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Theorem caucvgprlemcanl 7634
Description: Lemma for cauappcvgprlemladdrl 7647. 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 7609 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
21adantl 277 . . 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 7537 . . . 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 7537 . . . . 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 7527 . . . 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 411 . . 3  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P. )
12 caucvgprlemcanl.q . . . 4  |-  ( ph  ->  Q  e.  Q. )
13 nqprlu 7537 . . . 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 7568 . . . 4  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
1615adantl 277 . . 3  |-  ( (
ph  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
172, 5, 11, 14, 16caovord2d 6038 . 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 7541 . . 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 411 . 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 7551 . . . . 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 411 . . . 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 7551 . . . . . 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 411 . . . . 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 5885 . . . 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 4013 . . 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 7365 . . . . 5  |-  ( ( R  e.  Q.  /\  Q  e.  Q. )  ->  ( R  +Q  Q
)  e.  Q. )
273, 12, 26syl2anc 411 . . . 4  |-  ( ph  ->  ( R  +Q  Q
)  e.  Q. )
28 addclnq 7365 . . . . . . 7  |-  ( ( S  e.  Q.  /\  Q  e.  Q. )  ->  ( S  +Q  Q
)  e.  Q. )
297, 12, 28syl2anc 411 . . . . . 6  |-  ( ph  ->  ( S  +Q  Q
)  e.  Q. )
30 nqprlu 7537 . . . . . 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 7527 . . . . 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 411 . . . 4  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  Q
) } ,  {
u  |  ( S  +Q  Q )  <Q  u } >. )  e.  P. )
34 nqprl 7541 . . . 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 411 . . 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 7569 . . . . 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 1238 . . . 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 4012 . . 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 220 . 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 221 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   {cab 2163   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   Q.cnq 7270    +Q cplq 7272    <Q cltq 7275   P.cnp 7281    +P. cpp 7283    <P cltp 7285
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iplp 7458  df-iltp 7460
This theorem is referenced by:  cauappcvgprlemladdrl  7647  caucvgprlemladdrl  7668
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