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Theorem subsubelfzo0 32712
Description: Subtracting a difference from a number which is not less than the difference results in a bounded nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
Assertion
Ref Expression
subsubelfzo0  |-  ( ( A  e.  ( 0..^ N )  /\  I  e.  ( 0..^ N )  /\  -.  I  < 
( N  -  A
) )  ->  (
I  -  ( N  -  A ) )  e.  ( 0..^ A ) )

Proof of Theorem subsubelfzo0
StepHypRef Expression
1 elfzo0 11840 . . . 4  |-  ( A  e.  ( 0..^ N )  <->  ( A  e. 
NN0  /\  N  e.  NN  /\  A  <  N
) )
2 elfzo0 11840 . . . . . 6  |-  ( I  e.  ( 0..^ N )  <->  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )
3 nnre 10538 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  RR )
433ad2ant2 1016 . . . . . . . . . . . . 13  |-  ( ( I  e.  NN0  /\  N  e.  NN  /\  I  <  N )  ->  N  e.  RR )
5 nn0re 10800 . . . . . . . . . . . . . 14  |-  ( A  e.  NN0  ->  A  e.  RR )
65adantr 463 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  A  <  N )  ->  A  e.  RR )
7 resubcl 9874 . . . . . . . . . . . . 13  |-  ( ( N  e.  RR  /\  A  e.  RR )  ->  ( N  -  A
)  e.  RR )
84, 6, 7syl2anr 476 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  ( N  -  A )  e.  RR )
9 nn0re 10800 . . . . . . . . . . . . . 14  |-  ( I  e.  NN0  ->  I  e.  RR )
1093ad2ant1 1015 . . . . . . . . . . . . 13  |-  ( ( I  e.  NN0  /\  N  e.  NN  /\  I  <  N )  ->  I  e.  RR )
1110adantl 464 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  I  e.  RR )
12 lenlt 9652 . . . . . . . . . . . . 13  |-  ( ( ( N  -  A
)  e.  RR  /\  I  e.  RR )  ->  ( ( N  -  A )  <_  I  <->  -.  I  <  ( N  -  A ) ) )
1312bicomd 201 . . . . . . . . . . . 12  |-  ( ( ( N  -  A
)  e.  RR  /\  I  e.  RR )  ->  ( -.  I  < 
( N  -  A
)  <->  ( N  -  A )  <_  I
) )
148, 11, 13syl2anc 659 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  ( -.  I  <  ( N  -  A )  <->  ( N  -  A )  <_  I
) )
1514biimpa 482 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  A  <  N )  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  < 
N ) )  /\  -.  I  <  ( N  -  A ) )  ->  ( N  -  A )  <_  I
)
16 nnz 10882 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN  ->  N  e.  ZZ )
17163ad2ant2 1016 . . . . . . . . . . . . . 14  |-  ( ( I  e.  NN0  /\  N  e.  NN  /\  I  <  N )  ->  N  e.  ZZ )
18 nn0z 10883 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN0  ->  A  e.  ZZ )
1918adantr 463 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN0  /\  A  <  N )  ->  A  e.  ZZ )
20 zsubcl 10902 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  A  e.  ZZ )  ->  ( N  -  A
)  e.  ZZ )
2117, 19, 20syl2anr 476 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  ( N  -  A )  e.  ZZ )
22 ltle 9662 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  N  e.  RR )  ->  ( A  <  N  ->  A  <_  N )
)
235, 4, 22syl2an 475 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN0  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  <  N ) )  ->  ( A  < 
N  ->  A  <_  N ) )
2423impancom 438 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  A  <  N )  -> 
( ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
)  ->  A  <_  N ) )
2524imp 427 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  A  <_  N )
26 subge0 10061 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  RR  /\  A  e.  RR )  ->  ( 0  <_  ( N  -  A )  <->  A  <_  N ) )
274, 6, 26syl2anr 476 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  (
0  <_  ( N  -  A )  <->  A  <_  N ) )
2825, 27mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  0  <_  ( N  -  A
) )
29 elnn0z 10873 . . . . . . . . . . . . 13  |-  ( ( N  -  A )  e.  NN0  <->  ( ( N  -  A )  e.  ZZ  /\  0  <_ 
( N  -  A
) ) )
3021, 28, 29sylanbrc 662 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  ( N  -  A )  e.  NN0 )
3130adantr 463 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
NN0  /\  A  <  N )  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  < 
N ) )  /\  -.  I  <  ( N  -  A ) )  ->  ( N  -  A )  e.  NN0 )
32 simplr1 1036 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
NN0  /\  A  <  N )  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  < 
N ) )  /\  -.  I  <  ( N  -  A ) )  ->  I  e.  NN0 )
33 nn0sub 10842 . . . . . . . . . . 11  |-  ( ( ( N  -  A
)  e.  NN0  /\  I  e.  NN0 )  -> 
( ( N  -  A )  <_  I  <->  ( I  -  ( N  -  A ) )  e.  NN0 ) )
3431, 32, 33syl2anc 659 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
NN0  /\  A  <  N )  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  < 
N ) )  /\  -.  I  <  ( N  -  A ) )  ->  ( ( N  -  A )  <_  I 
<->  ( I  -  ( N  -  A )
)  e.  NN0 )
)
3515, 34mpbid 210 . . . . . . . . 9  |-  ( ( ( ( A  e. 
NN0  /\  A  <  N )  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  < 
N ) )  /\  -.  I  <  ( N  -  A ) )  ->  ( I  -  ( N  -  A
) )  e.  NN0 )
36 elnn0uz 11119 . . . . . . . . 9  |-  ( ( I  -  ( N  -  A ) )  e.  NN0  <->  ( I  -  ( N  -  A
) )  e.  (
ZZ>= `  0 ) )
3735, 36sylib 196 . . . . . . . 8  |-  ( ( ( ( A  e. 
NN0  /\  A  <  N )  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  < 
N ) )  /\  -.  I  <  ( N  -  A ) )  ->  ( I  -  ( N  -  A
) )  e.  (
ZZ>= `  0 ) )
3819adantr 463 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  A  e.  ZZ )
3938adantr 463 . . . . . . . 8  |-  ( ( ( ( A  e. 
NN0  /\  A  <  N )  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  < 
N ) )  /\  -.  I  <  ( N  -  A ) )  ->  A  e.  ZZ )
409adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  NN0  /\  N  e.  NN )  ->  I  e.  RR )
4140adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN0  /\  ( I  e.  NN0  /\  N  e.  NN ) )  ->  I  e.  RR )
423adantl 464 . . . . . . . . . . . . . . . . 17  |-  ( ( I  e.  NN0  /\  N  e.  NN )  ->  N  e.  RR )
4342adantl 464 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN0  /\  ( I  e.  NN0  /\  N  e.  NN ) )  ->  N  e.  RR )
4442, 5, 7syl2anr 476 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN0  /\  ( I  e.  NN0  /\  N  e.  NN ) )  ->  ( N  -  A )  e.  RR )
4541, 43, 44ltsub1d 10157 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  ( I  e.  NN0  /\  N  e.  NN ) )  ->  ( I  <  N  <->  ( I  -  ( N  -  A
) )  <  ( N  -  ( N  -  A ) ) ) )
46 nncn 10539 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  N  e.  CC )
4746adantl 464 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  e.  NN0  /\  N  e.  NN )  ->  N  e.  CC )
48 nn0cn 10801 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  NN0  ->  A  e.  CC )
49 nncan 9839 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  CC  /\  A  e.  CC )  ->  ( N  -  ( N  -  A )
)  =  A )
5047, 48, 49syl2anr 476 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  NN0  /\  ( I  e.  NN0  /\  N  e.  NN ) )  ->  ( N  -  ( N  -  A ) )  =  A )
5150breq2d 4451 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  NN0  /\  ( I  e.  NN0  /\  N  e.  NN ) )  ->  ( (
I  -  ( N  -  A ) )  <  ( N  -  ( N  -  A
) )  <->  ( I  -  ( N  -  A ) )  < 
A ) )
5251biimpd 207 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  NN0  /\  ( I  e.  NN0  /\  N  e.  NN ) )  ->  ( (
I  -  ( N  -  A ) )  <  ( N  -  ( N  -  A
) )  ->  (
I  -  ( N  -  A ) )  <  A ) )
5345, 52sylbid 215 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN0  /\  ( I  e.  NN0  /\  N  e.  NN ) )  ->  ( I  <  N  ->  ( I  -  ( N  -  A ) )  < 
A ) )
5453ex 432 . . . . . . . . . . . . 13  |-  ( A  e.  NN0  ->  ( ( I  e.  NN0  /\  N  e.  NN )  ->  ( I  <  N  ->  ( I  -  ( N  -  A )
)  <  A )
) )
5554adantr 463 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  A  <  N )  -> 
( ( I  e. 
NN0  /\  N  e.  NN )  ->  ( I  <  N  ->  (
I  -  ( N  -  A ) )  <  A ) ) )
5655com3l 81 . . . . . . . . . . 11  |-  ( ( I  e.  NN0  /\  N  e.  NN )  ->  ( I  <  N  ->  ( ( A  e. 
NN0  /\  A  <  N )  ->  ( I  -  ( N  -  A ) )  < 
A ) ) )
57563impia 1191 . . . . . . . . . 10  |-  ( ( I  e.  NN0  /\  N  e.  NN  /\  I  <  N )  ->  (
( A  e.  NN0  /\  A  <  N )  ->  ( I  -  ( N  -  A
) )  <  A
) )
5857impcom 428 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  A  <  N )  /\  ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
) )  ->  (
I  -  ( N  -  A ) )  <  A )
5958adantr 463 . . . . . . . 8  |-  ( ( ( ( A  e. 
NN0  /\  A  <  N )  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  < 
N ) )  /\  -.  I  <  ( N  -  A ) )  ->  ( I  -  ( N  -  A
) )  <  A
)
6037, 39, 593jca 1174 . . . . . . 7  |-  ( ( ( ( A  e. 
NN0  /\  A  <  N )  /\  ( I  e.  NN0  /\  N  e.  NN  /\  I  < 
N ) )  /\  -.  I  <  ( N  -  A ) )  ->  ( ( I  -  ( N  -  A ) )  e.  ( ZZ>= `  0 )  /\  A  e.  ZZ  /\  ( I  -  ( N  -  A )
)  <  A )
)
6160exp31 602 . . . . . 6  |-  ( ( A  e.  NN0  /\  A  <  N )  -> 
( ( I  e. 
NN0  /\  N  e.  NN  /\  I  <  N
)  ->  ( -.  I  <  ( N  -  A )  ->  (
( I  -  ( N  -  A )
)  e.  ( ZZ>= ` 
0 )  /\  A  e.  ZZ  /\  ( I  -  ( N  -  A ) )  < 
A ) ) ) )
622, 61syl5bi 217 . . . . 5  |-  ( ( A  e.  NN0  /\  A  <  N )  -> 
( I  e.  ( 0..^ N )  -> 
( -.  I  < 
( N  -  A
)  ->  ( (
I  -  ( N  -  A ) )  e.  ( ZZ>= `  0
)  /\  A  e.  ZZ  /\  ( I  -  ( N  -  A
) )  <  A
) ) ) )
63623adant2 1013 . . . 4  |-  ( ( A  e.  NN0  /\  N  e.  NN  /\  A  <  N )  ->  (
I  e.  ( 0..^ N )  ->  ( -.  I  <  ( N  -  A )  -> 
( ( I  -  ( N  -  A
) )  e.  (
ZZ>= `  0 )  /\  A  e.  ZZ  /\  (
I  -  ( N  -  A ) )  <  A ) ) ) )
641, 63sylbi 195 . . 3  |-  ( A  e.  ( 0..^ N )  ->  ( I  e.  ( 0..^ N )  ->  ( -.  I  <  ( N  -  A
)  ->  ( (
I  -  ( N  -  A ) )  e.  ( ZZ>= `  0
)  /\  A  e.  ZZ  /\  ( I  -  ( N  -  A
) )  <  A
) ) ) )
65643imp 1188 . 2  |-  ( ( A  e.  ( 0..^ N )  /\  I  e.  ( 0..^ N )  /\  -.  I  < 
( N  -  A
) )  ->  (
( I  -  ( N  -  A )
)  e.  ( ZZ>= ` 
0 )  /\  A  e.  ZZ  /\  ( I  -  ( N  -  A ) )  < 
A ) )
66 elfzo2 11807 . 2  |-  ( ( I  -  ( N  -  A ) )  e.  ( 0..^ A )  <->  ( ( I  -  ( N  -  A ) )  e.  ( ZZ>= `  0 )  /\  A  e.  ZZ  /\  ( I  -  ( N  -  A )
)  <  A )
)
6765, 66sylibr 212 1  |-  ( ( A  e.  ( 0..^ N )  /\  I  e.  ( 0..^ N )  /\  -.  I  < 
( N  -  A
) )  ->  (
I  -  ( N  -  A ) )  e.  ( 0..^ A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481    < clt 9617    <_ cle 9618    - cmin 9796   NNcn 10531   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082  ..^cfzo 11799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800
This theorem is referenced by: (None)
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