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Theorem rebtwn2zlemstep 9998
Description: Lemma for rebtwn2z 10000. 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 9058 . . . . . . . 8  |-  ( m  e.  ZZ  ->  (
m  +  1 )  e.  ZZ )
21ad3antlr 484 . . . . . . 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 109 . . . . . . 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 510 . . . . . . . 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 508 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  ( m  + 
1 )  <  A
)  ->  m  e.  ZZ )
65zcnd 9142 . . . . . . . . . 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 7750 . . . . . . . . . 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 9305 . . . . . . . . . . 11  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  CC )
98ad4antr 485 . . . . . . . . . 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 7756 . . . . . . . . 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 7881 . . . . . . . . . 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 5758 . . . . . . . . 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 2150 . . . . . . . 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 3926 . . . . . . 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 3902 . . . . . . . . 9  |-  ( j  =  ( m  + 
1 )  ->  (
j  <  A  <->  ( m  +  1 )  < 
A ) )
16 oveq1 5749 . . . . . . . . . 10  |-  ( j  =  ( m  + 
1 )  ->  (
j  +  K )  =  ( ( m  +  1 )  +  K ) )
1716breq2d 3911 . . . . . . . . 9  |-  ( j  =  ( m  + 
1 )  ->  ( A  <  ( j  +  K )  <->  A  <  ( ( m  +  1 )  +  K ) ) )
1815, 17anbi12d 464 . . . . . . . 8  |-  ( j  =  ( m  + 
1 )  ->  (
( j  <  A  /\  A  <  ( j  +  K ) )  <-> 
( ( m  + 
1 )  <  A  /\  A  <  ( ( m  +  1 )  +  K ) ) ) )
1918rspcev 2763 . . . . . . 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 1199 . . . . . 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 508 . . . . . . 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 509 . . . . . . 7  |-  ( ( ( ( ( K  e.  ( ZZ>= `  2
)  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  (
m  <  A  /\  A  <  ( m  +  ( K  +  1
) ) ) )  /\  A  <  (
m  +  K ) )  ->  m  <  A )
23 simpr 109 . . . . . . 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 3902 . . . . . . . . 9  |-  ( j  =  m  ->  (
j  <  A  <->  m  <  A ) )
25 oveq1 5749 . . . . . . . . . 10  |-  ( j  =  m  ->  (
j  +  K )  =  ( m  +  K ) )
2625breq2d 3911 . . . . . . . . 9  |-  ( j  =  m  ->  ( A  <  ( j  +  K )  <->  A  <  ( m  +  K ) ) )
2724, 26anbi12d 464 . . . . . . . 8  |-  ( j  =  m  ->  (
( j  <  A  /\  A  <  ( j  +  K ) )  <-> 
( m  <  A  /\  A  <  ( m  +  K ) ) ) )
2827rspcev 2763 . . . . . . 7  |-  ( ( m  e.  ZZ  /\  ( m  <  A  /\  A  <  ( m  +  K ) ) )  ->  E. j  e.  ZZ  ( j  <  A  /\  A  <  ( j  +  K ) ) )
2921, 22, 23, 28syl12anc 1199 . . . . . 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 7749 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  1  e.  RR )
31 eluzelre 9304 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  RR )
3231ad3antrrr 483 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  K  e.  RR )
33 simplr 504 . . . . . . . . 9  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  m  e.  ZZ )
3433zred 9141 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  m  e.  RR )
35 1z 9048 . . . . . . . . . . 11  |-  1  e.  ZZ
36 eluzp1l 9318 . . . . . . . . . . 11  |-  ( ( 1  e.  ZZ  /\  K  e.  ( ZZ>= `  ( 1  +  1 ) ) )  -> 
1  <  K )
3735, 36mpan 420 . . . . . . . . . 10  |-  ( K  e.  ( ZZ>= `  (
1  +  1 ) )  ->  1  <  K )
38 df-2 8747 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3938fveq2i 5392 . . . . . . . . . 10  |-  ( ZZ>= ` 
2 )  =  (
ZZ>= `  ( 1  +  1 ) )
4037, 39eleq2s 2212 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  2
)  ->  1  <  K )
4140ad3antrrr 483 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  1  <  K )
4230, 32, 34, 41ltadd2dd 8152 . . . . . . 7  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
m  +  1 )  <  ( m  +  K ) )
4334, 30readdcld 7763 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
m  +  1 )  e.  RR )
4434, 32readdcld 7763 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  (
m  +  K )  e.  RR )
45 simpllr 508 . . . . . . . 8  |-  ( ( ( ( K  e.  ( ZZ>= `  2 )  /\  A  e.  RR )  /\  m  e.  ZZ )  /\  ( m  < 
A  /\  A  <  ( m  +  ( K  +  1 ) ) ) )  ->  A  e.  RR )
46 axltwlin 7800 . . . . . . . 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 1201 . . . . . . 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 772 . . . . 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 114 . . . 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 2526 . . 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 1163 . 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 3902 . . . 4  |-  ( m  =  j  ->  (
m  <  A  <->  j  <  A ) )
54 oveq1 5749 . . . . 5  |-  ( m  =  j  ->  (
m  +  K )  =  ( j  +  K ) )
5554breq2d 3911 . . . 4  |-  ( m  =  j  ->  ( A  <  ( m  +  K )  <->  A  <  ( j  +  K ) ) )
5653, 55anbi12d 464 . . 3  |-  ( m  =  j  ->  (
( m  <  A  /\  A  <  ( m  +  K ) )  <-> 
( j  <  A  /\  A  <  ( j  +  K ) ) ) )
5756cbvrexv 2632 . 2  |-  ( E. m  e.  ZZ  (
m  <  A  /\  A  <  ( m  +  K ) )  <->  E. j  e.  ZZ  ( j  < 
A  /\  A  <  ( j  +  K ) ) )
5852, 57sylibr 133 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 103    \/ wo 682    /\ w3a 947    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   CCcc 7586   RRcr 7587   1c1 7589    + caddc 7591    < clt 7768   2c2 8739   ZZcz 9022   ZZ>=cuz 9294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023  df-uz 9295
This theorem is referenced by:  rebtwn2zlemshrink  9999
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