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Theorem ltaprlem 8601
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
StepHypRef Expression
1 ltrelpr 8555 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 4690 . . . . 5  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simpld 447 . . . 4  |-  ( A 
<P  B  ->  A  e. 
P. )
4 ltexpri 8600 . . . . 5  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
5 addclpr 8575 . . . . . . . 8  |-  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  e.  P. )
6 ltaddpr 8591 . . . . . . . . . 10  |-  ( ( ( C  +P.  A
)  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x ) )
7 addasspr 8579 . . . . . . . . . . . 12  |-  ( ( C  +P.  A )  +P.  x )  =  ( C  +P.  ( A  +P.  x ) )
8 oveq2 5765 . . . . . . . . . . . 12  |-  ( ( A  +P.  x )  =  B  ->  ( C  +P.  ( A  +P.  x ) )  =  ( C  +P.  B
) )
97, 8syl5eq 2300 . . . . . . . . . . 11  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  +P.  x )  =  ( C  +P.  B ) )
109breq2d 3975 . . . . . . . . . 10  |-  ( ( A  +P.  x )  =  B  ->  (
( C  +P.  A
)  <P  ( ( C  +P.  A )  +P.  x )  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
116, 10syl5ib 212 . . . . . . . . 9  |-  ( ( A  +P.  x )  =  B  ->  (
( ( C  +P.  A )  e.  P.  /\  x  e.  P. )  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) )
1211exp3a 427 . . . . . . . 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 2632 . . . . 5  |-  ( E. x  e.  P.  ( A  +P.  x )  =  B  ->  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A )  <P 
( C  +P.  B
) ) )
164, 15syl 17 . . . 4  |-  ( A 
<P  B  ->  ( ( C  e.  P.  /\  A  e.  P. )  ->  ( C  +P.  A
)  <P  ( C  +P.  B ) ) )
173, 16sylan2i 639 . . 3  |-  ( A 
<P  B  ->  ( ( C  e.  P.  /\  A  <P  B )  -> 
( C  +P.  A
)  <P  ( C  +P.  B ) ) )
1817exp3a 427 . 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517   class class class wbr 3963  (class class class)co 5757   P.cnp 8414    +P. cpp 8416    <P cltp 8418
This theorem is referenced by:  ltapr  8602
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-omul 6417  df-er 6593  df-ni 8429  df-pli 8430  df-mi 8431  df-lti 8432  df-plpq 8465  df-mpq 8466  df-ltpq 8467  df-enq 8468  df-nq 8469  df-erq 8470  df-plq 8471  df-mq 8472  df-1nq 8473  df-rq 8474  df-ltnq 8475  df-np 8538  df-plp 8540  df-ltp 8542
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