  P. )
4134, 36, 38, 2, 23, 40, 4caov411d 5714 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( X  .P.  V
)  +P.  ( Y  .P.  U ) )  +P.  ( ( Z  .P.  U )  +P.  ( W  .P.  V ) ) )  =  ( ( ( Z  .P.  U
)  +P.  ( Y  .P.  U ) )  +P.  ( ( X  .P.  V )  +P.  ( W  .P.  V ) ) ) )
4226, 5, 9, 14, 4, 28caovdir2d 5705 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( Z  +P.  Y )  .P. 
U )  =  ( ( Z  .P.  U
)  +P.  ( Y  .P.  U ) ) )
4326, 13, 17, 6, 4, 28caovdir2d 5705 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( X  +P.  W )  .P. 
V )  =  ( ( X  .P.  V
)  +P.  ( W  .P.  V ) ) )
4442, 43oveq12d 5558 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( Z  +P.  Y
)  .P.  U )  +P.  ( ( X  +P.  W )  .P.  V ) )  =  ( ( ( Z  .P.  U
)  +P.  ( Y  .P.  U ) )  +P.  ( ( X  .P.  V )  +P.  ( W  .P.  V ) ) ) )
4541, 44eqtr4d 2091 . . . 4  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( X  .P.  V
)  +P.  ( Y  .P.  U ) )  +P.  ( ( Z  .P.  U )  +P.  ( W  .P.  V ) ) )  =  ( ( ( Z  +P.  Y
)  .P.  U )  +P.  ( ( X  +P.  W )  .P.  V ) ) )
4632, 45breq12d 3805 . . 3  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( ( X  .P.  U )  +P.  ( Y  .P.  V ) )  +P.  ( ( Z  .P.  V )  +P.  ( W  .P.  U
) ) )  <P 
( ( ( X  .P.  V )  +P.  ( Y  .P.  U
) )  +P.  (
( Z  .P.  U
)  +P.  ( W  .P.  V ) ) )  <-> 
( ( ( X  +P.  W )  .P. 
U )  +P.  (
( Z  +P.  Y
)  .P.  V )
)  <P  ( ( ( Z  +P.  Y )  .P.  U )  +P.  ( ( X  +P.  W )  .P.  V ) ) ) )
4729, 20eqeltrd 2130 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( X  +P.  W )  .P. 
U )  e.  P. )
4830, 12eqeltrd 2130 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( Z  +P.  Y )  .P. 
V )  e.  P. )
49 addclpr 6693 . . . . . . 7  |-  ( ( Z  e.  P.  /\  Y  e.  P. )  ->  ( Z  +P.  Y
)  e.  P. )
505, 9, 49syl2anc 397 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( Z  +P.  Y )  e.  P. )
51 mulclpr 6728 . . . . . 6  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  U  e.  P. )  ->  ( ( Z  +P.  Y )  .P.  U )  e.  P. )
5250, 14, 51syl2anc 397 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( Z  +P.  Y )  .P. 
U )  e.  P. )
53 addclpr 6693 . . . . . . 7  |-  ( ( X  e.  P.  /\  W  e.  P. )  ->  ( X  +P.  W
)  e.  P. )
5413, 17, 53syl2anc 397 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( X  +P.  W )  e.  P. )
55 mulclpr 6728 . . . . . 6  |-  ( ( ( X  +P.  W
)  e.  P.  /\  V  e.  P. )  ->  ( ( X  +P.  W )  .P.  V )  e.  P. )
5654, 6, 55syl2anc 397 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( X  +P.  W )  .P. 
V )  e.  P. )
57 addextpr 6777 . . . . 5  |-  ( ( ( ( ( X  +P.  W )  .P. 
U )  e.  P.  /\  ( ( Z  +P.  Y )  .P.  V )  e.  P. )  /\  ( ( ( Z  +P.  Y )  .P. 
U )  e.  P.  /\  ( ( X  +P.  W )  .P.  V )  e.  P. ) )  ->  ( ( ( ( X  +P.  W
)  .P.  U )  +P.  ( ( Z  +P.  Y )  .P.  V ) )  <P  ( (
( Z  +P.  Y
)  .P.  U )  +P.  ( ( X  +P.  W )  .P.  V ) )  ->  ( (
( X  +P.  W
)  .P.  U )  <P  ( ( Z  +P.  Y )  .P.  U )  \/  ( ( Z  +P.  Y )  .P. 
V )  <P  (
( X  +P.  W
)  .P.  V )
) ) )
5847, 48, 52, 56, 57syl22anc 1147 . . . 4  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( ( X  +P.  W )  .P.  U )  +P.  ( ( Z  +P.  Y )  .P. 
V ) )  <P 
( ( ( Z  +P.  Y )  .P. 
U )  +P.  (
( X  +P.  W
)  .P.  V )
)  ->  ( (
( X  +P.  W
)  .P.  U )  <P  ( ( Z  +P.  Y )  .P.  U )  \/  ( ( Z  +P.  Y )  .P. 
V )  <P  (
( X  +P.  W
)  .P.  V )
) ) )
59 mulcomprg 6736 . . . . . . . . 9  |-  ( ( ( X  +P.  W
)  e.  P.  /\  U  e.  P. )  ->  ( ( X  +P.  W )  .P.  U )  =  ( U  .P.  ( X  +P.  W ) ) )
60593adant2 934 . . . . . . . 8  |-  ( ( ( X  +P.  W
)  e.  P.  /\  ( Z  +P.  Y )  e.  P.  /\  U  e.  P. )  ->  (
( X  +P.  W
)  .P.  U )  =  ( U  .P.  ( X  +P.  W ) ) )
61 mulcomprg 6736 . . . . . . . . 9  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  U  e.  P. )  ->  ( ( Z  +P.  Y )  .P.  U )  =  ( U  .P.  ( Z  +P.  Y ) ) )
62613adant1 933 . . . . . . . 8  |-  ( ( ( X  +P.  W
)  e.  P.  /\  ( Z  +P.  Y )  e.  P.  /\  U  e.  P. )  ->  (
( Z  +P.  Y
)  .P.  U )  =  ( U  .P.  ( Z  +P.  Y ) ) )
6360, 62breq12d 3805 . . . . . . 7  |-  ( ( ( X  +P.  W
)  e.  P.  /\  ( Z  +P.  Y )  e.  P.  /\  U  e.  P. )  ->  (
( ( X  +P.  W )  .P.  U ) 
<P  ( ( Z  +P.  Y )  .P.  U )  <-> 
( U  .P.  ( X  +P.  W ) ) 
<P  ( U  .P.  ( Z  +P.  Y ) ) ) )
64 ltmprr 6798 . . . . . . 7  |-  ( ( ( X  +P.  W
)  e.  P.  /\  ( Z  +P.  Y )  e.  P.  /\  U  e.  P. )  ->  (
( U  .P.  ( X  +P.  W ) ) 
<P  ( U  .P.  ( Z  +P.  Y ) )  ->  ( X  +P.  W )  <P  ( Z  +P.  Y ) ) )
6563, 64sylbid 143 . . . . . 6  |-  ( ( ( X  +P.  W
)  e.  P.  /\  ( Z  +P.  Y )  e.  P.  /\  U  e.  P. )  ->  (
( ( X  +P.  W )  .P.  U ) 
<P  ( ( Z  +P.  Y )  .P.  U )  ->  ( X  +P.  W )  <P  ( Z  +P.  Y ) ) )
6654, 50, 14, 65syl3anc 1146 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( X  +P.  W
)  .P.  U )  <P  ( ( Z  +P.  Y )  .P.  U )  ->  ( X  +P.  W )  <P  ( Z  +P.  Y ) ) )
67 mulcomprg 6736 . . . . . . . 8  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  V  e.  P. )  ->  ( ( Z  +P.  Y )  .P.  V )  =  ( V  .P.  ( Z  +P.  Y ) ) )
6850, 6, 67syl2anc 397 . . . . . . 7  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( Z  +P.  Y )  .P. 
V )  =  ( V  .P.  ( Z  +P.  Y ) ) )
69 mulcomprg 6736 . . . . . . . 8  |-  ( ( ( X  +P.  W
)  e.  P.  /\  V  e.  P. )  ->  ( ( X  +P.  W )  .P.  V )  =  ( V  .P.  ( X  +P.  W ) ) )
7054, 6, 69syl2anc 397 . . . . . . 7  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( X  +P.  W )  .P. 
V )  =  ( V  .P.  ( X  +P.  W ) ) )
7168, 70breq12d 3805 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( Z  +P.  Y
)  .P.  V )  <P  ( ( X  +P.  W )  .P.  V )  <-> 
( V  .P.  ( Z  +P.  Y ) ) 
<P  ( V  .P.  ( X  +P.  W ) ) ) )
72 ltmprr 6798 . . . . . . 7  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  ( X  +P.  W )  e.  P.  /\  V  e.  P. )  ->  (
( V  .P.  ( Z  +P.  Y ) ) 
<P  ( V  .P.  ( X  +P.  W ) )  ->  ( Z  +P.  Y )  <P  ( X  +P.  W ) ) )
7350, 54, 6, 72syl3anc 1146 . . . . . 6  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( V  .P.  ( Z  +P.  Y ) )  <P  ( V  .P.  ( X  +P.  W ) )  ->  ( Z  +P.  Y )  <P 
( X  +P.  W
) ) )
7471, 73sylbid 143 . . . . 5  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( Z  +P.  Y
)  .P.  V )  <P  ( ( X  +P.  W )  .P.  V )  ->  ( Z  +P.  Y )  <P  ( X  +P.  W ) ) )
7566, 74orim12d 710 . . . 4  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( ( X  +P.  W )  .P.  U ) 
<P  ( ( Z  +P.  Y )  .P.  U )  \/  ( ( Z  +P.  Y )  .P. 
V )  <P  (
( X  +P.  W
)  .P.  V )
)  ->  ( ( X  +P.  W )  <P 
( Z  +P.  Y
)  \/  ( Z  +P.  Y )  <P 
( X  +P.  W
) ) ) )
7658, 75syld 44 . . 3  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( ( X  +P.  W )  .P.  U )  +P.  ( ( Z  +P.  Y )  .P. 
V ) )  <P 
( ( ( Z  +P.  Y )  .P. 
U )  +P.  (
( X  +P.  W
)  .P.  V )
)  ->  ( ( X  +P.  W )  <P 
( Z  +P.  Y
)  \/  ( Z  +P.  Y )  <P 
( X  +P.  W
) ) ) )
7746, 76sylbid 143 . 2  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( ( X  .P.  U )  +P.  ( Y  .P.  V ) )  +P.  ( ( Z  .P.  V )  +P.  ( W  .P.  U
) ) )  <P 
( ( ( X  .P.  V )  +P.  ( Y  .P.  U
) )  +P.  (
( Z  .P.  U
)  +P.  ( W  .P.  V ) ) )  ->  ( ( X  +P.  W )  <P 
( Z  +P.  Y
)  \/  ( Z  +P.  Y )  <P 
( X  +P.  W
) ) ) )
78 addcomprg 6734 . . . . 5  |-  ( ( Z  e.  P.  /\  Y  e.  P. )  ->  ( Z  +P.  Y
)  =  ( Y  +P.  Z ) )
795, 9, 78syl2anc 397 . . . 4  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( Z  +P.  Y )  =  ( Y  +P.  Z ) )
8079breq2d 3804 . . 3  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( X  +P.  W )  <P 
( Z  +P.  Y
)  <->  ( X  +P.  W )  <P  ( Y  +P.  Z ) ) )
81 addcomprg 6734 . . . . 5  |-  ( ( X  e.  P.  /\  W  e.  P. )  ->  ( X  +P.  W
)  =  ( W  +P.  X ) )
8213, 17, 81syl2anc 397 . . . 4  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( X  +P.  W )  =  ( W  +P.  X ) )
8382breq2d 3804 . . 3  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( ( Z  +P.  Y )  <P 
( X  +P.  W
)  <->  ( Z  +P.  Y )  <P  ( W  +P.  X ) ) )
8480, 83orbi12d 717 . 2  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( X  +P.  W
)  <P  ( Z  +P.  Y )  \/  ( Z  +P.  Y )  <P 
( X  +P.  W
) )  <->  ( ( X  +P.  W )  <P 
( Y  +P.  Z
)  \/  ( Z  +P.  Y )  <P 
( W  +P.  X
) ) ) )
8577, 84sylibd 142 1  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  ( (
( ( X  .P.  U )  +P.  ( Y  .P.  V ) )  +P.  ( ( Z  .P.  V )  +P.  ( W  .P.  U
) ) )  <P 
( ( ( X  .P.  V )  +P.  ( Y  .P.  U
) )  +P.  (
( Z  .P.  U
)  +P.  ( W  .P.  V ) ) )  ->  ( ( X  +P.  W )  <P 
( Y  +P.  Z
)  \/  ( Z  +P.  Y )  <P 
( W  +P.  X
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   class class class wbr 3792  (class class class)co 5540   P.cnp 6447    +P. cpp 6449    .P. cmp 6450    <P cltp 6451
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 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
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 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-imp 6625  df-iltp 6626
This theorem is referenced by:  mulextsr1  6923
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