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Theorem mulextsr1lem 7810
Description: Lemma for mulextsr1 7811. (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 7608 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
21adantl 277 . . . . . 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 7567 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  e.  P. )
43adantl 277 . . . . . . 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 1025 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  Z  e.  P. )
6 simp3r 1028 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  V  e.  P. )
7 mulclpr 7602 . . . . . . . 8  |-  ( ( Z  e.  P.  /\  V  e.  P. )  ->  ( Z  .P.  V
)  e.  P. )
85, 6, 7syl2anc 411 . . . . . . 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 1024 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  Y  e.  P. )
10 mulclpr 7602 . . . . . . . 8  |-  ( ( Y  e.  P.  /\  V  e.  P. )  ->  ( Y  .P.  V
)  e.  P. )
119, 6, 10syl2anc 411 . . . . . . 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 6051 . . . . . 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 1023 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  X  e.  P. )
14 simp3l 1027 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  U  e.  P. )
15 mulclpr 7602 . . . . . . . 8  |-  ( ( X  e.  P.  /\  U  e.  P. )  ->  ( X  .P.  U
)  e.  P. )
1613, 14, 15syl2anc 411 . . . . . . 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 1026 . . . . . . . 8  |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. )
)  ->  W  e.  P. )
18 mulclpr 7602 . . . . . . . 8  |-  ( ( W  e.  P.  /\  U  e.  P. )  ->  ( W  .P.  U
)  e.  P. )
1917, 14, 18syl2anc 411 . . . . . . 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 6051 . . . . . 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 6054 . . . . 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 7609 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
( f  +P.  g
)  +P.  h )  =  ( f  +P.  ( g  +P.  h
) ) )
2322adantl 277 . . . . . 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 6083 . . . . 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 7618 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  .P.  ( g  +P.  h ) )  =  ( ( f  .P.  g )  +P.  (
f  .P.  h )
) )
2625adantl 277 . . . . . . 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 7610 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  .P.  g
)  =  ( g  .P.  f ) )
2827adantl 277 . . . . . . 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 6074 . . . . . 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 6074 . . . . . 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 5915 . . . . 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 2232 . . . 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 7602 . . . . . . 7  |-  ( ( X  e.  P.  /\  V  e.  P. )  ->  ( X  .P.  V
)  e.  P. )
3413, 6, 33syl2anc 411 . . . . . 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 7602 . . . . . . 7  |-  ( ( Y  e.  P.  /\  U  e.  P. )  ->  ( Y  .P.  U
)  e.  P. )
369, 14, 35syl2anc 411 . . . . . 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 7602 . . . . . . 7  |-  ( ( Z  e.  P.  /\  U  e.  P. )  ->  ( Z  .P.  U
)  e.  P. )
385, 14, 37syl2anc 411 . . . . . 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 7602 . . . . . . 7  |-  ( ( W  e.  P.  /\  V  e.  P. )  ->  ( W  .P.  V
)  e.  P. )
4017, 6, 39syl2anc 411 . . . . . 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 6083 . . . . 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 6074 . . . . . 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 6074 . . . . . 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 5915 . . . . 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 2225 . . . 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 4031 . . 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 2266 . . . . 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 2266 . . . . 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 7567 . . . . . . 7  |-  ( ( Z  e.  P.  /\  Y  e.  P. )  ->  ( Z  +P.  Y
)  e.  P. )
505, 9, 49syl2anc 411 . . . . . 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 7602 . . . . . 6  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  U  e.  P. )  ->  ( ( Z  +P.  Y )  .P.  U )  e.  P. )
5250, 14, 51syl2anc 411 . . . . 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 7567 . . . . . . 7  |-  ( ( X  e.  P.  /\  W  e.  P. )  ->  ( X  +P.  W
)  e.  P. )
5413, 17, 53syl2anc 411 . . . . . 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 7602 . . . . . 6  |-  ( ( ( X  +P.  W
)  e.  P.  /\  V  e.  P. )  ->  ( ( X  +P.  W )  .P.  V )  e.  P. )
5654, 6, 55syl2anc 411 . . . . 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 7651 . . . . 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 1250 . . . 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 7610 . . . . . . . . 9  |-  ( ( ( X  +P.  W
)  e.  P.  /\  U  e.  P. )  ->  ( ( X  +P.  W )  .P.  U )  =  ( U  .P.  ( X  +P.  W ) ) )
60593adant2 1018 . . . . . . . 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 7610 . . . . . . . . 9  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  U  e.  P. )  ->  ( ( Z  +P.  Y )  .P.  U )  =  ( U  .P.  ( Z  +P.  Y ) ) )
62613adant1 1017 . . . . . . . 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 4031 . . . . . . 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 7672 . . . . . . 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 150 . . . . . 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 1249 . . . . 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 7610 . . . . . . . 8  |-  ( ( ( Z  +P.  Y
)  e.  P.  /\  V  e.  P. )  ->  ( ( Z  +P.  Y )  .P.  V )  =  ( V  .P.  ( Z  +P.  Y ) ) )
6850, 6, 67syl2anc 411 . . . . . . 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 7610 . . . . . . . 8  |-  ( ( ( X  +P.  W
)  e.  P.  /\  V  e.  P. )  ->  ( ( X  +P.  W )  .P.  V )  =  ( V  .P.  ( X  +P.  W ) ) )
7054, 6, 69syl2anc 411 . . . . . . 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 4031 . . . . . 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 7672 . . . . . . 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 1249 . . . . . 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 150 . . . . 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 787 . . . 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 150 . 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 7608 . . . . 5  |-  ( ( Z  e.  P.  /\  Y  e.  P. )  ->  ( Z  +P.  Y
)  =  ( Y  +P.  Z ) )
795, 9, 78syl2anc 411 . . . 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 4030 . . 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 7608 . . . . 5  |-  ( ( X  e.  P.  /\  W  e.  P. )  ->  ( X  +P.  W
)  =  ( W  +P.  X ) )
8213, 17, 81syl2anc 411 . . . 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 4030 . . 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 794 . 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 149 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 104    \/ wo 709    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5897   P.cnp 7321    +P. cpp 7323    .P. cmp 7324    <P cltp 7325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-2o 6443  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-enq0 7454  df-nq0 7455  df-0nq0 7456  df-plq0 7457  df-mq0 7458  df-inp 7496  df-i1p 7497  df-iplp 7498  df-imp 7499  df-iltp 7500
This theorem is referenced by:  mulextsr1  7811
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