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Theorem prarloclem 7478
Description: A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly spaced rational numbers from 
A to  A  +Q  ( N  .Q  P ) (which are in the lower and upper cuts, respectively, of a real number), there are a pair of numbers, two positions apart in the even spacing, which straddle the cut. (Contributed by Jim Kingdon, 22-Oct-2019.)
Assertion
Ref Expression
prarloclem  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  E. j  e.  om  ( ( A +Q0  ( [ <. j ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( j  +o  2o ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) )
Distinct variable groups:    A, j    j, L    j, N    P, j    U, j

Proof of Theorem prarloclem
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prarloclem5 7477 . 2  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  E. x  e.  om  E. y  e. 
om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) )
2 prarloclem4 7475 . . . 4  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  P  e.  Q. )  ->  ( E. x  e.  om  E. y  e.  om  (
( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
)  ->  E. j  e.  om  ( ( A +Q0  ( [ <. j ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( j  +o  2o ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) ) )
323ad2antr2 1163 . . 3  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N ) )  ->  ( E. x  e.  om  E. y  e.  om  (
( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
)  ->  E. j  e.  om  ( ( A +Q0  ( [ <. j ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( j  +o  2o ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) ) )
433adant3 1017 . 2  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  ( E. x  e.  om  E. y  e.  om  (
( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
)  ->  E. j  e.  om  ( ( A +Q0  ( [ <. j ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( j  +o  2o ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) ) )
51, 4mpd 13 1  |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L )  /\  ( N  e.  N.  /\  P  e.  Q.  /\  1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  E. j  e.  om  ( ( A +Q0  ( [ <. j ,  1o >. ] ~Q0 ·Q0  P )
)  e.  L  /\  ( A  +Q  ( [ <. ( j  +o  2o ) ,  1o >. ]  ~Q  .Q  P
) )  e.  U
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    e. wcel 2148   E.wrex 2456   <.cop 3594   class class class wbr 4000   omcom 4585  (class class class)co 5868   1oc1o 6403   2oc2o 6404    +o coa 6407   [cec 6526   N.cnpi 7249    <N clti 7252    ~Q ceq 7256   Q.cnq 7257    +Q cplq 7259    .Q cmq 7260   ~Q0 ceq0 7263   +Q0 cplq0 7266   ·Q0 cmq0 7267   P.cnp 7268
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 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
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 4285  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-1o 6410  df-2o 6411  df-oadd 6414  df-omul 6415  df-er 6528  df-ec 6530  df-qs 6534  df-ni 7281  df-pli 7282  df-mi 7283  df-lti 7284  df-plpq 7321  df-mpq 7322  df-enq 7324  df-nqqs 7325  df-plqqs 7326  df-mqqs 7327  df-ltnqqs 7330  df-enq0 7401  df-nq0 7402  df-0nq0 7403  df-plq0 7404  df-mq0 7405  df-inp 7443
This theorem is referenced by:  prarloc  7480
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