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Theorem mulextsr1lem 7522
Description: Lemma for mulextsr1 7523. (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 7334 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
21adantl 273 . . . . . 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 7293 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
43adantl 273 . . . . . . 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 990 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  Z  e.  P. )
6 simp3r 993 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  V  e.  P. )
7 mulclpr 7328 . . . . . . . 8  |-  ( ( Z  e.  P.  /\  V  e.  P. )  ->  ( Z  .P.  V
)  e.  P. )
85, 6, 7syl2anc 406 . . . . . . 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 989 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  Y  e.  P. )
10 mulclpr 7328 . . . . . . . 8  |-  ( ( Y  e.  P.  /\  V  e.  P. )  ->  ( Y  .P.  V
)  e.  P. )
119, 6, 10syl2anc 406 . . . . . . 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 5878 . . . . . 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 988 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  X  e.  P. )
14 simp3l 992 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  U  e.  P. )
15 mulclpr 7328 . . . . . . . 8  |-  ( ( X  e.  P.  /\  U  e.  P. )  ->  ( X  .P.  U
)  e.  P. )
1613, 14, 15syl2anc 406 . . . . . . 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 991 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  W  e.  P. )
18 mulclpr 7328 . . . . . . . 8  |-  ( ( W  e.  P.  /\  U  e.  P. )  ->  ( W  .P.  U
)  e.  P. )
1917, 14, 18syl2anc 406 . . . . . . 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 5878 . . . . . 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 5881 . . . . 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 7335 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
2322adantl 273 . . . . . 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 5910 . . . . 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 7344 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
) )
2625adantl 273 . . . . . . 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 7336 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  =  ( g  .P.  f ) )
2827adantl 273 . . . . . . 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 5901 . . . . . 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 5901 . . . . . 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 5746 . . . . 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 2157 . . . 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 7328 . . . . . . 7  |-  ( ( X  e.  P.  /\  V  e.  P. )  ->  ( X  .P.  V
)  e.  P. )
3413, 6, 33syl2anc 406 . . . . . 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 7328 . . . . . . 7  |-  ( ( Y  e.  P.  /\  U  e.  P. )  ->  ( Y  .P.  U
)  e.  P. )
369, 14, 35syl2anc 406 . . . . . 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 7328 . . . . . . 7  |-  ( ( Z  e.  P.  /\  U  e.  P. )  ->  ( Z  .P.  U
)  e.  P. )
385, 14, 37syl2anc 406 . . . . . 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 7328 . . . . . . 7  |-  ( ( W  e.  P.  /\  V  e.  P. )  ->  ( W  .P.  V
)  e.  P. )
4017, 6, 39syl2anc 406 . . . . . 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 5910 . . . . 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 5901 . . . . . 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 5901 . . . . . 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 5746 . . . . 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 2150 . . . 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 3908 . . 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 2191 . . . . 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 2191 . . . . 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 7293 . . . . . . 7  |-  ( ( Z  e.  P.  /\  Y  e.  P. )  ->  ( Z  +P.  Y
)  e.  P. )
505, 9, 49syl2anc 406 . . . . . 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 7328 . . . . . 6  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  U  e.  P. )  ->  ( ( Z  +P.  Y )  .P.  U )  e.  P. )
5250, 14, 51syl2anc 406 . . . . 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 7293 . . . . . . 7  |-  ( ( X  e.  P.  /\  W  e.  P. )  ->  ( X  +P.  W
)  e.  P. )
5413, 17, 53syl2anc 406 . . . . . 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 7328 . . . . . 6  |-  ( ( ( X  +P.  W
)  e.  P.  /\  V  e.  P. )  ->  ( ( X  +P.  W )  .P.  V )  e.  P. )
5654, 6, 55syl2anc 406 . . . . 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 7377 . . . . 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 1200 . . . 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 7336 . . . . . . . . 9  |-  ( ( ( X  +P.  W
)  e.  P.  /\  U  e.  P. )  ->  ( ( X  +P.  W )  .P.  U )  =  ( U  .P.  ( X  +P.  W ) ) )
60593adant2 983 . . . . . . . 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 7336 . . . . . . . . 9  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  U  e.  P. )  ->  ( ( Z  +P.  Y )  .P.  U )  =  ( U  .P.  ( Z  +P.  Y ) ) )
62613adant1 982 . . . . . . . 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 3908 . . . . . . 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 7398 . . . . . . 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 149 . . . . . 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 1199 . . . . 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 7336 . . . . . . . 8  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  V  e.  P. )  ->  ( ( Z  +P.  Y )  .P.  V )  =  ( V  .P.  ( Z  +P.  Y ) ) )
6850, 6, 67syl2anc 406 . . . . . . 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 7336 . . . . . . . 8  |-  ( ( ( X  +P.  W
)  e.  P.  /\  V  e.  P. )  ->  ( ( X  +P.  W )  .P.  V )  =  ( V  .P.  ( X  +P.  W ) ) )
7054, 6, 69syl2anc 406 . . . . . . 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 3908 . . . . . 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 7398 . . . . . . 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 1199 . . . . . 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 149 . . . . 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 758 . . . 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 45 . . 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 149 . 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 7334 . . . . 5  |-  ( ( Z  e.  P.  /\  Y  e.  P. )  ->  ( Z  +P.  Y
)  =  ( Y  +P.  Z ) )
795, 9, 78syl2anc 406 . . . 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 3907 . . 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 7334 . . . . 5  |-  ( ( X  e.  P.  /\  W  e.  P. )  ->  ( X  +P.  W
)  =  ( W  +P.  X ) )
8213, 17, 81syl2anc 406 . . . 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 3907 . . 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 765 . 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 148 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 103    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   class class class wbr 3895  (class class class)co 5728   P.cnp 7047    +P. cpp 7049    .P. cmp 7050    <P cltp 7051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
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 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-i1p 7223  df-iplp 7224  df-imp 7225  df-iltp 7226
This theorem is referenced by:  mulextsr1  7523
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