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Theorem rebtwn2zlemstep 9629
Description: Lemma for rebtwn2z 9631. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
Assertion
Ref Expression
rebtwn2zlemstep  |-  ( ( K  e.  ( ZZ>= ` 
2 )  /\  A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  K ) ) )
Distinct variable groups:    A, m    m, K

Proof of Theorem rebtwn2zlemstep
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 peano2z 8756 . . . . . . . 8  |-  ( m  e.  ZZ  ->  (
m  +  1 )  e.  ZZ )
21ad3antlr 477 . . . . . . 7  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  ( m  +  1 )  e.  ZZ )
3 simpr 108 . . . . . . 7  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  ( m  +  1 )  < 
A )
4 simplrr 503 . . . . . . . 8  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  A  <  ( m  +  ( K  +  1 ) ) )
5 simpllr 501 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  m  e.  ZZ )
65zcnd 8839 . . . . . . . . . 10  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  m  e.  CC )
7 1cnd 7483 . . . . . . . . . 10  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  1  e.  CC )
8 eluzelcn 8999 . . . . . . . . . . 11  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  CC )
98ad4antr 478 . . . . . . . . . 10  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  K  e.  CC )
106, 7, 9addassd 7489 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  ( (
m  +  1 )  +  K )  =  ( m  +  ( 1  +  K ) ) )
117, 9addcomd 7612 . . . . . . . . . 10  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  ( 1  +  K )  =  ( K  +  1 ) )
1211oveq2d 5650 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  ( m  +  ( 1  +  K ) )  =  ( m  +  ( K  +  1 ) ) )
1310, 12eqtrd 2120 . . . . . . . 8  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  ( (
m  +  1 )  +  K )  =  ( m  +  ( K  +  1 ) ) )
144, 13breqtrrd 3863 . . . . . . 7  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  A  <  ( ( m  +  1 )  +  K ) )
15 breq1 3840 . . . . . . . . 9  |-  ( j  =  ( m  + 
1 )  ->  (
j  <  A  <->  ( m  +  1 )  < 
A ) )
16 oveq1 5641 . . . . . . . . . 10  |-  ( j  =  ( m  + 
1 )  ->  (
j  +  K )  =  ( ( m  +  1 )  +  K ) )
1716breq2d 3849 . . . . . . . . 9  |-  ( j  =  ( m  + 
1 )  ->  ( A  <  ( j  +  K )  <->  A  <  ( ( m  +  1 )  +  K ) ) )
1815, 17anbi12d 457 . . . . . . . 8  |-  ( j  =  ( m  + 
1 )  ->  (
( j  <  A  /\  A  <  ( j  +  K ) )  <-> 
( ( m  + 
1 )  <  A  /\  A  <  ( ( m  +  1 )  +  K ) ) ) )
1918rspcev 2722 . . . . . . 7  |-  ( ( ( m  +  1 )  e.  ZZ  /\  ( ( m  + 
1 )  <  A  /\  A  <  ( ( m  +  1 )  +  K ) ) )  ->  E. j  e.  ZZ  ( j  < 
A  /\  A  <  ( j  +  K ) ) )
202, 3, 14, 19syl12anc 1172 . . . . . 6  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  E. j  e.  ZZ  ( j  < 
A  /\  A  <  ( j  +  K ) ) )
21 simpllr 501 . . . . . . 7  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  A  <  (
m  +  K ) )  ->  m  e.  ZZ )
22 simplrl 502 . . . . . . 7  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  A  <  (
m  +  K ) )  ->  m  <  A )
23 simpr 108 . . . . . . 7  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  A  <  (
m  +  K ) )  ->  A  <  ( m  +  K ) )
24 breq1 3840 . . . . . . . . 9  |-  ( j  =  m  ->  (
j  <  A  <->  m  <  A ) )
25 oveq1 5641 . . . . . . . . . 10  |-  ( j  =  m  ->  (
j  +  K )  =  ( m  +  K ) )
2625breq2d 3849 . . . . . . . . 9  |-  ( j  =  m  ->  ( A  <  ( j  +  K )  <->  A  <  ( m  +  K ) ) )
2724, 26anbi12d 457 . . . . . . . 8  |-  ( j  =  m  ->  (
( j  <  A  /\  A  <  ( j  +  K ) )  <-> 
( m  <  A  /\  A  <  ( m  +  K ) ) ) )
2827rspcev 2722 . . . . . . 7  |-  ( ( m  e.  ZZ  /\  ( m  <  A  /\  A  <  ( m  +  K ) ) )  ->  E. j  e.  ZZ  ( j  <  A  /\  A  <  ( j  +  K ) ) )
2921, 22, 23, 28syl12anc 1172 . . . . . 6  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  A  <  (
m  +  K ) )  ->  E. j  e.  ZZ  ( j  < 
A  /\  A  <  ( j  +  K ) ) )
30 1red 7482 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  1  e.  RR )
31 eluzelre 8998 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  RR )
3231ad3antrrr 476 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  K  e.  RR )
33 simplr 497 . . . . . . . . 9  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  m  e.  ZZ )
3433zred 8838 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  m  e.  RR )
35 1z 8746 . . . . . . . . . . 11  |-  1  e.  ZZ
36 eluzp1l 9012 . . . . . . . . . . 11  |-  ( ( 1  e.  ZZ  /\  K  e.  ( ZZ>= `  ( 1  +  1 ) ) )  -> 
1  <  K )
3735, 36mpan 415 . . . . . . . . . 10  |-  ( K  e.  ( ZZ>= `  (
1  +  1 ) )  ->  1  <  K )
38 df-2 8452 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3938fveq2i 5292 . . . . . . . . . 10  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
4037, 39eleq2s 2182 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  2
)  ->  1  <  K )
4140ad3antrrr 476 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  1  <  K )
4230, 32, 34, 41ltadd2dd 7879 . . . . . . 7  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
m  +  1 )  <  ( m  +  K ) )
4334, 30readdcld 7496 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
m  +  1 )  e.  RR )
4434, 32readdcld 7496 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
m  +  K )  e.  RR )
45 simpllr 501 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  A  e.  RR )
46 axltwlin 7533 . . . . . . . 8  |-  ( ( ( m  +  1 )  e.  RR  /\  ( m  +  K
)  e.  RR  /\  A  e.  RR )  ->  ( ( m  + 
1 )  <  (
m  +  K )  ->  ( ( m  +  1 )  < 
A  \/  A  < 
( m  +  K
) ) ) )
4743, 44, 45, 46syl3anc 1174 . . . . . . 7  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
( m  +  1 )  <  ( m  +  K )  -> 
( ( m  + 
1 )  <  A  \/  A  <  ( m  +  K ) ) ) )
4842, 47mpd 13 . . . . . 6  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
( m  +  1 )  <  A  \/  A  <  ( m  +  K ) ) )
4920, 29, 48mpjaodan 747 . . . . 5  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  E. j  e.  ZZ  ( j  < 
A  /\  A  <  ( j  +  K ) ) )
5049ex 113 . . . 4  |-  ( ( ( K  e.  (
ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  ->  ( ( m  <  A  /\  A  <  ( m  +  ( K  +  1 ) ) )  ->  E. j  e.  ZZ  ( j  < 
A  /\  A  <  ( j  +  K ) ) ) )
5150rexlimdva 2489 . . 3  |-  ( ( K  e.  ( ZZ>= ` 
2 )  /\  A  e.  RR )  ->  ( E. m  e.  ZZ  ( m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) )  ->  E. j  e.  ZZ  ( j  <  A  /\  A  <  ( j  +  K ) ) ) )
52513impia 1140 . 2  |-  ( ( K  e.  ( ZZ>= ` 
2 )  /\  A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  E. j  e.  ZZ  ( j  < 
A  /\  A  <  ( j  +  K ) ) )
53 breq1 3840 . . . 4  |-  ( m  =  j  ->  (
m  <  A  <->  j  <  A ) )
54 oveq1 5641 . . . . 5  |-  ( m  =  j  ->  (
m  +  K )  =  ( j  +  K ) )
5554breq2d 3849 . . . 4  |-  ( m  =  j  ->  ( A  <  ( m  +  K )  <->  A  <  ( j  +  K ) ) )
5653, 55anbi12d 457 . . 3  |-  ( m  =  j  ->  (
( m  <  A  /\  A  <  ( m  +  K ) )  <-> 
( j  <  A  /\  A  <  ( j  +  K ) ) ) )
5756cbvrexv 2591 . 2  |-  ( E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  K ) )  <->  E. j  e.  ZZ  ( j  < 
A  /\  A  <  ( j  +  K ) ) )
5852, 57sylibr 132 1  |-  ( ( K  e.  ( ZZ>= ` 
2 )  /\  A  e.  RR  /\  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  E. m  e.  ZZ  ( m  < 
A  /\  A  <  ( m  +  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   CCcc 7327   RRcr 7328   1c1 7330    + caddc 7332    < clt 7501   2c2 8444   ZZcz 8720   ZZ>=cuz 8988
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-uz 8989
This theorem is referenced by:  rebtwn2zlemshrink  9630
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