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Theorem addlocprlemgt 7306
Description: Lemma for addlocpr 7308. The  ( D  +Q  E
)  <Q  Q 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
addlocprlemgt  |-  ( ph  ->  ( ( D  +Q  E )  <Q  Q  ->  R  e.  ( 2nd `  ( A  +P.  B
) ) ) )

Proof of Theorem addlocprlemgt
StepHypRef Expression
1 addlocprlem.a . . . . . . 7  |-  ( ph  ->  A  e.  P. )
2 addlocprlem.b . . . . . . 7  |-  ( ph  ->  B  e.  P. )
3 addlocprlem.qr . . . . . . 7  |-  ( ph  ->  Q  <Q  R )
4 addlocprlem.p . . . . . . 7  |-  ( ph  ->  P  e.  Q. )
5 addlocprlem.qppr . . . . . . 7  |-  ( ph  ->  ( Q  +Q  ( P  +Q  P ) )  =  R )
6 addlocprlem.dlo . . . . . . 7  |-  ( ph  ->  D  e.  ( 1st `  A ) )
7 addlocprlem.uup . . . . . . 7  |-  ( ph  ->  U  e.  ( 2nd `  A ) )
8 addlocprlem.du . . . . . . 7  |-  ( ph  ->  U  <Q  ( D  +Q  P ) )
9 addlocprlem.elo . . . . . . 7  |-  ( ph  ->  E  e.  ( 1st `  B ) )
10 addlocprlem.tup . . . . . . 7  |-  ( ph  ->  T  e.  ( 2nd `  B ) )
11 addlocprlem.et . . . . . . 7  |-  ( ph  ->  T  <Q  ( E  +Q  P ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11addlocprlemeqgt 7304 . . . . . 6  |-  ( ph  ->  ( U  +Q  T
)  <Q  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
1312adantr 272 . . . . 5  |-  ( (
ph  /\  ( D  +Q  E )  <Q  Q )  ->  ( U  +Q  T )  <Q  (
( D  +Q  E
)  +Q  ( P  +Q  P ) ) )
14 prop 7247 . . . . . . . . . . . 12  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
151, 14syl 14 . . . . . . . . . . 11  |-  ( ph  -> 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P. )
16 elprnql 7253 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  D  e.  ( 1st `  A ) )  ->  D  e.  Q. )
1715, 6, 16syl2anc 406 . . . . . . . . . 10  |-  ( ph  ->  D  e.  Q. )
18 prop 7247 . . . . . . . . . . . 12  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
192, 18syl 14 . . . . . . . . . . 11  |-  ( ph  -> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P. )
20 elprnql 7253 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  E  e.  ( 1st `  B ) )  ->  E  e.  Q. )
2119, 9, 20syl2anc 406 . . . . . . . . . 10  |-  ( ph  ->  E  e.  Q. )
22 addclnq 7147 . . . . . . . . . 10  |-  ( ( D  e.  Q.  /\  E  e.  Q. )  ->  ( D  +Q  E
)  e.  Q. )
2317, 21, 22syl2anc 406 . . . . . . . . 9  |-  ( ph  ->  ( D  +Q  E
)  e.  Q. )
24 ltrelnq 7137 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
2524brel 4559 . . . . . . . . . . 11  |-  ( Q 
<Q  R  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )
263, 25syl 14 . . . . . . . . . 10  |-  ( ph  ->  ( Q  e.  Q.  /\  R  e.  Q. )
)
2726simpld 111 . . . . . . . . 9  |-  ( ph  ->  Q  e.  Q. )
28 addclnq 7147 . . . . . . . . . 10  |-  ( ( P  e.  Q.  /\  P  e.  Q. )  ->  ( P  +Q  P
)  e.  Q. )
294, 4, 28syl2anc 406 . . . . . . . . 9  |-  ( ph  ->  ( P  +Q  P
)  e.  Q. )
30 ltanqg 7172 . . . . . . . . 9  |-  ( ( ( D  +Q  E
)  e.  Q.  /\  Q  e.  Q.  /\  ( P  +Q  P )  e. 
Q. )  ->  (
( D  +Q  E
)  <Q  Q  <->  ( ( P  +Q  P )  +Q  ( D  +Q  E
) )  <Q  (
( P  +Q  P
)  +Q  Q ) ) )
3123, 27, 29, 30syl3anc 1199 . . . . . . . 8  |-  ( ph  ->  ( ( D  +Q  E )  <Q  Q  <->  ( ( P  +Q  P )  +Q  ( D  +Q  E
) )  <Q  (
( P  +Q  P
)  +Q  Q ) ) )
32 addcomnqg 7153 . . . . . . . . . 10  |-  ( ( ( P  +Q  P
)  e.  Q.  /\  ( D  +Q  E
)  e.  Q. )  ->  ( ( P  +Q  P )  +Q  ( D  +Q  E ) )  =  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
3329, 23, 32syl2anc 406 . . . . . . . . 9  |-  ( ph  ->  ( ( P  +Q  P )  +Q  ( D  +Q  E ) )  =  ( ( D  +Q  E )  +Q  ( P  +Q  P
) ) )
34 addcomnqg 7153 . . . . . . . . . 10  |-  ( ( ( P  +Q  P
)  e.  Q.  /\  Q  e.  Q. )  ->  ( ( P  +Q  P )  +Q  Q
)  =  ( Q  +Q  ( P  +Q  P ) ) )
3529, 27, 34syl2anc 406 . . . . . . . . 9  |-  ( ph  ->  ( ( P  +Q  P )  +Q  Q
)  =  ( Q  +Q  ( P  +Q  P ) ) )
3633, 35breq12d 3910 . . . . . . . 8  |-  ( ph  ->  ( ( ( P  +Q  P )  +Q  ( D  +Q  E
) )  <Q  (
( P  +Q  P
)  +Q  Q )  <-> 
( ( D  +Q  E )  +Q  ( P  +Q  P ) ) 
<Q  ( Q  +Q  ( P  +Q  P ) ) ) )
3731, 36bitrd 187 . . . . . . 7  |-  ( ph  ->  ( ( D  +Q  E )  <Q  Q  <->  ( ( D  +Q  E )  +Q  ( P  +Q  P
) )  <Q  ( Q  +Q  ( P  +Q  P ) ) ) )
3837biimpa 292 . . . . . 6  |-  ( (
ph  /\  ( D  +Q  E )  <Q  Q )  ->  ( ( D  +Q  E )  +Q  ( P  +Q  P
) )  <Q  ( Q  +Q  ( P  +Q  P ) ) )
395breq2d 3909 . . . . . . 7  |-  ( ph  ->  ( ( ( D  +Q  E )  +Q  ( P  +Q  P
) )  <Q  ( Q  +Q  ( P  +Q  P ) )  <->  ( ( D  +Q  E )  +Q  ( P  +Q  P
) )  <Q  R ) )
4039adantr 272 . . . . . 6  |-  ( (
ph  /\  ( D  +Q  E )  <Q  Q )  ->  ( ( ( D  +Q  E )  +Q  ( P  +Q  P ) )  <Q 
( Q  +Q  ( P  +Q  P ) )  <-> 
( ( D  +Q  E )  +Q  ( P  +Q  P ) ) 
<Q  R ) )
4138, 40mpbid 146 . . . . 5  |-  ( (
ph  /\  ( D  +Q  E )  <Q  Q )  ->  ( ( D  +Q  E )  +Q  ( P  +Q  P
) )  <Q  R )
4213, 41jca 302 . . . 4  |-  ( (
ph  /\  ( D  +Q  E )  <Q  Q )  ->  ( ( U  +Q  T )  <Q 
( ( D  +Q  E )  +Q  ( P  +Q  P ) )  /\  ( ( D  +Q  E )  +Q  ( P  +Q  P
) )  <Q  R ) )
43 ltsonq 7170 . . . . 5  |-  <Q  Or  Q.
4443, 24sotri 4902 . . . 4  |-  ( ( ( U  +Q  T
)  <Q  ( ( D  +Q  E )  +Q  ( P  +Q  P
) )  /\  (
( D  +Q  E
)  +Q  ( P  +Q  P ) ) 
<Q  R )  ->  ( U  +Q  T )  <Q  R )
4542, 44syl 14 . . 3  |-  ( (
ph  /\  ( D  +Q  E )  <Q  Q )  ->  ( U  +Q  T )  <Q  R )
461, 7jca 302 . . . . 5  |-  ( ph  ->  ( A  e.  P.  /\  U  e.  ( 2nd `  A ) ) )
472, 10jca 302 . . . . 5  |-  ( ph  ->  ( B  e.  P.  /\  T  e.  ( 2nd `  B ) ) )
4826simprd 113 . . . . 5  |-  ( ph  ->  R  e.  Q. )
49 addnqpru 7302 . . . . 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
) ) ) )
5046, 47, 48, 49syl21anc 1198 . . . 4  |-  ( ph  ->  ( ( U  +Q  T )  <Q  R  ->  R  e.  ( 2nd `  ( A  +P.  B
) ) ) )
5150adantr 272 . . 3  |-  ( (
ph  /\  ( D  +Q  E )  <Q  Q )  ->  ( ( U  +Q  T )  <Q  R  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
5245, 51mpd 13 . 2  |-  ( (
ph  /\  ( D  +Q  E )  <Q  Q )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) )
5352ex 114 1  |-  ( ph  ->  ( ( D  +Q  E )  <Q  Q  ->  R  e.  ( 2nd `  ( A  +P.  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   <.cop 3498   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    +Q cplq 7054    <Q cltq 7057   P.cnp 7063    +P. cpp 7065
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-inp 7238  df-iplp 7240
This theorem is referenced by:  addlocprlem  7307
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