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Theorem addlocprlemeqgt 7359
Description: Lemma for addlocpr 7363. 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 7302 . . . . . 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 7309 . . . . 5  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  U  e.  ( 2nd `  A ) )  ->  U  e.  Q. )
85, 6, 7syl2anc 408 . . . 4  |-  ( ph  ->  U  e.  Q. )
9 addlocprlem.dlo . . . . . 6  |-  ( ph  ->  D  e.  ( 1st `  A ) )
10 elprnql 7308 . . . . . 6  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  D  e.  ( 1st `  A ) )  ->  D  e.  Q. )
115, 9, 10syl2anc 408 . . . . 5  |-  ( ph  ->  D  e.  Q. )
12 addlocprlem.p . . . . 5  |-  ( ph  ->  P  e.  Q. )
13 addclnq 7202 . . . . 5  |-  ( ( D  e.  Q.  /\  P  e.  Q. )  ->  ( D  +Q  P
)  e.  Q. )
1411, 12, 13syl2anc 408 . . . 4  |-  ( ph  ->  ( D  +Q  P
)  e.  Q. )
15 addlocprlem.b . . . . . 6  |-  ( ph  ->  B  e.  P. )
16 prop 7302 . . . . . 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 7309 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  T  e.  ( 2nd `  B ) )  ->  T  e.  Q. )
2017, 18, 19syl2anc 408 . . . 4  |-  ( ph  ->  T  e.  Q. )
21 addlocprlem.elo . . . . . 6  |-  ( ph  ->  E  e.  ( 1st `  B ) )
22 elprnql 7308 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  E  e.  ( 1st `  B ) )  ->  E  e.  Q. )
2317, 21, 22syl2anc 408 . . . . 5  |-  ( ph  ->  E  e.  Q. )
24 addclnq 7202 . . . . 5  |-  ( ( E  e.  Q.  /\  P  e.  Q. )  ->  ( E  +Q  P
)  e.  Q. )
2523, 12, 24syl2anc 408 . . . 4  |-  ( ph  ->  ( E  +Q  P
)  e.  Q. )
26 lt2addnq 7231 . . . 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 1217 . . 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 429 . 2  |-  ( ph  ->  ( U  +Q  T
)  <Q  ( ( D  +Q  P )  +Q  ( E  +Q  P
) ) )
29 addcomnqg 7208 . . . 4  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3029adantl 275 . . 3  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q. ) )  -> 
( f  +Q  g
)  =  ( g  +Q  f ) )
31 addassnqg 7209 . . . 4  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
3231adantl 275 . . 3  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q.  /\  h  e.  Q. ) )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
33 addclnq 7202 . . . 4  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  e.  Q. )
3433adantl 275 . . 3  |-  ( (
ph  /\  ( f  e.  Q.  /\  g  e. 
Q. ) )  -> 
( f  +Q  g
)  e.  Q. )
3511, 12, 23, 30, 32, 12, 34caov4d 5958 . 2  |-  ( ph  ->  ( ( D  +Q  P )  +Q  ( E  +Q  P ) )  =  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
3628, 35breqtrd 3957 1  |-  ( ph  ->  ( U  +Q  T
)  <Q  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480   <.cop 3530   class class class wbr 3932   ` cfv 5126  (class class class)co 5777   1stc1st 6039   2ndc2nd 6040   Q.cnq 7107    +Q cplq 7109    <Q cltq 7112   P.cnp 7118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-eprel 4214  df-id 4218  df-po 4221  df-iso 4222  df-iord 4291  df-on 4293  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-irdg 6270  df-oadd 6320  df-omul 6321  df-er 6432  df-ec 6434  df-qs 6438  df-ni 7131  df-pli 7132  df-mi 7133  df-lti 7134  df-plpq 7171  df-enq 7174  df-nqqs 7175  df-plqqs 7176  df-ltnqqs 7180  df-inp 7293
This theorem is referenced by:  addlocprlemeq  7360  addlocprlemgt  7361
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