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Theorem addlocprlem 7307
Description: Lemma for addlocpr 7308. 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 7137 . . . . . 6  |-  <Q  C_  ( Q.  X.  Q. )
32brel 4559 . . . . 5  |-  ( Q 
<Q  R  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )
43simpld 111 . . . 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 7247 . . . . . 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 7253 . . . . 5  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  D  e.  ( 1st `  A ) )  ->  D  e.  Q. )
118, 9, 10syl2anc 406 . . . 4  |-  ( ph  ->  D  e.  Q. )
12 addlocprlem.b . . . . . 6  |-  ( ph  ->  B  e.  P. )
13 prop 7247 . . . . . 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 7253 . . . . 5  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  E  e.  ( 1st `  B ) )  ->  E  e.  Q. )
1714, 15, 16syl2anc 406 . . . 4  |-  ( ph  ->  E  e.  Q. )
18 addclnq 7147 . . . 4  |-  ( ( D  e.  Q.  /\  E  e.  Q. )  ->  ( D  +Q  E
)  e.  Q. )
1911, 17, 18syl2anc 406 . . 3  |-  ( ph  ->  ( D  +Q  E
)  e.  Q. )
20 nqtri3or 7168 . . 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 406 . 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 7303 . . . 4  |-  ( ph  ->  ( Q  <Q  ( D  +Q  E )  ->  Q  e.  ( 1st `  ( A  +P.  B
) ) ) )
29 orc 684 . . . 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 7305 . . . 4  |-  ( ph  ->  ( Q  =  ( D  +Q  E )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
32 olc 683 . . . 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 7306 . . . 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 1265 . 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 680    \/ w3o 944    = 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:  addlocpr  7308
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