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

Theorem ltaprlem 8910
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 8864 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 4917 . . . . 5  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simpld 446 . . . 4  |-  ( A 
<P  B  ->  A  e. 
P. )
4 ltexpri 8909 . . . . 5  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
5 addclpr 8884 . . . . . . . 8  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
6 ltaddpr 8900 . . . . . . . . . 10  |-  ( ( ( C  +P.  A
)  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x ) )
7 addasspr 8888 . . . . . . . . . . . 12  |-  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) )
8 oveq2 6080 . . . . . . . . . . . 12  |-  ( ( A  +P.  x )  =  B  ->  ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B
) )
97, 8syl5eq 2479 . . . . . . . . . . 11  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  B ) )
109breq2d 4216 . . . . . . . . . 10  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x )  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
116, 10syl5ib 211 . . . . . . . . 9  |-  ( ( A  +P.  x )  =  B  ->  (
( ( C  +P.  A )  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) )
1211exp3a 426 . . . . . . . 8  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  e.  P.  ->  ( x  e.  P.  ->  ( C  +P.  A ) 
<P  ( C  +P.  B
) ) ) )
135, 12syl5 30 . . . . . . 7  |-  ( ( A  +P.  x )  =  B  ->  (
( C  e.  P.  /\  A  e.  P. )  ->  ( x  e.  P.  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) ) )
1413com3r 75 . . . . . 6  |-  ( x  e.  P.  ->  (
( A  +P.  x
)  =  B  -> 
( ( C  e. 
P.  /\  A  e.  P. )  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) ) )
1514rexlimiv 2816 . . . . 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 637 . . 3  |-  ( A 
<P  B  ->  ( ( C  e.  P.  /\  A  <P  B )  -> 
( C  +P.  A
)  <P  ( C  +P.  B ) ) )
1817exp3a 426 . 2  |-  ( A 
<P  B  ->  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) ) )
1918pm2.43b 48 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 359    = wceq 1652    e. wcel 1725   E.wrex 2698   class class class wbr 4204  (class class class)co 6072   P.cnp 8723    +P. cpp 8725    <P cltp 8727
This theorem is referenced by:  ltapr  8911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-omul 6720  df-er 6896  df-ni 8738  df-pli 8739  df-mi 8740  df-lti 8741  df-plpq 8774  df-mpq 8775  df-ltpq 8776  df-enq 8777  df-nq 8778  df-erq 8779  df-plq 8780  df-mq 8781  df-1nq 8782  df-rq 8783  df-ltnq 8784  df-np 8847  df-plp 8849  df-ltp 8851
  Copyright terms: Public domain W3C validator