ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprlemcanl Unicode version

Theorem caucvgprlemcanl 7618
Description: Lemma for cauappcvgprlemladdrl 7631. 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 7593 . . . 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 7521 . . . 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 7521 . . . . 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 7511 . . . 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 7521 . . . 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 7552 . . . 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 6034 . 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 7525 . . 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 7535 . . . . 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 7535 . . . . . 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 5881 . . . 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 4011 . . 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 7349 . . . . 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 7349 . . . . . . 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 7521 . . . . . 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 7511 . . . . 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 7525 . . . 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 7553 . . . . 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 4010 . . 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 2146   {cab 2161   <.cop 3592   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   1stc1st 6129   Q.cnq 7254    +Q cplq 7256    <Q cltq 7259   P.cnp 7265    +P. cpp 7267    <P cltp 7269
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-iplp 7442  df-iltp 7444
This theorem is referenced by:  cauappcvgprlemladdrl  7631  caucvgprlemladdrl  7652
  Copyright terms: Public domain W3C validator