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Theorem addlocprlemeq 7153
Description: Lemma for addlocpr 7156. 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 7152 . . . . 5  |-  ( ph  ->  ( U  +Q  T
)  <Q  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
1312adantr 271 . . . 4  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  ( U  +Q  T )  <Q 
( ( D  +Q  E )  +Q  ( P  +Q  P ) ) )
14 oveq1 5673 . . . . 5  |-  ( Q  =  ( D  +Q  E )  ->  ( Q  +Q  ( P  +Q  P ) )  =  ( ( D  +Q  E )  +Q  ( P  +Q  P ) ) )
155, 14sylan9req 2142 . . . 4  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  R  =  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
1613, 15breqtrrd 3877 . . 3  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  ( U  +Q  T )  <Q  R )
171, 7jca 301 . . . . 5  |-  ( ph  ->  ( A  e.  P.  /\  U  e.  ( 2nd `  A ) ) )
182, 10jca 301 . . . . 5  |-  ( ph  ->  ( B  e.  P.  /\  T  e.  ( 2nd `  B ) ) )
19 ltrelnq 6985 . . . . . . . 8  |-  <Q  C_  ( Q.  X.  Q. )
2019brel 4503 . . . . . . 7  |-  ( Q 
<Q  R  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )
2120simprd 113 . . . . . 6  |-  ( Q 
<Q  R  ->  R  e. 
Q. )
223, 21syl 14 . . . . 5  |-  ( ph  ->  R  e.  Q. )
23 addnqpru 7150 . . . . 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 1174 . . . 4  |-  ( ph  ->  ( ( U  +Q  T )  <Q  R  ->  R  e.  ( 2nd `  ( A  +P.  B
) ) ) )
2524adantr 271 . . 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 114 1  |-  ( ph  ->  ( Q  =  ( D  +Q  E )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900    +Q cplq 6902    <Q cltq 6905   P.cnp 6911    +P. cpp 6913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-inp 7086  df-iplp 7088
This theorem is referenced by:  addlocprlem  7155
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