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Theorem addlocprlemeqgt 6854
Description: Lemma for addlocpr 6858. This is a step used in both the  Q  =  ( D  +Q  E ) and  ( D  +Q  E
)  <Q  Q cases. (Contributed by Jim Kingdon, 7-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
addlocprlemeqgt  |-  ( ph  ->  ( U  +Q  T
)  <Q  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )

Proof of Theorem addlocprlemeqgt
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addlocprlem.du . . 3  |-  ( ph  ->  U  <Q  ( D  +Q  P ) )
2 addlocprlem.et . . 3  |-  ( ph  ->  T  <Q  ( E  +Q  P ) )
3 addlocprlem.a . . . . . 6  |-  ( ph  ->  A  e.  P. )
4 prop 6797 . . . . . 6  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
53, 4syl 14 . . . . 5  |-  ( ph  -> 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P. )
6 addlocprlem.uup . . . . 5  |-  ( ph  ->  U  e.  ( 2nd `  A ) )
7 elprnqu 6804 . . . . 5  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  U  e.  ( 2nd `  A ) )  ->  U  e.  Q. )
85, 6, 7syl2anc 403 . . . 4  |-  ( ph  ->  U  e.  Q. )
9 addlocprlem.dlo . . . . . 6  |-  ( ph  ->  D  e.  ( 1st `  A ) )
10 elprnql 6803 . . . . . 6  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  D  e.  ( 1st `  A ) )  ->  D  e.  Q. )
115, 9, 10syl2anc 403 . . . . 5  |-  ( ph  ->  D  e.  Q. )
12 addlocprlem.p . . . . 5  |-  ( ph  ->  P  e.  Q. )
13 addclnq 6697 . . . . 5  |-  ( ( D  e.  Q.  /\  P  e.  Q. )  ->  ( D  +Q  P
)  e.  Q. )
1411, 12, 13syl2anc 403 . . . 4  |-  ( ph  ->  ( D  +Q  P
)  e.  Q. )
15 addlocprlem.b . . . . . 6  |-  ( ph  ->  B  e.  P. )
16 prop 6797 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
1715, 16syl 14 . . . . 5  |-  ( ph  -> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P. )
18 addlocprlem.tup . . . . 5  |-  ( ph  ->  T  e.  ( 2nd `  B ) )
19 elprnqu 6804 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  T  e.  ( 2nd `  B ) )  ->  T  e.  Q. )
2017, 18, 19syl2anc 403 . . . 4  |-  ( ph  ->  T  e.  Q. )
21 addlocprlem.elo . . . . . 6  |-  ( ph  ->  E  e.  ( 1st `  B ) )
22 elprnql 6803 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  E  e.  ( 1st `  B ) )  ->  E  e.  Q. )
2317, 21, 22syl2anc 403 . . . . 5  |-  ( ph  ->  E  e.  Q. )
24 addclnq 6697 . . . . 5  |-  ( ( E  e.  Q.  /\  P  e.  Q. )  ->  ( E  +Q  P
)  e.  Q. )
2523, 12, 24syl2anc 403 . . . 4  |-  ( ph  ->  ( E  +Q  P
)  e.  Q. )
26 lt2addnq 6726 . . . 4  |-  ( ( ( U  e.  Q.  /\  ( D  +Q  P
)  e.  Q. )  /\  ( T  e.  Q.  /\  ( E  +Q  P
)  e.  Q. )
)  ->  ( ( U  <Q  ( D  +Q  P )  /\  T  <Q  ( E  +Q  P
) )  ->  ( U  +Q  T )  <Q 
( ( D  +Q  P )  +Q  ( E  +Q  P ) ) ) )
278, 14, 20, 25, 26syl22anc 1171 . . 3  |-  ( ph  ->  ( ( U  <Q  ( D  +Q  P )  /\  T  <Q  ( E  +Q  P ) )  ->  ( U  +Q  T )  <Q  (
( D  +Q  P
)  +Q  ( E  +Q  P ) ) ) )
281, 2, 27mp2and 424 . 2  |-  ( ph  ->  ( U  +Q  T
)  <Q  ( ( D  +Q  P )  +Q  ( E  +Q  P
) ) )
29 addcomnqg 6703 . . . 4  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3029adantl 271 . . 3  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q. ) )  -> 
( f  +Q  g
)  =  ( g  +Q  f ) )
31 addassnqg 6704 . . . 4  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
3231adantl 271 . . 3  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q.  /\  h  e.  Q. ) )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
33 addclnq 6697 . . . 4  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  e.  Q. )
3433adantl 271 . . 3  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q. ) )  -> 
( f  +Q  g
)  e.  Q. )
3511, 12, 23, 30, 32, 12, 34caov4d 5737 . 2  |-  ( ph  ->  ( ( D  +Q  P )  +Q  ( E  +Q  P ) )  =  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
3628, 35breqtrd 3829 1  |-  ( ph  ->  ( U  +Q  T
)  <Q  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434   <.cop 3419   class class class wbr 3805   ` cfv 4952  (class class class)co 5564   1stc1st 5817   2ndc2nd 5818   Q.cnq 6602    +Q cplq 6604    <Q cltq 6607   P.cnp 6613
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 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6626  df-pli 6627  df-mi 6628  df-lti 6629  df-plpq 6666  df-enq 6669  df-nqqs 6670  df-plqqs 6671  df-ltnqqs 6675  df-inp 6788
This theorem is referenced by:  addlocprlemeq  6855  addlocprlemgt  6856
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