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Theorem caucvgprlemcanl 6896
Description: Lemma for cauappcvgprlemladdrl 6909. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
Hypotheses
Ref Expression
caucvgprlemcanl.l  |-  ( ph  ->  L  e.  P. )
caucvgprlemcanl.s  |-  ( ph  ->  S  e.  Q. )
caucvgprlemcanl.r  |-  ( ph  ->  R  e.  Q. )
caucvgprlemcanl.q  |-  ( ph  ->  Q  e.  Q. )
Assertion
Ref Expression
caucvgprlemcanl  |-  ( ph  ->  ( ( R  +Q  Q )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  Q ) } ,  { u  |  ( S  +Q  Q )  <Q  u } >. ) )  <->  R  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ) )
Distinct variable groups:    Q, l, u    R, l, u    S, l, u
Allowed substitution hints:    ph( u, l)    L( u, l)

Proof of Theorem caucvgprlemcanl
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltaprg 6871 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
21adantl 271 . . 3  |-  ( (
ph  /\  ( f  e.  P.  /\  g  e. 
P.  /\  h  e.  P. ) )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
3 caucvgprlemcanl.r . . . 4  |-  ( ph  ->  R  e.  Q. )
4 nqprlu 6799 . . . 4  |-  ( R  e.  Q.  ->  <. { l  |  l  <Q  R } ,  { u  |  R  <Q  u } >.  e.  P. )
53, 4syl 14 . . 3  |-  ( ph  -> 
<. { l  |  l 
<Q  R } ,  {
u  |  R  <Q  u } >.  e.  P. )
6 caucvgprlemcanl.l . . . 4  |-  ( ph  ->  L  e.  P. )
7 caucvgprlemcanl.s . . . . 5  |-  ( ph  ->  S  e.  Q. )
8 nqprlu 6799 . . . . 5  |-  ( S  e.  Q.  ->  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >.  e.  P. )
97, 8syl 14 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )
10 addclpr 6789 . . . 4  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )  ->  ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. )  e.  P. )
116, 9, 10syl2anc 403 . . 3  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P. )
12 caucvgprlemcanl.q . . . 4  |-  ( ph  ->  Q  e.  Q. )
13 nqprlu 6799 . . . 4  |-  ( Q  e.  Q.  ->  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  e.  P. )
1412, 13syl 14 . . 3  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  e.  P. )
15 addcomprg 6830 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
1615adantl 271 . . 3  |-  ( (
ph  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
172, 5, 11, 14, 16caovord2d 5701 . 2  |-  ( ph  ->  ( <. { l  |  l  <Q  R } ,  { u  |  R  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  <->  (
<. { l  |  l 
<Q  R } ,  {
u  |  R  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  ( ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. )  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )
18 nqprl 6803 . . 3  |-  ( ( R  e.  Q.  /\  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P. )  ->  ( R  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) )  <->  <. { l  |  l  <Q  R } ,  { u  |  R  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )
193, 11, 18syl2anc 403 . 2  |-  ( ph  ->  ( R  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  <->  <. { l  |  l  <Q  R } ,  { u  |  R  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )
20 addnqpr 6813 . . . . 5  |-  ( ( R  e.  Q.  /\  Q  e.  Q. )  -> 
<. { l  |  l 
<Q  ( R  +Q  Q
) } ,  {
u  |  ( R  +Q  Q )  <Q  u } >.  =  ( <. { l  |  l 
<Q  R } ,  {
u  |  R  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
213, 12, 20syl2anc 403 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  ( R  +Q  Q
) } ,  {
u  |  ( R  +Q  Q )  <Q  u } >.  =  ( <. { l  |  l 
<Q  R } ,  {
u  |  R  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
22 addnqpr 6813 . . . . . 6  |-  ( ( S  e.  Q.  /\  Q  e.  Q. )  -> 
<. { l  |  l 
<Q  ( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >.  =  ( <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
237, 12, 22syl2anc 403 . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  ( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >.  =  ( <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
2423oveq2d 5559 . . . 4  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >. )  =  ( L  +P.  ( <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) )
2521, 24breq12d 3806 . . 3  |-  ( ph  ->  ( <. { l  |  l  <Q  ( R  +Q  Q ) } ,  { u  |  ( R  +Q  Q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  Q ) } ,  { u  |  ( S  +Q  Q )  <Q  u } >. )  <->  ( <. { l  |  l  <Q  R } ,  { u  |  R  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  ( L  +P.  ( <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) ) )
26 addclnq 6627 . . . . 5  |-  ( ( R  e.  Q.  /\  Q  e.  Q. )  ->  ( R  +Q  Q
)  e.  Q. )
273, 12, 26syl2anc 403 . . . 4  |-  ( ph  ->  ( R  +Q  Q
)  e.  Q. )
28 addclnq 6627 . . . . . . 7  |-  ( ( S  e.  Q.  /\  Q  e.  Q. )  ->  ( S  +Q  Q
)  e.  Q. )
297, 12, 28syl2anc 403 . . . . . 6  |-  ( ph  ->  ( S  +Q  Q
)  e.  Q. )
30 nqprlu 6799 . . . . . 6  |-  ( ( S  +Q  Q )  e.  Q.  ->  <. { l  |  l  <Q  ( S  +Q  Q ) } ,  { u  |  ( S  +Q  Q
)  <Q  u } >.  e. 
P. )
3129, 30syl 14 . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  ( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >.  e.  P. )
32 addclpr 6789 . . . . 5  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  ( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >.  e.  P. )  ->  ( L  +P.  <. { l  |  l 
<Q  ( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >. )  e.  P. )
336, 31, 32syl2anc 403 . . . 4  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >. )  e.  P. )
34 nqprl 6803 . . . 4  |-  ( ( ( R  +Q  Q
)  e.  Q.  /\  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  Q ) } ,  { u  |  ( S  +Q  Q
)  <Q  u } >. )  e.  P. )  -> 
( ( R  +Q  Q )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  Q ) } ,  { u  |  ( S  +Q  Q )  <Q  u } >. ) )  <->  <. { l  |  l  <Q  ( R  +Q  Q ) } ,  { u  |  ( R  +Q  Q
)  <Q  u } >.  <P 
( L  +P.  <. { l  |  l  <Q 
( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >. ) ) )
3527, 33, 34syl2anc 403 . . 3  |-  ( ph  ->  ( ( R  +Q  Q )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  Q ) } ,  { u  |  ( S  +Q  Q )  <Q  u } >. ) )  <->  <. { l  |  l  <Q  ( R  +Q  Q ) } ,  { u  |  ( R  +Q  Q
)  <Q  u } >.  <P 
( L  +P.  <. { l  |  l  <Q 
( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >. ) ) )
36 addassprg 6831 . . . . 5  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P.  /\ 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  e.  P. )  ->  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  =  ( L  +P.  ( <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )
376, 9, 14, 36syl3anc 1170 . . . 4  |-  ( ph  ->  ( ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. )  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  =  ( L  +P.  ( <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )
3837breq2d 3805 . . 3  |-  ( ph  ->  ( ( <. { l  |  l  <Q  R } ,  { u  |  R  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <->  (
<. { l  |  l 
<Q  R } ,  {
u  |  R  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  ( L  +P.  ( <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) ) )
3925, 35, 383bitr4d 218 . 2  |-  ( ph  ->  ( ( R  +Q  Q )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  Q ) } ,  { u  |  ( S  +Q  Q )  <Q  u } >. ) )  <->  ( <. { l  |  l  <Q  R } ,  { u  |  R  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  ( ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. )  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )
4017, 19, 393bitr4rd 219 1  |-  ( ph  ->  ( ( R  +Q  Q )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  Q ) } ,  { u  |  ( S  +Q  Q )  <Q  u } >. ) )  <->  R  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   {cab 2068   <.cop 3409   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   1stc1st 5796   Q.cnq 6532    +Q cplq 6534    <Q cltq 6537   P.cnp 6543    +P. cpp 6545    <P cltp 6547
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-2o 6066  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-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  cauappcvgprlemladdrl  6909  caucvgprlemladdrl  6930
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