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Theorem ltaprlem 9418
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 9372 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 5047 . . . . 5  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simpld 459 . . . 4  |-  ( A 
<P  B  ->  A  e. 
P. )
4 ltexpri 9417 . . . . 5  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
5 addclpr 9392 . . . . . . . 8  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
6 ltaddpr 9408 . . . . . . . . . 10  |-  ( ( ( C  +P.  A
)  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x ) )
7 addasspr 9396 . . . . . . . . . . . 12  |-  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) )
8 oveq2 6290 . . . . . . . . . . . 12  |-  ( ( A  +P.  x )  =  B  ->  ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B
) )
97, 8syl5eq 2520 . . . . . . . . . . 11  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  B ) )
109breq2d 4459 . . . . . . . . . 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 2949 . . . . 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 1379    e. wcel 1767   E.wrex 2815   class class class wbr 4447  (class class class)co 6282   P.cnp 9233    +P. cpp 9235    <P cltp 9237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ni 9246  df-pli 9247  df-mi 9248  df-lti 9249  df-plpq 9282  df-mpq 9283  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-plq 9288  df-mq 9289  df-1nq 9290  df-rq 9291  df-ltnq 9292  df-np 9355  df-plp 9357  df-ltp 9359
This theorem is referenced by:  ltapr  9419
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