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Theorem ltaprlem 7567
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 7562 . . . 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 525 . . . . . 6  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  C  e.  P. )
4 ltrelpr 7454 . . . . . . . . . 10  |-  <P  C_  ( P.  X.  P. )
54brel 4661 . . . . . . . . 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 7486 . . . . . 6  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
103, 8, 9syl2anc 409 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  A )  e.  P. )
11 simprl 526 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  x  e.  P. )
12 ltaddpr 7546 . . . . 5  |-  ( ( ( C  +P.  A
)  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x ) )
1310, 11, 12syl2anc 409 . . . 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 7528 . . . . . 6  |-  ( ( C  e.  P.  /\  A  e.  P.  /\  x  e.  P. )  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  ( A  +P.  x ) ) )
153, 8, 11, 14syl3anc 1233 . . . . 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 5858 . . . . . 6  |-  ( ( A  +P.  x )  =  B  ->  ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B
) )
1716ad2antll 488 . . . . 5  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  ( A  +P.  x
) )  =  ( C  +P.  B ) )
1815, 17eqtrd 2203 . . . 4  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  B ) )
1913, 18breqtrd 4013 . . 3  |-  ( ( ( A  <P  B  /\  C  e.  P. )  /\  ( x  e.  P.  /\  ( A  +P.  x
)  =  B ) )  ->  ( C  +P.  A )  <P  ( C  +P.  B ) )
202, 19rexlimddv 2592 . 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 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3987  (class class class)co 5850   P.cnp 7240    +P. cpp 7242    <P cltp 7244
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-2o 6393  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-enq0 7373  df-nq0 7374  df-0nq0 7375  df-plq0 7376  df-mq0 7377  df-inp 7415  df-iplp 7417  df-iltp 7419
This theorem is referenced by:  ltaprg  7568
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