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Theorem addlocprlem 7533
Description: Lemma for addlocpr 7534. 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 7363 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4678 . . . . 5  |-  ( Q 
<Q  R  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )
43simpld 112 . . . 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 7473 . . . . . 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 7479 . . . . 5  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  D  e.  ( 1st `  A ) )  ->  D  e.  Q. )
118, 9, 10syl2anc 411 . . . 4  |-  ( ph  ->  D  e.  Q. )
12 addlocprlem.b . . . . . 6  |-  ( ph  ->  B  e.  P. )
13 prop 7473 . . . . . 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 7479 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  E  e.  ( 1st `  B ) )  ->  E  e.  Q. )
1714, 15, 16syl2anc 411 . . . 4  |-  ( ph  ->  E  e.  Q. )
18 addclnq 7373 . . . 4  |-  ( ( D  e.  Q.  /\  E  e.  Q. )  ->  ( D  +Q  E
)  e.  Q. )
1911, 17, 18syl2anc 411 . . 3  |-  ( ph  ->  ( D  +Q  E
)  e.  Q. )
20 nqtri3or 7394 . . 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 411 . 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 7529 . . . 4  |-  ( ph  ->  ( Q  <Q  ( D  +Q  E )  ->  Q  e.  ( 1st `  ( A  +P.  B
) ) ) )
29 orc 712 . . . 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 7531 . . . 4  |-  ( ph  ->  ( Q  =  ( D  +Q  E )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
32 olc 711 . . . 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 7532 . . . 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 1304 . 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 708    \/ w3o 977    = wceq 1353    e. wcel 2148   <.cop 3595   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   1stc1st 6138   2ndc2nd 6139   Q.cnq 7278    +Q cplq 7280    <Q cltq 7283   P.cnp 7289    +P. cpp 7291
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464  df-iplp 7466
This theorem is referenced by:  addlocpr  7534
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