ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltaprlem Unicode version

Theorem ltaprlem 6870
Description: Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)
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 ltexpri 6865 . . . 4  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
21adantr 270 . . 3  |-  ( ( A  <P  B  /\  C  e.  P. )  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
3 simplr 497 . . . . . 6  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  C  e.  P. )
4 ltrelpr 6757 . . . . . . . . . 10  |-  <P  C_  ( P.  X.  P. )
54brel 4418 . . . . . . . . 9  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
65simpld 110 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
76adantr 270 . . . . . . 7  |-  ( ( A  <P  B  /\  C  e.  P. )  ->  A  e.  P. )
87adantr 270 . . . . . 6  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  A  e.  P. )
9 addclpr 6789 . . . . . 6  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
103, 8, 9syl2anc 403 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  A )  e.  P. )
11 simprl 498 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  x  e.  P. )
12 ltaddpr 6849 . . . . 5  |-  ( ( ( C  +P.  A
)  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x ) )
1310, 11, 12syl2anc 403 . . . 4  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  A )  <P  (
( C  +P.  A
)  +P.  x )
)
14 addassprg 6831 . . . . . 6  |-  ( ( C  e.  P.  /\  A  e.  P.  /\  x  e.  P. )  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
153, 8, 11, 14syl3anc 1170 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
16 oveq2 5551 . . . . . 6  |-  ( ( A  +P.  x )  =  B  ->  ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B
) )
1716ad2antll 475 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  ( A  +P.  x
) )  =  ( C  +P.  B ) )
1815, 17eqtrd 2114 . . . 4  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) )
1913, 18breqtrd 3817 . . 3  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  A )  <P  ( C  +P.  B ) )
202, 19rexlimddv 2482 . 2  |-  ( ( A  <P  B  /\  C  e.  P. )  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) )
2120expcom 114 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 102    = wceq 1285    e. wcel 1434   E.wrex 2350   class class class wbr 3793  (class class class)co 5543   P.cnp 6543    +P. cpp 6545    <P cltp 6547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  ltaprg  6871
  Copyright terms: Public domain W3C validator