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Theorem ltaprlem 7440
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 7435 . . . 4  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
21adantr 274 . . 3  |-  ( ( A  <P  B  /\  C  e.  P. )  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
3 simplr 519 . . . . . 6  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  C  e.  P. )
4 ltrelpr 7327 . . . . . . . . . 10  |-  <P  C_  ( P.  X.  P. )
54brel 4591 . . . . . . . . 9  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
65simpld 111 . . . . . . . 8  |-  ( A 
<P  B  ->  A  e. 
P. )
76adantr 274 . . . . . . 7  |-  ( ( A  <P  B  /\  C  e.  P. )  ->  A  e.  P. )
87adantr 274 . . . . . 6  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  A  e.  P. )
9 addclpr 7359 . . . . . 6  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
103, 8, 9syl2anc 408 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  A )  e.  P. )
11 simprl 520 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  x  e.  P. )
12 ltaddpr 7419 . . . . 5  |-  ( ( ( C  +P.  A
)  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x ) )
1310, 11, 12syl2anc 408 . . . 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 7401 . . . . . 6  |-  ( ( C  e.  P.  /\  A  e.  P.  /\  x  e.  P. )  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
153, 8, 11, 14syl3anc 1216 . . . . 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 5782 . . . . . 6  |-  ( ( A  +P.  x )  =  B  ->  ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B
) )
1716ad2antll 482 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  ( A  +P.  x
) )  =  ( C  +P.  B ) )
1815, 17eqtrd 2172 . . . 4  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) )
1913, 18breqtrd 3954 . . 3  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  A )  <P  ( C  +P.  B ) )
202, 19rexlimddv 2554 . 2  |-  ( ( A  <P  B  /\  C  e.  P. )  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) )
2120expcom 115 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 103    = wceq 1331    e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   P.cnp 7113    +P. cpp 7115    <P cltp 7117
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7126  df-pli 7127  df-mi 7128  df-lti 7129  df-plpq 7166  df-mpq 7167  df-enq 7169  df-nqqs 7170  df-plqqs 7171  df-mqqs 7172  df-1nqqs 7173  df-rq 7174  df-ltnqqs 7175  df-enq0 7246  df-nq0 7247  df-0nq0 7248  df-plq0 7249  df-mq0 7250  df-inp 7288  df-iplp 7290  df-iltp 7292
This theorem is referenced by:  ltaprg  7441
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