MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltaprlem Unicode version

Theorem ltaprlem 8664
Description: Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltaprlem  |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )

Proof of Theorem ltaprlem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ltrelpr 8618 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 4736 . . . . 5  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simpld 445 . . . 4  |-  ( A 
<P  B  ->  A  e. 
P. )
4 ltexpri 8663 . . . . 5  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
5 addclpr 8638 . . . . . . . 8  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
6 ltaddpr 8654 . . . . . . . . . 10  |-  ( ( ( C  +P.  A
)  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x ) )
7 addasspr 8642 . . . . . . . . . . . 12  |-  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) )
8 oveq2 5828 . . . . . . . . . . . 12  |-  ( ( A  +P.  x )  =  B  ->  ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B
) )
97, 8syl5eq 2328 . . . . . . . . . . 11  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  B ) )
109breq2d 4036 . . . . . . . . . 10  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x )  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
116, 10syl5ib 210 . . . . . . . . 9  |-  ( ( A  +P.  x )  =  B  ->  (
( ( C  +P.  A )  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) )
1211exp3a 425 . . . . . . . 8  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  e.  P.  ->  ( x  e.  P.  ->  ( C  +P.  A ) 
<P  ( C  +P.  B
) ) ) )
135, 12syl5 28 . . . . . . 7  |-  ( ( A  +P.  x )  =  B  ->  (
( C  e.  P.  /\  A  e.  P. )  ->  ( x  e.  P.  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) ) )
1413com3r 73 . . . . . 6  |-  ( x  e.  P.  ->  (
( A  +P.  x
)  =  B  -> 
( ( C  e. 
P.  /\  A  e.  P. )  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) ) )
1514rexlimiv 2662 . . . . 5  |-  ( E. x  e.  P.  ( A  +P.  x )  =  B  ->  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
164, 15syl 15 . . . 4  |-  ( A 
<P  B  ->  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) )
173, 16sylan2i 636 . . 3  |-  ( A 
<P  B  ->  ( ( C  e.  P.  /\  A  <P  B )  -> 
( C  +P.  A
)  <P  ( C  +P.  B ) ) )
1817exp3a 425 . 2  |-  ( A 
<P  B  ->  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) ) )
1918pm2.43b 46 1  |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   E.wrex 2545   class class class wbr 4024  (class class class)co 5820   P.cnp 8477    +P. cpp 8479    <P cltp 8481
This theorem is referenced by:  ltapr  8665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-omul 6480  df-er 6656  df-ni 8492  df-pli 8493  df-mi 8494  df-lti 8495  df-plpq 8528  df-mpq 8529  df-ltpq 8530  df-enq 8531  df-nq 8532  df-erq 8533  df-plq 8534  df-mq 8535  df-1nq 8536  df-rq 8537  df-ltnq 8538  df-np 8601  df-plp 8603  df-ltp 8605
  Copyright terms: Public domain W3C validator