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Theorem addlocprlem 6787
Description: Lemma for addlocpr 6788. The result, in deduction form. (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
addlocprlem  |-  ( ph  ->  ( Q  e.  ( 1st `  ( A  +P.  B ) )  \/  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )

Proof of Theorem addlocprlem
StepHypRef Expression
1 addlocprlem.qr . . . 4  |-  ( ph  ->  Q  <Q  R )
2 ltrelnq 6617 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4418 . . . . 5  |-  ( Q 
<Q  R  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )
43simpld 110 . . . 4  |-  ( Q 
<Q  R  ->  Q  e. 
Q. )
51, 4syl 14 . . 3  |-  ( ph  ->  Q  e.  Q. )
6 addlocprlem.a . . . . . 6  |-  ( ph  ->  A  e.  P. )
7 prop 6727 . . . . . 6  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
86, 7syl 14 . . . . 5  |-  ( ph  -> 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P. )
9 addlocprlem.dlo . . . . 5  |-  ( ph  ->  D  e.  ( 1st `  A ) )
10 elprnql 6733 . . . . 5  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  D  e.  ( 1st `  A ) )  ->  D  e.  Q. )
118, 9, 10syl2anc 403 . . . 4  |-  ( ph  ->  D  e.  Q. )
12 addlocprlem.b . . . . . 6  |-  ( ph  ->  B  e.  P. )
13 prop 6727 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
1412, 13syl 14 . . . . 5  |-  ( ph  -> 
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P. )
15 addlocprlem.elo . . . . 5  |-  ( ph  ->  E  e.  ( 1st `  B ) )
16 elprnql 6733 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  E  e.  ( 1st `  B ) )  ->  E  e.  Q. )
1714, 15, 16syl2anc 403 . . . 4  |-  ( ph  ->  E  e.  Q. )
18 addclnq 6627 . . . 4  |-  ( ( D  e.  Q.  /\  E  e.  Q. )  ->  ( D  +Q  E
)  e.  Q. )
1911, 17, 18syl2anc 403 . . 3  |-  ( ph  ->  ( D  +Q  E
)  e.  Q. )
20 nqtri3or 6648 . . 3  |-  ( ( Q  e.  Q.  /\  ( D  +Q  E
)  e.  Q. )  ->  ( Q  <Q  ( D  +Q  E )  \/  Q  =  ( D  +Q  E )  \/  ( D  +Q  E
)  <Q  Q ) )
215, 19, 20syl2anc 403 . 2  |-  ( ph  ->  ( Q  <Q  ( D  +Q  E )  \/  Q  =  ( D  +Q  E )  \/  ( D  +Q  E
)  <Q  Q ) )
22 addlocprlem.p . . . . 5  |-  ( ph  ->  P  e.  Q. )
23 addlocprlem.qppr . . . . 5  |-  ( ph  ->  ( Q  +Q  ( P  +Q  P ) )  =  R )
24 addlocprlem.uup . . . . 5  |-  ( ph  ->  U  e.  ( 2nd `  A ) )
25 addlocprlem.du . . . . 5  |-  ( ph  ->  U  <Q  ( D  +Q  P ) )
26 addlocprlem.tup . . . . 5  |-  ( ph  ->  T  e.  ( 2nd `  B ) )
27 addlocprlem.et . . . . 5  |-  ( ph  ->  T  <Q  ( E  +Q  P ) )
286, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27addlocprlemlt 6783 . . . 4  |-  ( ph  ->  ( Q  <Q  ( D  +Q  E )  ->  Q  e.  ( 1st `  ( A  +P.  B
) ) ) )
29 orc 666 . . . 4  |-  ( Q  e.  ( 1st `  ( A  +P.  B ) )  ->  ( Q  e.  ( 1st `  ( A  +P.  B ) )  \/  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
3028, 29syl6 33 . . 3  |-  ( ph  ->  ( Q  <Q  ( D  +Q  E )  -> 
( Q  e.  ( 1st `  ( A  +P.  B ) )  \/  R  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
316, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27addlocprlemeq 6785 . . . 4  |-  ( ph  ->  ( Q  =  ( D  +Q  E )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
32 olc 665 . . . 4  |-  ( R  e.  ( 2nd `  ( A  +P.  B ) )  ->  ( Q  e.  ( 1st `  ( A  +P.  B ) )  \/  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
3331, 32syl6 33 . . 3  |-  ( ph  ->  ( Q  =  ( D  +Q  E )  ->  ( Q  e.  ( 1st `  ( A  +P.  B ) )  \/  R  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
346, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27addlocprlemgt 6786 . . . 4  |-  ( ph  ->  ( ( D  +Q  E )  <Q  Q  ->  R  e.  ( 2nd `  ( A  +P.  B
) ) ) )
3534, 32syl6 33 . . 3  |-  ( ph  ->  ( ( D  +Q  E )  <Q  Q  -> 
( Q  e.  ( 1st `  ( A  +P.  B ) )  \/  R  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
3630, 33, 353jaod 1236 . 2  |-  ( ph  ->  ( ( Q  <Q  ( D  +Q  E )  \/  Q  =  ( D  +Q  E )  \/  ( D  +Q  E )  <Q  Q )  ->  ( Q  e.  ( 1st `  ( A  +P.  B ) )  \/  R  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
3721, 36mpd 13 1  |-  ( ph  ->  ( Q  e.  ( 1st `  ( A  +P.  B ) )  \/  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    \/ w3o 919    = wceq 1285    e. wcel 1434   <.cop 3409   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532    +Q cplq 6534    <Q cltq 6537   P.cnp 6543    +P. cpp 6545
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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-inp 6718  df-iplp 6720
This theorem is referenced by:  addlocpr  6788
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