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Theorem addlocprlemeq 7545
Description: Lemma for addlocpr 7548. 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 7544 . . . . 5  |-  ( ph  ->  ( U  +Q  T
)  <Q  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
1312adantr 276 . . . 4  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  ( U  +Q  T )  <Q 
( ( D  +Q  E )  +Q  ( P  +Q  P ) ) )
14 oveq1 5895 . . . . 5  |-  ( Q  =  ( D  +Q  E )  ->  ( Q  +Q  ( P  +Q  P ) )  =  ( ( D  +Q  E )  +Q  ( P  +Q  P ) ) )
155, 14sylan9req 2241 . . . 4  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  R  =  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
1613, 15breqtrrd 4043 . . 3  |-  ( (
ph  /\  Q  =  ( D  +Q  E
) )  ->  ( U  +Q  T )  <Q  R )
171, 7jca 306 . . . . 5  |-  ( ph  ->  ( A  e.  P.  /\  U  e.  ( 2nd `  A ) ) )
182, 10jca 306 . . . . 5  |-  ( ph  ->  ( B  e.  P.  /\  T  e.  ( 2nd `  B ) ) )
19 ltrelnq 7377 . . . . . . . 8  |-  <Q  C_  ( Q.  X.  Q. )
2019brel 4690 . . . . . . 7  |-  ( Q 
<Q  R  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )
2120simprd 114 . . . . . 6  |-  ( Q 
<Q  R  ->  R  e. 
Q. )
223, 21syl 14 . . . . 5  |-  ( ph  ->  R  e.  Q. )
23 addnqpru 7542 . . . . 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 1247 . . . 4  |-  ( ph  ->  ( ( U  +Q  T )  <Q  R  ->  R  e.  ( 2nd `  ( A  +P.  B
) ) ) )
2524adantr 276 . . 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 115 1  |-  ( ph  ->  ( Q  =  ( D  +Q  E )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   1stc1st 6152   2ndc2nd 6153   Q.cnq 7292    +Q cplq 7294    <Q cltq 7297   P.cnp 7303    +P. cpp 7305
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-inp 7478  df-iplp 7480
This theorem is referenced by:  addlocprlem  7547
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