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Theorem addlocprlemlt 7560
Description: Lemma for addlocpr 7565. The  Q  <Q  ( D  +Q  E ) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
Hypotheses
Ref Expression
addlocprlem.a  |-  ( ph  ->  A  e.  P. )
addlocprlem.b  |-  ( ph  ->  B  e.  P. )
addlocprlem.qr  |-  ( ph  ->  Q  <Q  R )
addlocprlem.p  |-  ( ph  ->  P  e.  Q. )
addlocprlem.qppr  |-  ( ph  ->  ( Q  +Q  ( P  +Q  P ) )  =  R )
addlocprlem.dlo  |-  ( ph  ->  D  e.  ( 1st `  A ) )
addlocprlem.uup  |-  ( ph  ->  U  e.  ( 2nd `  A ) )
addlocprlem.du  |-  ( ph  ->  U  <Q  ( D  +Q  P ) )
addlocprlem.elo  |-  ( ph  ->  E  e.  ( 1st `  B ) )
addlocprlem.tup  |-  ( ph  ->  T  e.  ( 2nd `  B ) )
addlocprlem.et  |-  ( ph  ->  T  <Q  ( E  +Q  P ) )
Assertion
Ref Expression
addlocprlemlt  |-  ( ph  ->  ( Q  <Q  ( D  +Q  E )  ->  Q  e.  ( 1st `  ( A  +P.  B
) ) ) )

Proof of Theorem addlocprlemlt
StepHypRef Expression
1 addlocprlem.a . . 3  |-  ( ph  ->  A  e.  P. )
2 addlocprlem.dlo . . 3  |-  ( ph  ->  D  e.  ( 1st `  A ) )
31, 2jca 306 . 2  |-  ( ph  ->  ( A  e.  P.  /\  D  e.  ( 1st `  A ) ) )
4 addlocprlem.b . . 3  |-  ( ph  ->  B  e.  P. )
5 addlocprlem.elo . . 3  |-  ( ph  ->  E  e.  ( 1st `  B ) )
64, 5jca 306 . 2  |-  ( ph  ->  ( B  e.  P.  /\  E  e.  ( 1st `  B ) ) )
7 addlocprlem.qr . . 3  |-  ( ph  ->  Q  <Q  R )
8 ltrelnq 7394 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
98brel 4696 . . . 4  |-  ( Q 
<Q  R  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )
109simpld 112 . . 3  |-  ( Q 
<Q  R  ->  Q  e. 
Q. )
117, 10syl 14 . 2  |-  ( ph  ->  Q  e.  Q. )
12 addnqprl 7558 . 2  |-  ( ( ( ( A  e. 
P.  /\  D  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  E  e.  ( 1st `  B
) ) )  /\  Q  e.  Q. )  ->  ( Q  <Q  ( D  +Q  E )  ->  Q  e.  ( 1st `  ( A  +P.  B
) ) ) )
133, 6, 11, 12syl21anc 1248 1  |-  ( ph  ->  ( Q  <Q  ( D  +Q  E )  ->  Q  e.  ( 1st `  ( A  +P.  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   class class class wbr 4018   ` cfv 5235  (class class class)co 5896   1stc1st 6163   2ndc2nd 6164   Q.cnq 7309    +Q cplq 7311    <Q cltq 7314   P.cnp 7320    +P. cpp 7322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-rq 7381  df-ltnqqs 7382  df-inp 7495  df-iplp 7497
This theorem is referenced by:  addlocprlem  7564
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