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Theorem ltmprr 7562
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 7558 . . . . 5  |-  ( C  e.  P.  ->  E. y  e.  P.  ( C  .P.  y )  =  1P )
213ad2ant3 1005 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  E. y  e.  P.  ( C  .P.  y )  =  1P )
32adantr 274 . . 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 7533 . . . . 5  |-  ( ( C  .P.  A ) 
<P  ( C  .P.  B
)  ->  E. x  e.  P.  ( ( C  .P.  A )  +P.  x )  =  ( C  .P.  B ) )
54ad2antlr 481 . . . 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 523 . . . . . . 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 994 . . . . . 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 525 . . . . . . 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 521 . . . . . . 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 7492 . . . . . . 7  |-  ( ( y  e.  P.  /\  x  e.  P. )  ->  ( y  .P.  x
)  e.  P. )
118, 9, 10syl2anc 409 . . . . . 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 7517 . . . . . 6  |-  ( ( A  e.  P.  /\  ( y  .P.  x
)  e.  P. )  ->  A  <P  ( A  +P.  ( y  .P.  x
) ) )
137, 11, 12syl2anc 409 . . . . 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 522 . . . . . . 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 5840 . . . . . 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 996 . . . . . . . . 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 7492 . . . . . . . . 9  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  .P.  A
)  e.  P. )
1816, 7, 17syl2anc 409 . . . . . . . 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 7508 . . . . . . . 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 1220 . . . . . . 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 7501 . . . . . . . . 9  |-  ( ( y  e.  P.  /\  C  e.  P.  /\  A  e.  P. )  ->  (
( y  .P.  C
)  .P.  A )  =  ( y  .P.  ( C  .P.  A
) ) )
228, 16, 7, 21syl3anc 1220 . . . . . . . 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 5839 . . . . . . 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 7500 . . . . . . . . . . . 12  |-  ( ( y  e.  P.  /\  C  e.  P. )  ->  ( y  .P.  C
)  =  ( C  .P.  y ) )
258, 16, 24syl2anc 409 . . . . . . . . . . 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 526 . . . . . . . . . . 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 2190 . . . . . . . . . 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 5839 . . . . . . . . 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 7474 . . . . . . . . . . . 12  |-  1P  e.  P.
30 mulcomprg 7500 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  .P.  1P )  =  ( 1P  .P.  A ) )
3129, 30mpan2 422 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  ( 1P  .P.  A
) )
32 1idpr 7512 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  A )
3331, 32eqtr3d 2192 . . . . . . . . . 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 2190 . . . . . . . 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 5839 . . . . . . 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 2196 . . . . . 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 5839 . . . . . . 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 995 . . . . . . . 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 7501 . . . . . . . 8  |-  ( ( y  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  ->  (
( y  .P.  C
)  .P.  B )  =  ( y  .P.  ( C  .P.  B
) ) )
418, 16, 39, 40syl3anc 1220 . . . . . . 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 7500 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  1P  e.  P. )  -> 
( B  .P.  1P )  =  ( 1P  .P.  B ) )
4329, 42mpan2 422 . . . . . . . . 9  |-  ( B  e.  P.  ->  ( B  .P.  1P )  =  ( 1P  .P.  B
) )
44 1idpr 7512 . . . . . . . . 9  |-  ( B  e.  P.  ->  ( B  .P.  1P )  =  B )
4543, 44eqtr3d 2192 . . . . . . . 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 2198 . . . . . 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 2198 . . . . 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 3990 . . . 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 2579 . . 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 2579 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( C  .P.  A
)  <P  ( C  .P.  B ) )  ->  A  <P  B )
5251ex 114 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 103    /\ w3a 963    = wceq 1335    e. wcel 2128   E.wrex 2436   class class class wbr 3965  (class class class)co 5824   P.cnp 7211   1Pc1p 7212    +P. cpp 7213    .P. cmp 7214    <P cltp 7215
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-i1p 7387  df-iplp 7388  df-imp 7389  df-iltp 7390
This theorem is referenced by:  mulextsr1lem  7700
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