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Theorem ltmprr 6797
Description: Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
Assertion
Ref Expression
ltmprr  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( C  .P.  A
)  <P  ( C  .P.  B )  ->  A  <P  B ) )

Proof of Theorem ltmprr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 recexpr 6793 . . . . 5  |-  ( C  e.  P.  ->  E. y  e.  P.  ( C  .P.  y )  =  1P )
213ad2ant3 938 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  E. y  e.  P.  ( C  .P.  y )  =  1P )
32adantr 265 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  .P.  A
)  <P  ( C  .P.  B ) )  ->  E. y  e.  P.  ( C  .P.  y )  =  1P )
4 ltexpri 6768 . . . . 5  |-  ( ( C  .P.  A ) 
<P  ( C  .P.  B
)  ->  E. x  e.  P.  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) )
54ad2antlr 466 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  .P.  A )  <P  ( C  .P.  B ) )  /\  ( y  e.  P.  /\  ( C  .P.  y
)  =  1P ) )  ->  E. x  e.  P.  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) )
6 simplll 493 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. ) )
76simp1d 927 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  A  e.  P. )
8 simplrl 495 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  y  e.  P. )
9 simprl 491 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  x  e.  P. )
10 mulclpr 6727 . . . . . . 7  |-  ( ( y  e.  P.  /\  x  e.  P. )  ->  ( y  .P.  x
)  e.  P. )
118, 9, 10syl2anc 397 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
y  .P.  x )  e.  P. )
12 ltaddpr 6752 . . . . . 6  |-  ( ( A  e.  P.  /\  ( y  .P.  x
)  e.  P. )  ->  A  <P  ( A  +P.  ( y  .P.  x
) ) )
137, 11, 12syl2anc 397 . . . . 5  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  A  <P  ( A  +P.  (
y  .P.  x )
) )
14 simprr 492 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
( C  .P.  A
)  +P.  x )  =  ( C  .P.  B ) )
1514oveq2d 5555 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
y  .P.  ( ( C  .P.  A )  +P.  x ) )  =  ( y  .P.  ( C  .P.  B ) ) )
166simp3d 929 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  C  e.  P. )
17 mulclpr 6727 . . . . . . . . 9  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  .P.  A
)  e.  P. )
1816, 7, 17syl2anc 397 . . . . . . . 8  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  ( C  .P.  A )  e. 
P. )
19 distrprg 6743 . . . . . . . 8  |-  ( ( y  e.  P.  /\  ( C  .P.  A )  e.  P.  /\  x  e.  P. )  ->  (
y  .P.  ( ( C  .P.  A )  +P.  x ) )  =  ( ( y  .P.  ( C  .P.  A
) )  +P.  (
y  .P.  x )
) )
208, 18, 9, 19syl3anc 1146 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
y  .P.  ( ( C  .P.  A )  +P.  x ) )  =  ( ( y  .P.  ( C  .P.  A
) )  +P.  (
y  .P.  x )
) )
21 mulassprg 6736 . . . . . . . . 9  |-  ( ( y  e.  P.  /\  C  e.  P.  /\  A  e.  P. )  ->  (
( y  .P.  C
)  .P.  A )  =  ( y  .P.  ( C  .P.  A
) ) )
228, 16, 7, 21syl3anc 1146 . . . . . . . 8  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
( y  .P.  C
)  .P.  A )  =  ( y  .P.  ( C  .P.  A
) ) )
2322oveq1d 5554 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
( ( y  .P. 
C )  .P.  A
)  +P.  ( y  .P.  x ) )  =  ( ( y  .P.  ( C  .P.  A
) )  +P.  (
y  .P.  x )
) )
24 mulcomprg 6735 . . . . . . . . . . . 12  |-  ( ( y  e.  P.  /\  C  e.  P. )  ->  ( y  .P.  C
)  =  ( C  .P.  y ) )
258, 16, 24syl2anc 397 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
y  .P.  C )  =  ( C  .P.  y ) )
26 simplrr 496 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  ( C  .P.  y )  =  1P )
2725, 26eqtrd 2088 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
y  .P.  C )  =  1P )
2827oveq1d 5554 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
( y  .P.  C
)  .P.  A )  =  ( 1P  .P.  A ) )
29 1pr 6709 . . . . . . . . . . . 12  |-  1P  e.  P.
30 mulcomprg 6735 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  .P.  1P )  =  ( 1P  .P.  A ) )
3129, 30mpan2 409 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  ( 1P  .P.  A
) )
32 1idpr 6747 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  A )
3331, 32eqtr3d 2090 . . . . . . . . . 10  |-  ( A  e.  P.  ->  ( 1P  .P.  A )  =  A )
347, 33syl 14 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  ( 1P  .P.  A )  =  A )
3528, 34eqtrd 2088 . . . . . . . 8  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
( y  .P.  C
)  .P.  A )  =  A )
3635oveq1d 5554 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
( ( y  .P. 
C )  .P.  A
)  +P.  ( y  .P.  x ) )  =  ( A  +P.  (
y  .P.  x )
) )
3720, 23, 363eqtr2d 2094 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
y  .P.  ( ( C  .P.  A )  +P.  x ) )  =  ( A  +P.  (
y  .P.  x )
) )
3827oveq1d 5554 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
( y  .P.  C
)  .P.  B )  =  ( 1P  .P.  B ) )
396simp2d 928 . . . . . . . 8  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  B  e.  P. )
40 mulassprg 6736 . . . . . . . 8  |-  ( ( y  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  ->  (
( y  .P.  C
)  .P.  B )  =  ( y  .P.  ( C  .P.  B
) ) )
418, 16, 39, 40syl3anc 1146 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
( y  .P.  C
)  .P.  B )  =  ( y  .P.  ( C  .P.  B
) ) )
42 mulcomprg 6735 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  1P  e.  P. )  -> 
( B  .P.  1P )  =  ( 1P  .P.  B ) )
4329, 42mpan2 409 . . . . . . . . 9  |-  ( B  e.  P.  ->  ( B  .P.  1P )  =  ( 1P  .P.  B
) )
44 1idpr 6747 . . . . . . . . 9  |-  ( B  e.  P.  ->  ( B  .P.  1P )  =  B )
4543, 44eqtr3d 2090 . . . . . . . 8  |-  ( B  e.  P.  ->  ( 1P  .P.  B )  =  B )
4639, 45syl 14 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  ( 1P  .P.  B )  =  B )
4738, 41, 463eqtr3d 2096 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  (
y  .P.  ( C  .P.  B ) )  =  B )
4815, 37, 473eqtr3d 2096 . . . . 5  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  ( A  +P.  ( y  .P.  x ) )  =  B )
4913, 48breqtrd 3815 . . . 4  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e. 
P. )  /\  ( C  .P.  A )  <P 
( C  .P.  B
) )  /\  (
y  e.  P.  /\  ( C  .P.  y )  =  1P ) )  /\  ( x  e. 
P.  /\  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) ) )  ->  A  <P  B )
505, 49rexlimddv 2454 . . 3  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  .P.  A )  <P  ( C  .P.  B ) )  /\  ( y  e.  P.  /\  ( C  .P.  y
)  =  1P ) )  ->  A  <P  B )
513, 50rexlimddv 2454 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  .P.  A
)  <P  ( C  .P.  B ) )  ->  A  <P  B )
5251ex 112 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( C  .P.  A
)  <P  ( C  .P.  B )  ->  A  <P  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3791  (class class class)co 5539   P.cnp 6446   1Pc1p 6447    +P. cpp 6448    .P. cmp 6449    <P cltp 6450
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 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
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 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-imp 6624  df-iltp 6625
This theorem is referenced by:  mulextsr1lem  6921
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