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Theorem fzonmapblen 9563
Description: The result of subtracting a nonnegative integer from a positive integer and adding another nonnegative integer which is less than the first one is less then the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.)
Assertion
Ref Expression
fzonmapblen  |-  ( ( A  e.  ( 0..^ N )  /\  B  e.  ( 0..^ N )  /\  B  <  A
)  ->  ( B  +  ( N  -  A ) )  < 
N )

Proof of Theorem fzonmapblen
StepHypRef Expression
1 elfzo0 9558 . . . 4  |-  ( A  e.  ( 0..^ N )  <->  ( A  e. 
NN0  /\  N  e.  NN  /\  A  <  N
) )
2 nn0re 8652 . . . . . 6  |-  ( A  e.  NN0  ->  A  e.  RR )
3 nnre 8401 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR )
42, 3anim12i 331 . . . . 5  |-  ( ( A  e.  NN0  /\  N  e.  NN )  ->  ( A  e.  RR  /\  N  e.  RR ) )
543adant3 963 . . . 4  |-  ( ( A  e.  NN0  /\  N  e.  NN  /\  A  <  N )  ->  ( A  e.  RR  /\  N  e.  RR ) )
61, 5sylbi 119 . . 3  |-  ( A  e.  ( 0..^ N )  ->  ( A  e.  RR  /\  N  e.  RR ) )
7 elfzoelz 9523 . . . 4  |-  ( B  e.  ( 0..^ N )  ->  B  e.  ZZ )
87zred 8838 . . 3  |-  ( B  e.  ( 0..^ N )  ->  B  e.  RR )
9 simpr 108 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  B  e.  RR )
10 simpll 496 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  A  e.  RR )
11 resubcl 7725 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  A  e.  RR )  ->  ( N  -  A
)  e.  RR )
1211ancoms 264 . . . . . . . 8  |-  ( ( A  e.  RR  /\  N  e.  RR )  ->  ( N  -  A
)  e.  RR )
1312adantr 270 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( N  -  A )  e.  RR )
149, 10, 13ltadd1d 7991 . . . . . 6  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( B  < 
A  <->  ( B  +  ( N  -  A
) )  <  ( A  +  ( N  -  A ) ) ) )
1514biimpa 290 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( B  +  ( N  -  A ) )  < 
( A  +  ( N  -  A ) ) )
16 recn 7454 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
17 recn 7454 . . . . . . . . 9  |-  ( N  e.  RR  ->  N  e.  CC )
1816, 17anim12i 331 . . . . . . . 8  |-  ( ( A  e.  RR  /\  N  e.  RR )  ->  ( A  e.  CC  /\  N  e.  CC ) )
1918adantr 270 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( A  e.  CC  /\  N  e.  CC ) )
2019adantr 270 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( A  e.  CC  /\  N  e.  CC ) )
21 pncan3 7669 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  +  ( N  -  A ) )  =  N )
2220, 21syl 14 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( A  +  ( N  -  A ) )  =  N )
2315, 22breqtrd 3861 . . . 4  |-  ( ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  /\  B  <  A )  ->  ( B  +  ( N  -  A ) )  < 
N )
2423ex 113 . . 3  |-  ( ( ( A  e.  RR  /\  N  e.  RR )  /\  B  e.  RR )  ->  ( B  < 
A  ->  ( B  +  ( N  -  A ) )  < 
N ) )
256, 8, 24syl2an 283 . 2  |-  ( ( A  e.  ( 0..^ N )  /\  B  e.  ( 0..^ N ) )  ->  ( B  <  A  ->  ( B  +  ( N  -  A ) )  < 
N ) )
26253impia 1140 1  |-  ( ( A  e.  ( 0..^ N )  /\  B  e.  ( 0..^ N )  /\  B  <  A
)  ->  ( B  +  ( N  -  A ) )  < 
N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327   RRcr 7328   0cc0 7329    + caddc 7332    < clt 7501    - cmin 7632   NNcn 8394   NN0cn0 8643  ..^cfzo 9518
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-csb 2932  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-iun 3727  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-1st 5893  df-2nd 5894  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-n0 8644  df-z 8721  df-uz 8989  df-fz 9394  df-fzo 9519
This theorem is referenced by: (None)
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