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Theorem rebtwn2zlemstep 9339
Description: Lemma for rebtwn2z 9341. 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 8468 . . . . . . . 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 8551 . . . . . . . . . 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 7197 . . . . . . . . . 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 8711 . . . . . . . . . . 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 7203 . . . . . . . . 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 7326 . . . . . . . . . 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 5559 . . . . . . . . 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 2114 . . . . . . . 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 3819 . . . . . . 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 3796 . . . . . . . . 9  |-  ( j  =  ( m  + 
1 )  ->  (
j  <  A  <->  ( m  +  1 )  < 
A ) )
16 oveq1 5550 . . . . . . . . . 10  |-  ( j  =  ( m  + 
1 )  ->  (
j  +  K )  =  ( ( m  +  1 )  +  K ) )
1716breq2d 3805 . . . . . . . . 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 2702 . . . . . . 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 1168 . . . . . 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 3796 . . . . . . . . 9  |-  ( j  =  m  ->  (
j  <  A  <->  m  <  A ) )
25 oveq1 5550 . . . . . . . . . 10  |-  ( j  =  m  ->  (
j  +  K )  =  ( m  +  K ) )
2625breq2d 3805 . . . . . . . . 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 2702 . . . . . . 7  |-  ( ( m  e.  ZZ  /\  ( m  <  A  /\  A  <  ( m  +  K ) ) )  ->  E. j  e.  ZZ  ( j  <  A  /\  A  <  ( j  +  K ) ) )
2921, 22, 23, 28syl12anc 1168 . . . . . 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 7196 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  1  e.  RR )
31 eluzelre 8710 . . . . . . . . 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 8550 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  m  e.  RR )
35 1z 8458 . . . . . . . . . . 11  |-  1  e.  ZZ
36 eluzp1l 8724 . . . . . . . . . . 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 8165 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3938fveq2i 5212 . . . . . . . . . 10  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
4037, 39eleq2s 2174 . . . . . . . . 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 7593 . . . . . . 7  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
m  +  1 )  <  ( m  +  K ) )
4334, 30readdcld 7210 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
m  +  1 )  e.  RR )
4434, 32readdcld 7210 . . . . . . . 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 7247 . . . . . . . 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 1170 . . . . . . 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 745 . . . . 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 2478 . . 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 1136 . 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 3796 . . . 4  |-  ( m  =  j  ->  (
m  <  A  <->  j  <  A ) )
54 oveq1 5550 . . . . 5  |-  ( m  =  j  ->  (
m  +  K )  =  ( j  +  K ) )
5554breq2d 3805 . . . 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 2579 . 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 662    /\ w3a 920    = wceq 1285    e. wcel 1434   E.wrex 2350   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   CCcc 7041   RRcr 7042   1c1 7044    + caddc 7046    < clt 7215   2c2 8156   ZZcz 8432   ZZ>=cuz 8700
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-2 8165  df-n0 8356  df-z 8433  df-uz 8701
This theorem is referenced by:  rebtwn2zlemshrink  9340
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