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

Theorem ltaprlem 9316
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 9270 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 4987 . . . . 5  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simpld 459 . . . 4  |-  ( A 
<P  B  ->  A  e. 
P. )
4 ltexpri 9315 . . . . 5  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
5 addclpr 9290 . . . . . . . 8  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
6 ltaddpr 9306 . . . . . . . . . 10  |-  ( ( ( C  +P.  A
)  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x ) )
7 addasspr 9294 . . . . . . . . . . . 12  |-  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) )
8 oveq2 6200 . . . . . . . . . . . 12  |-  ( ( A  +P.  x )  =  B  ->  ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B
) )
97, 8syl5eq 2504 . . . . . . . . . . 11  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  B ) )
109breq2d 4404 . . . . . . . . . 10  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x )  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
116, 10syl5ib 219 . . . . . . . . 9  |-  ( ( A  +P.  x )  =  B  ->  (
( ( C  +P.  A )  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) )
1211expd 436 . . . . . . . 8  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  e.  P.  ->  ( x  e.  P.  ->  ( C  +P.  A ) 
<P  ( C  +P.  B
) ) ) )
135, 12syl5 32 . . . . . . 7  |-  ( ( A  +P.  x )  =  B  ->  (
( C  e.  P.  /\  A  e.  P. )  ->  ( x  e.  P.  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) ) )
1413com3r 79 . . . . . 6  |-  ( x  e.  P.  ->  (
( A  +P.  x
)  =  B  -> 
( ( C  e. 
P.  /\  A  e.  P. )  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) ) )
1514rexlimiv 2933 . . . . 5  |-  ( E. x  e.  P.  ( A  +P.  x )  =  B  ->  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
164, 15syl 16 . . . 4  |-  ( A 
<P  B  ->  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) )
173, 16sylan2i 655 . . 3  |-  ( A 
<P  B  ->  ( ( C  e.  P.  /\  A  <P  B )  -> 
( C  +P.  A
)  <P  ( C  +P.  B ) ) )
1817expd 436 . 2  |-  ( A 
<P  B  ->  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) ) )
1918pm2.43b 50 1  |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796   class class class wbr 4392  (class class class)co 6192   P.cnp 9129    +P. cpp 9131    <P cltp 9133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-omul 7027  df-er 7203  df-ni 9144  df-pli 9145  df-mi 9146  df-lti 9147  df-plpq 9180  df-mpq 9181  df-ltpq 9182  df-enq 9183  df-nq 9184  df-erq 9185  df-plq 9186  df-mq 9187  df-1nq 9188  df-rq 9189  df-ltnq 9190  df-np 9253  df-plp 9255  df-ltp 9257
This theorem is referenced by:  ltapr  9317
  Copyright terms: Public domain W3C validator