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Theorem caucvgprlemcanl 7420
Description: Lemma for cauappcvgprlemladdrl 7433. 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 7395 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
21adantl 275 . . 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 7323 . . . 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 7323 . . . . 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 7313 . . . 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 408 . . 3  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P. )
12 caucvgprlemcanl.q . . . 4  |-  ( ph  ->  Q  e.  Q. )
13 nqprlu 7323 . . . 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 7354 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
1615adantl 275 . . 3  |-  ( (
ph  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
172, 5, 11, 14, 16caovord2d 5908 . 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 7327 . . 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 408 . 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 7337 . . . . 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 408 . . . 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 7337 . . . . . 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 408 . . . . 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 5758 . . . 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 3912 . . 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 7151 . . . . 5  |-  ( ( R  e.  Q.  /\  Q  e.  Q. )  ->  ( R  +Q  Q
)  e.  Q. )
273, 12, 26syl2anc 408 . . . 4  |-  ( ph  ->  ( R  +Q  Q
)  e.  Q. )
28 addclnq 7151 . . . . . . 7  |-  ( ( S  e.  Q.  /\  Q  e.  Q. )  ->  ( S  +Q  Q
)  e.  Q. )
297, 12, 28syl2anc 408 . . . . . 6  |-  ( ph  ->  ( S  +Q  Q
)  e.  Q. )
30 nqprlu 7323 . . . . . 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 7313 . . . . 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 408 . . . 4  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >. )  e.  P. )
34 nqprl 7327 . . . 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 408 . . 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 7355 . . . . 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 1201 . . . 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 3911 . . 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 219 . 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 220 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   {cab 2103   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   1stc1st 6004   Q.cnq 7056    +Q cplq 7058    <Q cltq 7061   P.cnp 7067    +P. cpp 7069    <P cltp 7071
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-iltp 7246
This theorem is referenced by:  cauappcvgprlemladdrl  7433  caucvgprlemladdrl  7454
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