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Theorem addlocprlemeq 6837
Description: Lemma for addlocpr 6840. The  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
addlocprlemeq  |-  ( ph  ->  ( Q  =  ( D  +Q  E )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )

Proof of Theorem addlocprlemeq
StepHypRef Expression
1 addlocprlem.a . . . . . 6  |-  ( ph  ->  A  e.  P. )
2 addlocprlem.b . . . . . 6  |-  ( ph  ->  B  e.  P. )
3 addlocprlem.qr . . . . . 6  |-  ( ph  ->  Q  <Q  R )
4 addlocprlem.p . . . . . 6  |-  ( ph  ->  P  e.  Q. )
5 addlocprlem.qppr . . . . . 6  |-  ( ph  ->  ( Q  +Q  ( P  +Q  P ) )  =  R )
6 addlocprlem.dlo . . . . . 6  |-  ( ph  ->  D  e.  ( 1st `  A ) )
7 addlocprlem.uup . . . . . 6  |-  ( ph  ->  U  e.  ( 2nd `  A ) )
8 addlocprlem.du . . . . . 6  |-  ( ph  ->  U  <Q  ( D  +Q  P ) )
9 addlocprlem.elo . . . . . 6  |-  ( ph  ->  E  e.  ( 1st `  B ) )
10 addlocprlem.tup . . . . . 6  |-  ( ph  ->  T  e.  ( 2nd `  B ) )
11 addlocprlem.et . . . . . 6  |-  ( ph  ->  T  <Q  ( E  +Q  P ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11addlocprlemeqgt 6836 . . . . 5  |-  ( ph  ->  ( U  +Q  T
)  <Q  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
1312adantr 270 . . . 4  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  ( U  +Q  T )  <Q 
( ( D  +Q  E )  +Q  ( P  +Q  P ) ) )
14 oveq1 5570 . . . . 5  |-  ( Q  =  ( D  +Q  E )  ->  ( Q  +Q  ( P  +Q  P ) )  =  ( ( D  +Q  E )  +Q  ( P  +Q  P ) ) )
155, 14sylan9req 2136 . . . 4  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  R  =  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
1613, 15breqtrrd 3831 . . 3  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  ( U  +Q  T )  <Q  R )
171, 7jca 300 . . . . 5  |-  ( ph  ->  ( A  e.  P.  /\  U  e.  ( 2nd `  A ) ) )
182, 10jca 300 . . . . 5  |-  ( ph  ->  ( B  e.  P.  /\  T  e.  ( 2nd `  B ) ) )
19 ltrelnq 6669 . . . . . . . 8  |-  <Q  C_  ( Q.  X.  Q. )
2019brel 4438 . . . . . . 7  |-  ( Q 
<Q  R  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )
2120simprd 112 . . . . . 6  |-  ( Q 
<Q  R  ->  R  e. 
Q. )
223, 21syl 14 . . . . 5  |-  ( ph  ->  R  e.  Q. )
23 addnqpru 6834 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  U  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  T  e.  ( 2nd `  B
) ) )  /\  R  e.  Q. )  ->  ( ( U  +Q  T )  <Q  R  ->  R  e.  ( 2nd `  ( A  +P.  B
) ) ) )
2417, 18, 22, 23syl21anc 1169 . . . 4  |-  ( ph  ->  ( ( U  +Q  T )  <Q  R  ->  R  e.  ( 2nd `  ( A  +P.  B
) ) ) )
2524adantr 270 . . 3  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  (
( U  +Q  T
)  <Q  R  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
2616, 25mpd 13 . 2  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) )
2726ex 113 1  |-  ( ph  ->  ( Q  =  ( D  +Q  E )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   1stc1st 5816   2ndc2nd 5817   Q.cnq 6584    +Q cplq 6586    <Q cltq 6589   P.cnp 6595    +P. cpp 6597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-mpq 6649  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-mqqs 6654  df-1nqqs 6655  df-rq 6656  df-ltnqqs 6657  df-inp 6770  df-iplp 6772
This theorem is referenced by:  addlocprlem  6839
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